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Contents
Preface ................................................................... vii
Portrait ofPhilippe Cl´ement.............................................. viii
Philippe Cl´ement: Curriculum Vitae ...................................... ix
Gabriella Caristi and Enzo Mitidieri
Harnack Inequality and Applications to
Solutions of Biharmonic Equations ................................... 1
Carsten Carstensen
Cl´ement Interpolation and Its Role in
Adaptive Finite Element Error Control .............................. 27
Sandra Cerrai
Ergodic Properties of Reaction-diﬀusion Equations
Perturbed by a Degenerate Multiplicative Noise ...................... 45
Giuseppe Da Prato and Alessandra Lunardi
Kolmogorov Operators of Hamiltonian Systems
Perturbed by Noise .................................................. 61
A.F.M. ter Elst, Derek W. Robinson, Adam Sikora and Yueping Zhu
Dirichlet Forms and Degenerate Elliptic Operators . . . . . . . . . . . . . . . . . . . 73
Onno van Gaans
On R-boundedness of Unions of Sets of Operators . . . . . . . . . . . . . . . . . . . 97
Matthias Geißert, Horst Heck and Matthias Hieber
On the Equation divu=gand Bogovski˘ı’s Operator
in Sobolev Spaces of Negative Order ................................. 113
F. den Hollander
Renormalization of Interacting Diﬀusions:
A Program and Four Examples ...................................... 123
Tuomas P. Hyt¨onen
Reduced Mihlin-Lizorkin Multiplier Theorem
in Vector-valuedL
p
Spaces .......................................... 137

vi Contents
Stig-Olof Londen
Interpolation Spaces for Initial Values of
Abstract Fractional Diﬀerential Equations ........................... 153
Noboru Okazawa
Semilinear Elliptic Problems Associated with
the Complex Ginzburg-Landau Equation ............................ 169
Jan Pr¨uss and Gieri Simonett
Operator-valued Symbols for Elliptic and
Parabolic Problems on Wedges ...................................... 189
Jan Pr¨uss and Mathias Wilke
MaximalL
p-regularity and Long-time Behavior
of the Non-isothermal Cahn-Hilliard Equation
with Dynamic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Jacques Rappaz
Numerical Approximation of PDEs and Cl´ement’s Interpolation . . . . . . 237
Erik G.F. Thomas
On Prohorov’s Criterion for Projective Limits ........................ 251
Lutz Weis
TheH
∞
Holomorphic Functional Calculus
for Sectorial Operators – a Survey ................................... 263

Preface
The present volume is dedicated to Philippe Cl´ement on the occasion of his re-
tirement in December 2004. It has its origin in the workshop “Partial Diﬀeren-
tial Equations and Functional Analysis” (Delft, November 29–December 1, 2004)
which was held to celebrate Philippe’s profound contributions in various areas of
Mathematical Analysis.
The articles presented here oﬀer a panorama of current developments in the
theory of partial diﬀerential equations as well as applications to such diverse ar-
eas as numerical analysis of PDEs, Volterra equations, evolution equations,H
∞
-
calculus, elliptic systems, mathematical physics, and stochastic analysis. They re-
ﬂect Philippe’s interests very well and indeed several of the authors have collabo-
rated with him in the course of his career.
The editors gratefully acknowledge the ﬁnancial support of the Royal Nether-
lands Academy of Arts and Sciences, the Netherlands Organization for Scientiﬁc
Research, and the Thomas Stieltjes Institute for organizing the workshop. They
also thank Thomas Hempﬂing for the pleasant collaboration during the prepara-
tion of this volume.
Last but not least the editors, all members of his former group, thank Philippe
for his constant inspiration and for sharing his enthusiasm in mathematics with
them.
The Editors

This volume is dedicated toPhilippe Cl´ement
on the occasion of his retirement.

Philippe Cl´ement: Curriculum Vitae
Philippe Cl´ement, born on 9 January 1943 in Billens, Switzerland, started his study
in Physics at the Ecole Polytechnique de l’ Universit´e de Lausanne (now EPFL) in
1962 and obtained the degree of Physicist-Engineer in 1967. During that period he
discovered that his true interest was much more in Mathematics and he obtained
the License des Sciences Math´ematiques in 1968 from the University of Lausanne.
Hereafter he started to work on his Ph.D. thesis in the area of Numerical Analysis
at the EPFL with J. Descloux as supervisor. He defended his thesis, “M´ethode des
´el´ements ﬁnis appliqu´ee `a des probl`emes variationnels de type ind´eﬁni”, in Febru-
ary 1974. Some of the results were published in his seminal paper “Approximation
by ﬁnite element functions using local regularization”(Rev.Fran¸caise Automat.
Informat. Recherche Op´erationelle, RAIRO Analyse Num´erique 9, 1975, R-2, 77–
84). In this paper he introduced what is nowadays known in the literature as
the Cl´ement-type interpolation operators, which play a key role in the analysis of
adaptive ﬁnite element methods.
In the period 1972–74 Philippe was First Assistant at the Department of
Mathematics of the EPFL and under the inﬂuence of B. Zwahlen he became in-
terested in Nonlinear Analysis. It was a very stimulating and inspiring time and
environment for him, in particular, he met at various workshops Amann, Aubin,
Da Prato, Grisvard, Tartar and others. The years 1974–77 Philippe continued his
mathematical work, supported by the Swiss National Foundation for Scientiﬁc
Research, in Madison (USA), ﬁrst as Honorary Fellow at the Mathematics De-
partment, later as a Research Staﬀ Member at the Mathematics Research Center,
of the University of Wisconsin. In that period he came into contact with Crandall
and Rabinowitz and worked on nonlinear elliptic problems. Together with Nohel
and Londen, he started to be involved in nonlinear Volterra equations.
In 1977 Philippe moved to the University of Technology in Delft, were he was
appointed as Associate Professor. In 1980 he became full professor and in 1985 he
obtained the Chair in Functional Analysis in Delft. His main areas of interest and
research were (and still are) the theory of evolution equations, operator semigroups
as well as the Volterra equations and elliptic problems mentioned before. In partic-
ular, he was involved in problems concerning maximal regularity and problems re-
lated to functional calculus. Philippe is widely recognized for his important contri-
butions in these areas. The very stimulating seminars in Delft on the theory of semi-
groups have resulted in the book “One-Parameter Semigroups” (Cl´ement, Heij-
mans et al.). In recent years his interests also include stochastic integral equations.

Operator Theory:
Advances and Applications, Vol. 168, 1–26
c∅2006 Birkh¨auser Verlag Basel/Switzerland
Harnack Inequality and Applications to
Solutions of Biharmonic Equations
Gabriella Caristi and Enzo Mitidieri
This paper is dedicated to our friend Philippe Cl´ement for “not killing birds”
Abstract.We prove Harnack type inequalities for linear biharmonic equa-
tions containing a Kato potential. Various applications to local boundedness,
H¨older continuity and universal estimates of solutions for biharmonic equa-
tions are presented.
1. Introduction
During the last decade considerable attention has been paid to solutions of the
time-independent Schr¨odinger equation
−∆u=V(x)u, x∈Ω⊆R
N
, (1.1)
where Ω is an open subset ofR
N
and the potentialVbelongs to the Kato class
K
N,1
loc
(Ω
[8], Chiarenza, Fabes and Garofalo [7], Hinz and Kalf [13], Serrin and Zou [22],
Simader [23] and the references therein. Following diﬀerent approaches these au-
thors have studied the regularity of the solutions and proved a Harnack-type
inequality. Such inequality in turn can be used to prove results such as strong
maximum principles, removable point singularities, existence of solutions for the
Dirichlet problem, Liouville theorems anduniversal estimateson solutions for non-
linear equations.
In the present paper we shall discuss weak solutions of the following fourth-
order elliptic equation
∆
2
u=Vu,x∈Ω⊆R
N
, (1.2)
where the potentialVis nonnegative in Ω and belongs toK
N,2
loc
(Ω
Kato class of potentials associated to the biharmonic operator. See Deﬁnition
2.2 in the next section. We point out that Kato classes of potentials associated
to polyharmonic operators and some generalizations have been used in a series

2 G. Caristi and E. Mitidieri
of works by Bachar et al. see [3], [4] and Maagli et al. [14]. In these papers the
authors prove various interesting results including 3G type Theorems and existence
of positive solutions for second order and semilinear polyharmonic equations.
Here we are interested in several kindsof results of qualitative nature and
mainly some of the consequences that can be deduced from Harnack inequality that
we are going to prove during the course. The ﬁrst one is the regularity problem
which includes the local boundedness and the H¨older continuity of solutions. The
second kind of results are Harnack-type inequalities. The prototype version of these
states: There exist constantsC=C(N)andr>0 depending on Ω and norms of
Vsuch that all solutions of (1.2) withu≥0,−∆u≥0 in Ω (i.e., in the sense of
distributions in Ω) satisfy
sup
B
r/2
u(x)≤Cinf
B
r/2
u(x), (1.3)
whereB
r/2denotes any ball contained in Ω.
Our interest in these results was stimulated by studying certain nonlinear
biharmonic equations and their isolated singularities (see [5], [25]). For several
questions concerning the nature of non-removable singularities and the behavior
of a positive solution in a neighborhood of the isolated singularity it is customary
to assume thatV∈L
q
loc
(Ω) forq>
N
4
.
On the other hand the technique for provingL
p
-estimates for allp<∞,
relating the suputo certain integrals involving the solution of (1.2) as employed by
Serrin [20], [21], Stampacchia [26], Trudinger [27] for second order elliptic equations
and by Mandras [15] in the context of weakly coupled linear elliptic systems, seems
not to be easily extendable for this kind of equations.
In order to derive a localapriorimajorization for the supu, we shall follow
two diﬀerent approaches. The ﬁrst one was introduced by Simader in [23] and it
is based on representation formulae of solutions and the 3Gtheorem of Zhao [30],
while the latter have been used by Chiarenza et al [7] and its main ingredients are
Caccioppoli-type inequalities and maximum principle arguments for the operator
∆
2
−V.
We remark that the Harnack inequality (1.3) admits a rather simple proof
in the special caseV≡0, that is for the biharmonic equation ∆
2
u=0inΩ.In
fact, due to the special type of mean value formulas for biharmonic functions (see
formula (3.4)0of Simader [24])
u(x)=
N+1
2
1
|BR(x)|
ﬀ
BR(x)
u(y)
ﬃ
(N+2)−(N+3)
|y−x|
R
γ
dy
it turns out that forx0∈ΩandB2r(x0)⊂Ω
sup
B
r/2(x0)
|u(x)|≤
c(N)
|Br(x0)|
ﬀ
Br(x0)
|u(y)|dy,

Harnack Inequality and Biharmonic Equations 3
wherec(N)=2
N−1
(N+1)(2N+ 5). On the other hand both conditionsu≥0
and−∆u≥0 imply (see [11])
inf
B
r/2(x0)
u(x)≥
(2/3)
N
|Br(x0)|
ﬀ
Br(x0)
u(y)dy,
hence inequality (1.3) follows immediately. We observe that in general there is
no hope of obtaining a Harnack inequality for solutions of (1.2) under the only
assumption that they are nonnegative. A simple example in this direction is given
byu(x)=x
2
1in Ω =B2(0). It is interesting to note that the sign conditions
(−∆)
m
u≥0inΩ,form=1,...,k−1, can be already found in the classical
book of Nicolescu [18], p. 16, in the context of polyharmonic equations of the type
∆
k
u=0inΩ,k∈N(see also [2]).
This paper is organized as follows. Section 2 contains few preliminary facts,
the proof of the representation formula (see Lemma 2.6) (following Simader [23])
for weak solutions of the problem,
(−∆)
m
u=Vu,x∈Ω⊆R
N
, (1.4)
whereV∈K
N,m
loc
(Ω,N >2m, m≥2,and some remarks on Green functions
for Schr¨odinger biharmonic operators. In Section 3 we prove the results on local
boundedness and continuity of solutions of (1.2) and a related Harnack inequality
(1.3), (see Theorem 3.6). Namely, for eachp∈(0,∞) there exists a constant
C=C(p)andr>0 depending on Ω and some local norms ofVsuch that
sup
B
r/2(x0)
|u(x)|≤C
α
1
|Br(x0)|
ﬀ
Br(x0)
|u(y)|
p
dy
ρ
1/p
. (1.5)
In Section 4 we brieﬂy use the approach by Chiarenza, Fabes and Garofalo
[7], adapted to fourth order problems of the type (1.2). The main outcome of this
technique is an estimate of the modulus of continuity of the solutions is given in
Theorem 4.12 and the H¨older continuity is established in Theorem 4.13 for a class
of potentialsVwhich includes those that belongs to the Lebesgue spacesL
p
loc
(Ω)
forp>
N
4
. Finally Section 5 is devoted to some consequences of Theorem 4.9. The
ﬁrst application is another proof of Theorem 3.6 while Theorem 5.2 concerns the
behavior at inﬁnity of solutionsu∈L
p
∂
R
N
δ
. Also, in Theorem 5.1 and Theorem
5.3 we prove respectively a Harnack inequality of the type (1.3) and a modiﬁed
version of it in the limiting caseV=O
θ
1
|x|
4

and Ω =BR(0)\{0}.
We conclude the paper with an application touniversal estimatesfor a semi-
linear biharmonic equation in a general domain (see Theorem 5.5). This result
can be considered a step towards to the general understanding of the existence of
universal estimatesfor solutions of elliptic systems containing non-linearities with
growth below the so calledﬁrst critical hyperbola(see [16], [17]).

4 G. Caristi and E. Mitidieri
2. Notation and preliminaries
In the sequel, Ω will denote a nonempty open subset ofR
N
.IfΩ1⊂Ω2⊆R
N
areopensets,wewriteΩ1⊂⊂Ω2if and only if Ω1is compact andΩ1⊂Ω2.For
x∈R
N
andr>0wewriteBr(x) to denote the open ball of centerxand radius
rand∂Br(x) to denote its boundary.
In what follows we assume thatjis a given nonnegative function inC
∞
0(R
N
)
such thatj(z)=0if|z|≥1 and:
∅
R
N
j(x)dx=1.
For anyω>0wedeﬁnej∅(x)=ω
−N
j(ω
−1
x).
Deﬁnition 2.1.Given any functionf∈L
q
(Ωwith1≤q<∞, for anyω>0we
deﬁne themolliﬁed functionby
f∅(x)=
∅
R
N
j∅(x−y)f(y)dy.
From the deﬁnition it follows thatf∅∈C
∞
(Ω∪L
q
(Ω f∅−f
L
q
(Ω)→0,
asω→0.
Deﬁnition 2.2.GivenN>2m,theKato classK
N,m
loc
(Ωis the set of functions
V∈L
1
loc
(Ωsuch that for any compact setK⊂Ωthe quantity
φV(t, K)= sup
x∈R
N
∅
Bt(x)
|V(y)|χK(y)
|x−y|
N−2m
dy
is ﬁnite(here,χKdenotes the characteristic function ofK)and
lim
t→0
φV(t, K)=0.
Example2.3.Any function satisfying a local Stummel condition. We recall that
Vsatisﬁes alocal Stummel-conditionif it is measurable in Ω and there exists
γ∈(0,4) such that for eachω⊂⊂Ω there exists a constantDω>0 such that
sup
x∈R
N
∅
ω∩B1(x)
|V(y)|
2
|x−y|
N−4m+γ
dy≤Dω.
We claim that if this condition is satisﬁed, then,V∈K
N,m
loc
(Ω
BR(x0)⊂⊂Ωand0<t≤1wehave
∅
Bt(x)∩BR(x0)
|V(y)|
|x−y|
N−2m
dy
≤
α
∅
Bt(x)∩BR(x0)
|V(y)|
2
|x−y|
N−4m+γ
dy
ρ1
2
α
∅
Bt(x)
dy
|x−y|
N−γ
ρ1
2
≤D
1
2
BR(x0)

σN
Nγ
1
2
t
γ
2.
Therefore Deﬁnition 2.2 is satisﬁed withφV(t)=C(D, γ)t
γ/2
.

Harnack Inequality and Biharmonic Equations 5
Example2.4.AnyVbelonging toL
α
loc
(Ω α>N/2m. In fact, in this case if
Bt(x)⊂⊂Ωwehave:
∅
Bt(x)
V(y)
|x−y|
N−2m
dy
≤
α
∅
Bt(x)
|V(y)|
α
dy
ρ1
α
α
∅
Bt(x)
1
|x−y|
(N−2m)α
∅dy
ρ1
α
∅
≤σ
1
α
∅
N
V
L
α
(Bt(x))t
2m−
N
α. (2.1)
Therefore, Deﬁnition 2.2 is satisﬁed withφV(t)=C(V)t
2m−
N
αfort>0.
Deﬁnition 2.5.We say thatu∈L
1
loc
(Ωis adistributional solutionof(1.4),
or thatuis a solution in the sense of distributions, ifu∈L
1
loc
(Ωand for any
ψ∈C
∞
0
(Ωwe have
∅
Ω
u(y)∆
m
ψ(y)dy=
∅
Ω
V(y)u(y)ψ(y)dy (2.2)
For anym≥1,N>2mandr>0weset
Φm,N(r)=Cm,Nr
2m−N
, (2.3)
where
Cm,N=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Γ(2−N/2)
2
2m−2
(m−1)!Γ(m+1−N/2)
,ifNis odd,
(−1)
m−1
(N/2−m−1)!
2
2m−2
(m−1)!(N/2−2)!
,ifNis even.
The function Φm,Nas function ofr=|x|,forx∈R
N
\{0},ispolyharmonicof
degreemandiscalledthefundamental solutionof the equation ∆
m
u= 0. Here,
the constantsCm,Nare chosen so that ∆
p
Φm,N=Φm−p,Nforp=1,...,m−1,
(see [2]). In the sequel, sometimes we will write Φm(x, y) instead of Φm,N(|y−x|).
The followingrepresentation formulaholds:
Lemma 2.6.LetN>2m.Ifu∈C
∞
0(R
N
), then for anyx∈supp(u)
u(x)=−ΩN
∅
R
N
Φm,N(|y−x|)∆
m
u(y)dy, (2.4)
whereΩN=((N−2)ωN)
−1
andωN=2π
N/2
/Γ(N/2)is the area of the unit
sphere.

6 G. Caristi and E. Mitidieri
Proof.In order to prove (2.4) we apply the second Green identity inDρ=R
N
\
Bρ(y) successively, add the results and obtain
u(x)=ΩN
m−1
ϕ
l=0
ﬀ
∂Dρ

∂Φl+1(x, y)
∂νy
∆
l
u(y)−
∂∆
l
u(y)
∂ν
Φl+1(x, y)

dy
−ΩN
ﬀ
Dρ
Φm(x, y)∆
m
u(y)dy. (2.5)
By a standard argument it can be proved that all the nonzero boundary terms
tend to 0 asρ→0. ﬀ
Throughout the paper, for the sake of clarity, we will assume thatm=2and
N>4, that is, we shall consider the problem
∆
2
u=V(x)u, x∈Ω⊆R
N
,N>4. (2.6)
The proofs of the extensions of the results to the general case can be obtained by
obvious modiﬁcations.
Lemma 2.7.Letg∈L
1
loc
(R
N
)andu∈L
1
loc
(R
N
)be a distributional solution of
∆
2
u=g. Then, for anyρ∈C
∞
0
(Ωthe following formula holds:
cNu(x)ρ(x)=−
ﬀ
Φ2(x, y)g(y)ρ(y)dy
+2
ﬀ
u(y)(∇∆ρ(y),∇Φ2(x, y))dy+2
ﬀ
∆
2
ρ(y)u(y)Φ2(x, y)dy
+2
ﬀ
∆(∇Φ2(x, y),∇ρ(y))u(y)dy+
ﬀ
(∇ρ(y),∇∆Φ2(x, y))u(y)dy
+2
ﬀ
u(y)∆Φ2(x, y)∆ρ(y)dy−
ﬀ
u(y)∆
2
ρ(y)Φ2(x, y)dy, (2.7)
wherecN=Ω
−1
N
.
Proof.For anyﬃ>0, letuﬀbe the molliﬁed function ofu. Then, givenρ∈C
∞
0
(Ω
we can apply Lemma 2.6 and get
cNuﬀ(x)ρ(x)=−
ﬀ
Ω
Φ2(x, y)∆
2
(uﬀρ)(y)dy. (2.8)
Now, we have
∆
2
(uﬀρ)=∆
2
(uﬀ)ρ+2(∇∆ρ,∇uﬀ)+2∆ρ∆uﬀ+2(∇∆uﬀ,∇ρ)
+2∆(∇ρ,∇uﬀ)+uﬀ∆
2
ρ, (2.9)
and we know that ∆
2
uﬀ=gﬀand thatgﬀ→ginL
1
loc
(Ω
integrate by parts, taking into account of the fact that all the integrals extended
to the boundary of Ω are equal to 0, since supp(ρ)⊂Ω. Then, we take the limit
asﬃ→0 and apply Lemma 2.8 below. ﬀ

Harnack Inequality and Biharmonic Equations 7
Lemma 2.8.LetΩ⊆R
N
be bounded,ρ∈C
0
0(Ω,g∈L
1
loc
(Ωand0<α<N .
Then,ifforanyn∈N,ﬃn>0andﬃn→0,asn→∞, there exists a subsequence,
which we still denote byﬃn, such that
ﬀ
Ω
gﬀn(y)ρ(y)|y−x|
−α
dy→
ﬀ
Ω
g(y)ρ(y)|y−x|
−α
dy,a.e. inR
N
.
We refer to [23] for its proof.
2.1. Green functions of Schr¨odinger biharmonic operators
We recall some properties of the Green functions associated to the following bound-
ary value problems for the biharmonic equation on the ballBr(0)⊂R
N
:
∆
2
u(x)=f(x),x∈Br(0) : (2.10)
that is, the Navier boundary value problem (N
u=∆u=0on∂Br(0), (2.11)
and the Dirichlet boundary value problem (D
u=
∂u
∂ν
=0on∂Br(0). (2.12)
It is well known that both problems admit a nonnegative Green function onBr(0),
which we denote respectively byG
N
r(x, y)andG
D
r(x, y). Moreover, the following
estimates hold:
Lemma 2.9 (3G-Lemma).LetG=G
N
rorG
D
r. Then, there exists a constantC1>0
depending only on the dimensionNsuch that for any ball B ofR
N
we have
G(x, z)G(z,y)
G(x, y)
≤C1
ε
|x−z|
4−N
+|z−y|
4−N
ψ
,
for allx, y, z∈B.
The proof of this lemma for the Dirichlet problem is contained in [12], while
for the Navier problem it can be straightforwardly obtained by iteration of the
3G-Lemma for the Laplace operator, see [30].
Now, consider instead of (2.10) the Schr¨odinger biharmonic equation:
∆
2
u(x)=V(x)u(x)+f(x),forx∈Br(0), (2.13)
whereVbelongs to the natural Kato classK
N,2
loc
(Br(0)) associated to ∆
2
.The
following result extends Lemma 2.3 of [6] to the biharmonic case.
Proposition 2.10.Assume thatV∈K
N,2
loc
(Br(0)).Then,thereexistsr1>0such
that if0<r<r 1, the problems(2.13)–(2.11)and(2.13)–(2.12)admit a nonneg-
ative Green function onBr(0).

8 G. Caristi and E. Mitidieri
Proof.Let us consider the Navier boundary value problem: the proof in the other
case is similar. SetA0(x, y)=G
N
r
(x, y) and deﬁne forn≥1
An(x, y)=
ﬀ
Br
G
N
r(x, z)V(z)An−1(z,y)dz.
We prove by induction that there existsr1>0 such that if 0<r<r1we have
|An(x, y)|≤
1
3
n
A0(x, y),n≥0. (2.14)
Ifn= 0, (2.14) holds by deﬁnition. Assume that forn>0 (2.14) is true, then we
have:
|An+1(x, y)|≤
1
3
n
ﬀ
Br
G
N
r
(x, z)V(z)G
N
r
(z,y)dz.
By assumptionV∈K
N,2
loc
(Br(0)) and then, we can apply Lemma 2.9 and obtain
that
|An+1(x, y)|≤
1
3
n
C1G
N
r
(x, y)
ﬀ
Br
V(z)
ε
|x−z|
4−N
+|z−y|
4−N
ψ
dz.(2.15)
SinceV∈K
N,2
loc
,givenﬃ>0thereexistsr1>0 such that if 0<r<r1,then
ﬀ
Br(y)
|V(y)|
|x−y|
N−4
dy < ﬃ.
Using this fact in (2.15), we get that
|An+1(x, y)|≤
1
3
n
2C1ﬃG
N
r
(x, y),
for anyr<r1.Ifwechooseﬃ=1/(6C1) we get (2.14). This inequality implies
that the series
A(x, y)=
∞
ϕ
n=0
An(x, y)
is convergent and that its sumA(x, y) satisﬁes for allx, y∈Br
A(x, y)=G
N
r
(x, y)+
ﬀ
Br
G
N
r
(x, z)V(z)A(z,y)dz (2.16)
and
1
2
G
N
r(x, y)≤A(x, y)≤2G
N
r(x, z). (2.17)
From the last inequality it follows thatA(·,·)≥0. Moreover, applying ∆
2
to both
sides of (2.16) for each ﬁxedy∈Br,wehaveforx∈Br
∆
2
A(x, y)=δy(x)+V(x)A(x, y)
whereδyis the Dirac function supported aty. Moreover, it is easy to check that
the boundary conditions are satisﬁed. ﬀ
Remark2.11.In particular, from (2.17) it follows thatA(·,·) is nonnegative on
suﬃciently small balls without any assumption on the sign or on the norm ofV.

Harnack Inequality and Biharmonic Equations 9
By a similar argument, it can also be proved that if we ﬁx the radius of the ball, for
instance, equal to 1, then, there exists a nonnegative Green function for problems
(2.10)–(2.11) and (2.10)–(2.12) ifφV(1,B1) is suﬃciently small.
3. Local boundedness and continuity of solutions
In this section we will prove the local boundedness and the continuity of distribu-
tional solutions of problem (2.6), extending in this way Theorem 2.4 of [23]. The
method of proofs is essentially the same.
First of all, we chooseη∈C
∞
0
(R) such that 0≤η(t)≤1,η(t)=1if
t∈[−1/2,1/2] andη(t)=0if|t|>1. Givenδ>0, we setηδ(z)=η(δ
−1
|z|), for
z∈R
N
.
Lemma 3.1.Assume thatV∈K
N,2
loc
(Ω.Letx∈Ωand0<δ<
1
4
dist(x, ∂Ω).
Then, for anyx=z
∅
Ω
|V(y)|ηδ(x−y)ηδ(z−y)
|x−y|
N−4
|y−z|
N−4
dy≤2
N−3
φV(δ, B3δ(x))
η4δ(x−z)
|x−z|
N−4
. (3.1)
Proof.First of all, we observe that ifσ=|x−z|>2δ, then it follows that
ηδ(x−y)ηδ(z−y)=0.Hence, we can assume that|x−z|<2δ.Consequently,
σ/2≤δ.
Deﬁne Ω1={y∈Ω:|y−x|≤σ/2}and Ω2={y∈Ω:|y−x|≥σ/2},
and denote the corresponding integrals byI1andI2.Fory∈Ω1we have that
|z−y|≥|z−x|−|x−y|≥σ/2 and therefore
I1≤
∅
Ω1

2
σ

N−4
|V(y)|ηδ(x−y)
|x−y|
N−4
dy≤

2
σ

N−4
φV(δ, B3δ(x)).
Similarly,
I2≤
∅
Ω2

2
σ

N−4
|V(y)|ηδ(y−z)
|y−z|
N−4
dy≤

2
σ

N−4
φV(δ, B3δ(x)).
Sinceη4δ(|x−z|)=1forσ≤2δ, these inequalities imply (3.1). ∅
Lemma 3.2.Assume thatV∈K
N,2
loc
(Ωand thatuis a solution of(2.6)in the
sense of distributions. Letx0∈Ω,R0>0be such thatB2R0(x0)⊂⊂Ω. Choose
0<δ0<R0/4such that
φV(δ)≡φV(δ, BR0(x0))<2
2−N
cN (3.2)
for0<δ≤δ0.Then,for0<R≤R0/2,0<δ≤δ0and for a.e.z∈BR(x0)we
have:
∅
Ω
|V(x)||u(x)|ηδ(x−z)
|x−z|
N−4
dx
≤2
N−4
δ
4−N
Vu
L
1
(BR+4δ(x0))+2Cδ
−N
u
L
1
(BR+2δ(x0)),(3.3)
whereCdepends onNand the choice ofη.

10 G. Caristi and E. Mitidieri
Proof.Letx∈BR0(x0)and0<δ0<R0/4. Sinceρ(·)=ηδ(|·−x|)∈C
∞
0(Ω
can apply the representation formula of Lemma 2.7 and obtain that:
cNu(x)=−
ﬀ
Φ2(x, y)V(y)u(y)ηδ(y−x)dy
+2
ﬀ
u(y)(∇∆yηδ(y−x),∇Φ2(x, y))dy
+2
ﬀ
∆
2
yηδ(y−x)u(y)Φ2(x, y)dy
+2
ﬀ
∆(∇Φ2(x, y),∇yηδ(y−x))u(y)dy
+
ﬀ
(∇yηδ(y−x),∇∆Φ2(x, y))u(y)dy
+2
ﬀ
u(y)∆Φ2(x, y)∆yηδ(y−x)dy
−
ﬀ
u(y)∆
2
ηδ(y−x)Φ2(x, y)dy. (3.4)
Taking into account of the deﬁnition ofηδand of Φ2and denoting byχthe
characteristic function of the interval [1/2,1], we get that for anyx∈BR(x0)
|u(x)|≤c
−1
N
ﬀ
Φ2(x, y)|V(y)||u(y)|ηδ(y−x)dy
+Cδ
−N
ﬀ
|u(y)|χ(δ
−1
|y−x|)dy, (3.5)
whereC=C(N,η)>0. Using (3.5) in the left-hand side of (3.3), we obtain
ﬀ
|V(x)||u(x)|ηδ(x−z)
|x−z|
N−4
dx
≤c
−1
N
ﬀﬀ
|V(x)||V(y)||u(y)|ηδ(y−x)ηδ(x−z)
|x−z|
N−4
|x−y|
N−4
dxdy
+Cδ
−N
ﬀ
|V(x)|ηδ(x−z)
|x−z|
N−4
ﬀ
|u(y)|χ(δ
−1
|y−x|)dxdy. (3.6)
Now, we proceed as in the proof of Lemma 2.3 of [23]. The last integral in (3.6)
can be estimated byφV(δ)|u|
L
1
(BR+2δ(x0)). Then, we apply Lemma 2.6 to the ﬁrst
integral and get:
c
−1
N
ﬀ
|V(y)||u(y)|
ﬀ
|V(x)|ηδ(y−x)ηδ(x−z)
|x−z|
N−4
|x−y|
N−4
dxdy
≤c
−1
N
2
N−3
ﬀﬀ
φV(δ, B3δ(x))
|V(y)||u(y)|η4δ(y−z)
|y−z|
N−4
dydx.(3.7)
We remark thatφV(δ, B3δ(x))≤φV(δ)andthat
η4δ(y−z)=ηδ(y−z)+(η4δ(y−z)−ηδ(y−z)),

Harnack Inequality and Biharmonic Equations 11
hence we get
c
−1
N
ﬀ
|V(y)||u(y)|
ﬀ
|V(x)|ηδ(y−x)ηδ(x−z)
|x−z|
N−4
|x−y|
N−4
dxdy
≤c
−1
N
2
N−3
φV(δ)
ﬀ
|V(x)||u(x)|ηδ(x−z)
|x−z|
N−4
dx
+c
−1
N
2
N−3
φV(δ)
ﬀ
|V(y)|(η4δ(y−z)−ηδ(y−z))
|y−z|
N−4
|u(y)|dy.(3.8)
Sinceη4δ(y−z)−ηδ(y−z)) = 0 if|y−z|≤δ/2andif|y−z|≥4δ, the last integral
in (3.8) can be estimated by

δ
2

4−N
Vu
L
1
(BR+4δ(x0)). (3.9)
By assumption, if 0<δ≤δ0,then
2c
−1
N
2
N−3
φV(δ)≤1
and hence, from (3.8 ﬀ
Theorem 3.3.Assume thatV∈K
N,2
loc
(Ωand thatuis a solution of(2.6)in
the sense of distributions. Letx0∈Ω,R0>0be such thatB2R0(x0)⊂⊂Ωand
assume that0<R1<R0/2is such that
φV(R1,B2R0(x0))≤(2cN)
−1
.
Then, for0<R≤R1,0<δ≤δ0(whereδ0is deﬁned in Lemma3.2)and
x∈BR(x0)the following estimate holds
|u(x)|≤2
N−4
δ
4−N
Vu
L
1
(BR+4δ(x0))+Cδ
−N
u
L
1
(BR+2δ(x0)), (3.10)
whereCdepends only onNand the choice ofη.
Proof.The proof follows directly from the proof of Lemma 2.7. ﬀ
Corollary 3.4.Assume thatV∈K
N,2
loc
(Ωand thatuis a solution of(2.6)in the
sense of distributions. Thenuis locally bounded.
The continuity ofufollows from its local boundedness. Indeed, in (3.4) all
the integrals which include derivatives ofηδcontain no singularity and therefore
they deﬁne continuous functions. Further, for any 0<ﬃ<, the function
hﬀ(x)=−
ﬀ
Ω\Bρ(x)
Φ2(x, y)V(y)u(y)ηδ(|y−x|)dy
is continuous and converges locally uniformly to the ﬁrst integral in (3.4) asﬃ→0.
In order to prove the Harnack inequality, we need the following result which
gives an estimate of the local · ∞-norm ofuin terms of its · 1-norm.

12 G. Caristi and E. Mitidieri
Proposition 3.5.Assume thatV∈K
N,2
loc
(Ωand thatuis a solution of(2.6)in
the sense of distributions. Letx0∈Ω,R0>0be such thatB2R0(x0)⊂⊂Ω.Then,
there exists0<R1≤R0/2such that for any0<R<R 1the following estimate
holds
u
L
∞
(BR(x0))≤CR
−N
u
L
1
(B2R(x0)), (3.11)
whereCdepends only onN.
Proof.LetR1≤R0/2 be as in Theorem 2.1. Then, for any 0<R<R 1:
φV(B2R(x0))≤2
−1
cN. (3.12)
We shall estimate Vu
L
1
(B
3R/2(x0))with the u
L
1
(B2R(x0)).Tothisaim,wetake
ρ∈C
∞
0
(R) such that 0≤ρ≤1,ρ(·)≡1in[0,3R/2] andρ(t)=0ift≥2R.
Further, we assume that|ρ
(i)
(t)|≤t
−i
fori=1,4andanyt. From Lemma 2.7 we
get that for anyx∈B2R(x0)
cN|u(x)|ρ(x)=
∅
|V(y)||u(y)|Φ2(x, y)ρ(|x−y|)dy
+
∅
|u(y)||R(x, y)|dy, (3.13)
whereRis deﬁned in terms of the derivatives ofρand of Φ2. Multiplying (3.13)
by|V(x)|and integrating overB≡B2R(x0)weget:
cN
∅
B
|V(x)||u(x)|ρ(x)dx
≤
∅ω∅
|V(x)|
|x−y|
N−4
dx
γ
|V(y)||u(y)|ρ(y)dy
+
∅
B
∅
3R
2
≤|x−y|≤2R
|u(y)|
|V(x)|
|x−y|
N−4
dxdy
≤φV(B2R(x0))
∅
B
|V(x)||u(x)|ρ(x)dx+φV(B2R(x0))
∅
B
|u(x)|dx.(3.14)
From (3.12) and (3.14) we deduce that:
∅
B
|V(x)||u(x)|ρ(x)dx≤C
∅
B
|u(x)|dx, (3.15)
which implies that
|V(u)|
L
1
(B3R
2
(x0))≤c
−1
N
|u|
L
1
(B2R(x0)). (3.16)
Now, we apply Theorem 2.1 withδ=R/8. From (3.10) and (3.16), it follows that
for anyx∈BR(x0):
|u(x)|≤2
N−4
δ
4−N
|V(u)|
L
1
(B3R
2
(x0))+Cδ
−N
|u|
L
1
(B2R(x0))
≤CR
N
|u|
L
1
(B2R(x0)), (3.17)
whereC=C(N). ∅

Harnack Inequality and Biharmonic Equations 13
3.1. The Harnack inequality
Theorem 3.6.Assume thatV∈K
N,2
loc
(Ω).Letube a nonnegative weak solution of
(2.6)such that−∆u≥0inΩ. Then there existC=C(N)andR0=R0(V)such
that for each0<R≤R0andB2R(x0)⊂⊂Ωwe have
sup
B
R/2(x0)
u≤Cinf
B
R/2(x0)
u, (3.18)
whereCindependent ofVandu.
Proof.Sinceuis nonnegative and satisﬁes−∆u≥0inΩ,thenwehave(see,
e.g., [11])
inf
B
R/2(x0)
u(x)≥C(N)
1
|BR|
ﬀ
BR(x0)
u(y)dy,
for allR>0 such thatB2R⊂Ω . Then, the conclusion follows from this inequality
and (3.11). ﬀ
Remark3.7.The condition−∆u≥0 in Ω is necessary for the validity of the above
Theorem as the following example shows: consider the functionu(x)=x
2
1
.Itis
nonnegative, satisﬁes ∆
2
u=0and∆u= 2, but (3.18) does not hold forx0and
anyR.
4. Local boundedness and continuity of solutions:
an alternative approach
In this section we shall prove The Harnack inequality following the approach of
[7], based onL
p
estimates. Assume thatm=2,N>4andthatV∈K
N,2
loc
(Ω
First of all, we recall the following result from [19]:
Lemma 4.1.Assume thatV∈K
N,2
loc
(Ω. Then, for everyﬃ>0there exists a
constantC=C(ﬃ, V)such that
ﬀ
Ω
|V|ψ
2
≤ﬃ
ﬀ
Ω
|∆ψ|
2
+C
ﬀ
Ω
|ψ|
2
,for allψ∈H
2,2
0
(Ω). (4.1)
For later use we need also the following slightly diﬀerent inequality. Ift>0
s∈(0,t), then:
ﬀ
Bt
|V|ψ
2
≤ﬃ
ﬀ
Bt
|∆ψ|
2
+
C(ﬃ, V)
(t−s)
4
ﬀ
Bt
|ψ|
2
,for allψ∈H
2,2
0
(Bt).(4.2)
WhenV∈L
N
4−δ(Ω),δ∈(0,4), a simple proof of the above embedding property
(4.2) can be obtained as follows. By the embedding
L
2N
N−4(Ω∩L
2
(Ω)λ→L
2N
N−4+δ(Ω
and by Sobolev’s inequality
ψ
L
2N
N−4(Ω)
≤c(N) ∆ψ
L
2
(Ω)
, for allψ∈H
2,2
0
(Ω,

14 G. Caristi and E. Mitidieri
it follows that for everyﬃ>0
ﬀ
Ω
|V|ψ
2
≤(t−s)
δ
V
L
N
4−δ(Ω)
(t−s)
−δ
ψ
2
L
2N
N−4+δ(Ω)
≤(t−s)
δ
V N
4−δ

ﬃ ψ
2
L
2N
N−4(Ω)
+C(δ)(t−s)
−4
ﬃ
−
4−δ
δ ψ
2
L
2
(Ω)

.
Hence, by takingﬃ1=ﬃ(t−s)
δ
V
L
N
4−δ(Ω)
we obtain
ﬀ
Ω
|V|ψ
2
≤ﬃ1
ﬀ
Ω
|∆ψ|
2
+
C
(t−s)
4
ﬀ
Ω
|ψ|
2
, (4.3)
whereCdepends onﬃ1,δand (t−s)
δ
V
N/4−δ
.
Remark4.2.As a consequence of (4.1) we deduce that, ifu∈H
2,2
loc
(Ω), then,
Vu∈L
1
loc
(Ω B2r⊂Ω and letϕ∈C
∞
0
(B2r) be such that 0≤ϕ≤1,
ϕ≡1onBr.Sinceuϕ∈H
2,2
0
(B2r)and
ﬀ
B2r
|V||u|ϕ
2
≤
ﬀ
B2r∩{|u|≤1}
|V||u|+
ﬀ
B2r∩{|u|>1}
|V||u|ϕ
2
≤
ﬀ
B2r
|V|+
ﬀ
B2r
|V||uϕ|
2
,
it follows thatVu∈L
1
(Br).
Deﬁnition 4.3.We say thatu∈H
2,2
loc
(Ωis aweak solutionof(2.6), if for any
ψ∈C
∞
0(Ωwe have
ﬀ
Ω
∆u(y)∆ψ(y)dy=
ﬀ
Ω
V(y)u(y)ψ(y)dy. (4.4)
The proof of the local boundedness of any weak solution of (2.6) will be proved
through a sequence of lemmas. The ﬁrst one states aCaccioppoli-type estimate([9]).
We shall omit its proof
Lemma 4.4.Letube a weak solution of(2.6).If0<s<tandBt⊂Ω, then there
exists a constantC=C(V)independent ofusuch that
ﬀ
Bs
|∆u|
2
≤
C
(t−s)
4
ﬀ
Bt
|u|
2
. (4.5)
Remark4.5.We note that also the following estimate holds:
ﬀ
Bs
|∇u|
2
≤
C
(t−s)
2
ﬀ
Bt
u
2
. (4.6)
whereC=C(φV).

Harnack Inequality and Biharmonic Equations 15
The following lemma is well known see for instance [7].
Lemma 4.6.Assume that there existθ∈(0,1),α≥0,a>0and a real continuous,
non-decreasing and positive functionI(·)deﬁned on(0,1]such that
I(s)≤a

1
(t−s)
nI(t)

θ
, (4.7)
holds for all0<s<t ≤1. Then, for everys1>0there exists a constant
C=C(θ, n, s1,a)independent ofIsuch that
I(s1)≤C. (4.8)
The next results provide a kind of reversed H¨older inequality.
Lemma 4.7.LetB2rbe a ball contained inΩand letube a weak solution of(2.6).
Then, there exists a constantC(φV)such that
α
1
|B
r/2|
Ω
B
r/2
u
2
ρ
1/2
≤C

1
|Br|
Ω
Br
|u|

. (4.9)
Proof.Without restriction we may assume that the center ofBris the origin. Let
ur(x)=u(rx). Thenuris a solution of ∆
2
ur=VrurinB2(0), whereVr(x)=
r
4
V(rx)satisﬁes
sup
x∈B2
Ω
B2
|Vr(y)|
|x−y|
N−4
dy≤sup
0<α<4r
sup
w∈Ω
Ω
Bα(w)
|V(z)|χΩ(z)
|w−z|
N−4
dz. (4.10)
SinceV∈K
N,2
loc
(Ω
zero asr→0. We may establish Lemma 4.7 by takingr=1,Ω=B2,whereB2
is centered at the origin. We may also assume that
Ω
B1
|u|=1.
Deﬁne for 0≤s≤1:
I(s)=
Ω
Bs
u
2

1/2
.
SinceL
1
(Ω)∩L
2N
N−4(Ω)λ→L
2
(Ω
I(s)≤
Ω
Bs
|u|
2N
N−4
N−4
2(N+4)
.
By the Sobolev inequality we obtain for 0≤s<t≤1
I(s)≤a(φV)
ω
1
(t−s)
2
I(t)
γN
N+4
. (4.11)
The conclusion follows from Lemma 4.6. Ω

16 G. Caristi and E. Mitidieri
Lemma 4.8.Letp∈(0,2),n≥0anda>0.Letv∈L
∞
(B1)be nonnegative and
sup
Bs
v
2
≤
a
(t−s)
n
ﬀ
Bt
v
2
(x)dx,
for every0<s<t≤1. Then, there exists a constantC(n, p, a)>0such that
sup
B
1/2
v
p
≤C
ﬀ
B1
v
p
(x)dx.
Proof.We may suppose thatv≡0. Putting
γ=
ﬀ
B1
v
p
(x)dx

−2/p
andI(s)=γ
ﬀ
Bs
v
2
,for 0<s<1,
we note that
sup
B
1/2
v
p
≤[6
n
aI(2/3)]
ﬀ
B1
v
p
(x)dx.
In order to complete the proof it is suﬃcient to boundI(2/3) by a constant
depending only onn, panda. To this end letα=
2
2−p
and 0<s<t≤1. We
obtain
I(s)
α
=γ
α
ﬀ
Bs
v
2−p
v
p

α
≤γ
α
sup
Bs
v
2
ﬀ
Bs
v
p

α
≤
aγ
α
(t−s)
n
ﬀ
Bs
v
p

αﬀ
Bt
v
2
≤
a
(t−s)
nI(t).
Finally, puttingθ=1/αands1=2/3, the conclusion follows from (4.7) and (4.8).
ﬀ
The following theorem states the local boundedness of weak solutions of (2.6).
Theorem 4.9.LetV∈K
N,2
loc
(Ω. Then, for eachp∈(0,∞)there existC=C(p)>
0andr1=r1(φV)>0such that ifuis a weak solution of(2.6), then, for each
0<R≤R1andB2R(x0)⊂Ω, we have
sup
B
R/2(x0)
|u(x)|≤C
α
1
|Br|
ﬀ
BR(x0)
|u(y)|
p
dy
ρ
1/p
. (4.12)
Proof.Without restriction we can assume thatx0=0.LetG(x, y) be the Green
function of the Dirichlet problem for equation (2.13) onB2R, see Proposition 2.10.
ChooseR/2≤s<t≤Rand a functionϕ∈C
∞
0
(B
t−(t−s)/4) such that 0≤ϕ≤1,
ϕ≡1onB
(t+s)/2,and
χ
χ
D
k
ϕ
χ
χ
≤c/(t−s)
k
,fork=1,2. We claim that for almost
allx∈B2Rthe following identity holds:
u(x)ϕ(x)=
ﬀ
B2R
∆yG(x, y)u(y)∆ϕ(y)dy−
ﬀ
B2R
G(x, y)∆u(y)∆ϕ(y)dy
+2
ﬀ
B2R
∆yG(x, y)(∇u(y),∇ϕ(y))dy−2
ﬀ
B2R
(∇yG(x, y),∇ϕ(y))∆u(y)dy.

18 G. Caristi and E. Mitidieri
Since
τ
Bt
G(x, y)dyis bounded, from the above estimates we obtain
sup
Bs
|u(x)|≤
c
(t−s)
5
∅
Bt
u(y)
2
dy

1/2
. (4.13)
Finally in virtue of Lemma 4.8 and estimate (4.13) we get the conclusion, for
p∈(0,2):
sup
B
r/2
|u(x)|≤C(p)
∅
Br
|u(y)|
p
dy

1/p
.
The casep∈(2,∞) follows applying H¨older inequality. ∅
The continuity of weak solutions can be proved as in the previous section.
4.1. The H¨older continuity
The proof of Theorem 4.12 below will be based on two lemmas, which are adap-
tations of the Lemmas 3.1 and 3.2 of [23]. The ﬁrst one is the following.
Lemma 4.10.Let0<β<1andx0,x∈R
N
,r>0be such that
λ:=

|x−x0|
r

1−β
≤1. (4.14)
Then, there exists a constantc>0such that forywith|y−x0|≥2r
1−β
|x−x0|
β
we have
χ
χ
χ
χ
χ
1
|y−x|
N−4
−
1
|y−x0|
N−4
χ
χ
χ
χ
χ
≤c

|x−x0|
r

1−β
1
|y−x0|
N−4
. (4.15)
Proof.Lety∈R
N
be such that
|y−x0|≥2r
1−β
|x−x0|
β
. (4.16)
Since|x−x0|=|x−x0|
β
|x−x0|
1−β
≤λ/2|y−x0|, it follows that
|y−x|≥|y−x0|−|x−x0|≥

1−
λ
2

|y−x0|.
From this inequality we get
χ
χ
χ
χ
χ
1
|y−x|
N−4
−
1
|y−x0|
N−4
χ
χ
χ
χ
χ
=||y−x0|−|y−x||
N−5
ϕ
k=0
|y−x0|
k−N+4
|y−x|
−k−1
≤
|x−x0|
|y−x0|
N−3
N−5
ϕ
k=0

1
1−λ/2

k+1
.
Hence, the conclusion follows withc=
η
N−5
k=0
2
k
. ∅

Harnack Inequality and Biharmonic Equations 19
Now, letx0∈Ω,R>0 be such thatB4R(x0)⊂Ωandletg∈K
N,2
loc
(Ω
nonnegative withφg(t)=φg(t, B2R(x0)). Letψ∈C
∞
0
(B2R(x0)), 0≤ψ≤1and
J(x):=cN
∅
g(y)ψ(y)
|y−x|
N−4
dy, x∈R
N
.
We know thatJ(·) is continuous onB2R(x0). The next lemma provides an estimate
of the local modulus of continuity ofJ.
Lemma 4.11.There exists a constantc>0such that for anyβ∈(0,1)andxsuch
that|x−x0|≤rwe have
|J(x)−J(x0)|≤c

|x−x0|
r

1−β
φg(2r)+2φg
θ
3r
1−β
|x−x0|
β

.(4.17)
Proof.Without loss of generality we may assume thatx0=0.Choose0<β<1
and let 0<|x|<r. It follows that
|J(x)−J(0)|≤cN
∅∅
g(y)χB2r(y)
χ
χ
χ|x−y|
4−N
−|y|
4−N
χ
χ
χdy
≤cN
∅
Br∩{y:|y|≥2r
1−β
|x|
β
}
···+cN
∅
Br∩{y:|y|≤2r
1−β
|x|
β
}
···:=J1(x)+J2(x).
Now, applying Lemma 4.10 toJ1we obtain
J1(x)≤c

|x|
r

1−β∅
g(y)χB2r(y)
|y|
N−4
dy≤c

|x|
r

1−β
φg(2r). (4.18)
ConsiderJ2(·). Since|y|≤2r
1−β
|x|
β
and|x|
1−β
≤r
1−β
we deduce that
|y−x|≤|x|+|y|≤|x|
β
θ
|x|
1−β
+2r
1−β

≤3r
1−β
|x|
β
,
and then,
J2(x)≤cN
∅
B2r∩{y:|y−x|≤3r
1−β
|x|
β
}
g(y)
|y−x|
N−4
dy
+cN
∅
B2r∩{y:|y|≤2r
1−β
|x|
β
}
g(y)
|y|
N−4
dy
≤2ηg
θ
3r
1−β
|x|
β

.
The conclusion now follows from this inequality and (4.18). ∅
Theorem 4.12.Letx0∈ΩandB2R(x0)⊂Ω.Ifuis a weak solution of(2.6),
then, chosen0<β<1there exists a constantcsuch that for|x−x0|<rwe have
|u(x)−u(x0)|≤
≤c
ζ

|x−x0|
r

1−β
(φV(2r)+1)+φV
α
3r

|x−x0|
r

β
∞∨
sup
B3r(x0)
|u|.

20 G. Caristi and E. Mitidieri
Proof.Without loss of generality we may assume thatx0=0andset B4r=
B4r(0). Letϕ∈C
∞
0
(B2r), 0≤ϕ≤1withϕ≡1onB
3r/2and
χ
χ
D
k
ϕ
χ
χ
≤c/r
|k|
,
fork=1,2. We have
u(x)ϕ(x)=
ﬀ
Φ2(x, y)V(y)u(y)ϕ(y)dy−
ﬀ
Φ2(x, y)∆u(y)∆ϕ(y)dy
+
ﬀ
∆yΦ2(x, y)u(y)∆ϕ(y)dy−2
ﬀ
∇yΦ2(x, y)·∇ϕ(y)∆u(y)dy
+2
ﬀ
∆yΦ2(x, y)∇u(y)·∇ϕ(y)dy=
5
ϕ
i=1
Ii(x).
Fixβ∈(0,1) and let|x|<r. From this identity we obtain
u(x)−u(0) =
ﬀ
(Φ2(x, y)−Φ2(0,y))V(y)u(y)ϕ(y)
−
ﬀ
(Φ2(x, y)−Φ2(0,y)) ∆u(y)∆ϕ(y)dy
+
ﬀ
(∆yΦ2(x, y)−∆yΦ2(0,y))u(y)∆ϕ(y)
−2
ﬀ
(∇yΦ2(x, y)−∇yΦ2(0,y))∇ϕ(y)∆u(y)
+2
ﬀ
(∆yΦ2(x, y)−∆yΦ2(0,y))∇u(y)∇ϕ(y). (4.19)
SinceVu∈K
N,2
loc
(B2r)andφVu(t)≤sup
B2r
|u|·φV(t), we can apply Lemma 4.11
to the ﬁrst term on the right-hand side of (4.19) and obtain
|I1(x)−I1(0)|≤
ζ
c

|x|
r

1−β
ηV(2r)+2ηV
θ
3r
1−β
|x|
β

∨
sup
B2r
|u|.(4.20)
To handle the other terms of (4.19), let us observe that all the derivatives ofϕ
vanish for|y|≤r. Deﬁne fort∈[0,1]φ(t):=|y−tx|
4−N
,wherey∈B2r\B
3r/2
and|x|<r. From the mean value theorem and|y−t0x|≥|y|−|x|≥r/2 it follows
that
φ(1)−φ(0) =
χ
χ
χ|y−x|
4−N
−|y|
4−N
χ
χ
χ=φ
δ
(t0)
≤(N−4)
|x|
|y−t0x|
N−3
≤(N−4) 2
N−3
|x|
r
N−3
,
wheret0∈(0,1). Therefore,
|Φ2(x, y)−Φ2(0,y)|≤c
|x|
r
N−3
≤c

|x|
r

1−β
1
r
N−4
. (4.21)

Harnack Inequality and Biharmonic Equations 21
The same argument shows also that
|∇yΦ2(x, y)−∇yΦ2(0,y)|≤c

|x|
r

1−β
1
r
N−3
, (4.22)
|∆yΦ2(x, y)−∆yΦ2(0,y)|≤c

|x|
r

1−β
1
r
N−2
. (4.23)
Now we can estimate the remaining terms of (4.19), that is|Ik(x)−Ik(0)|for
k=2,...,5, by using the bounds (4.21), (4.22) and (4.23). We ﬁnd that
|I2(x)−I2(0)|≤
c
r
2
ﬀ
|Φ2(x, y)−Φ2(0,y)||∆u(y)|dy
≤c

|x|
r

1−β
1
r
N−2
ﬀ
B2r
|∆u|≤c

|x|
r

1−β
r
2

1
|B2r|
ﬀ
B2r
|∆u|
2

1/2
,
and hence, by Lemma 4.4
|I2(x)−I2(0)|≤c

|x|
r

1−β
1
|B3r|
ﬀ
B3r
|u|
2

1/2
. (4.24)
It is also easy to check that
|I3(x)−I3(0)|≤c

|x|
r

1−β
1
|B3r|
ﬀ
B3r
|u|
2

1/2
, (4.25)
and
|I4(x)−I4(0)|+|I5(x)−I5(0)|≤c

|x|
r

1−β
1
|B3r|
ﬀ
B3r
|u|
2

1/2
.
Finally, this inequality together with the estimates (4.20), (4.24) and (4.25) com-
plete the proof. ﬀ
In order to conclude that the weak solutions of (2.6) are H¨older continuous
we restrict ourselves to a class of potentialsVsuch that
φV(t)≤Mt
α
(4.26)
for someα>0. Clearly, this class is not empty since, according to Example 2.3,
we know that ifVsatisﬁes a Stummel condition, then (4.26) holds withα=γ/2,
or ifVbelong toL
p
loc
withp>
N
4
, then (4.26) holds withα=4−
N
p
.
We point out that ifN≤3 all weak solutionsu∈H
2,2
loc
(Ω) of (2.6) are H¨older
continuous by Sobolev’s embedding.
Theorem 4.13.Letx0∈ΩandB4R(x0)⊂Ω.Letube a weak solution of(1.1).
Suppose that there existα>0and a constantMV=M(V,B4R)such that(4.26)
holds. Then, if|x−x0|≤R, we have
|u(x)−u(x0)|≤c
ν
[MV(3R)
α
+1]R
−
α
1+αsup
B3R(x0)
|u|
ι
|x−x0|
α
1+α
.(4.27)

22 G. Caristi and E. Mitidieri
Proof.SinceφV(2r)≤MV(2R)
α
and
φV
α
3r

|x−x0|
R

β
ρ
≤MV(3r)
α

|x−x0|
R

αβ
,
the conclusion follows directly from Theorem 4.12 by takingβ=
1
1+α
. Ω
5. Applications
In this section we shall present some consequences of Theorem 4.9. The ﬁrst is
another proof of Harnack inequality:
Theorem 5.1.Assume thatV∈K
N,2
(Ω).Letube a nonnegative weak solution of
(1.1)such that−∆u≥0inΩ. Then there existC=C(N)andr0=r0(ηV)such
that for each0<R≤R0andB2R⊂Ωwe have
sup
B
R/2
u≤Cinf
B
R/2
u.
Proof.Sinceuis nonnegative and satisﬁes−∆u≥0inΩ,thenwehave(see,
e.g., [11])
inf
B
r/2
u(x)≥C(N)
1
|Br|
Ω
Br
u(y)dy,
for allr>0 such thatB2r⊂Ω. The conclusion follows from this inequality and
Theorem 4.9 withp=1. Ω
Theorem 5.2.Letp:1≤p<∞andΩ=R
N
\BR(0).Ifu∈L
p
(Ω∩H
2,2
loc
(Ωis
asolutionof(1.1)withV∈K
N,2
(Ωnonnegative then:
lim
|x|→∞
|u(x)|=0
Proof.Without loss of generality we may assume thatuis continuous in Ω. By
Theorem 4.9, it follows that there existsr0=r0(ηV) such that for allx:|x|≥R+2,
for eachr∈(0,r0)and0<s<t< 1wehave
sup
Bsr(x)
|u(y)|
p
≤
C
r
N
(t−s)
N
Ω
Btr(x)
|u(z)|
p
dz.
Sinceu∈L
p
(Ω) the result follows. Ω
The next result concerns the limiting case for a Harnack’s theorem near the
possible isolated singularityx= 0 (compare with Theorem 3.1 of [10]).
Theorem 5.3.LetΩ=BR(0)\{0}and letV∈Cloc(Ω)be nonnegative and such
that for allx∈Ω
0≤V(x)≤
c1
|x|
4
,c1>0.
Letu∈H
2,2
loc
(Ωbe a solution of
π
∆
2
u=Vu onΩ,
u≥0,−∆u≥0inΩ.
(5.1)

Harnack Inequality and Biharmonic Equations 23
Then there exist two constantsCandϑ1depending onc1such that for each0<
ϑ≤ϑ1and0<ω<
1
2R
we have
sup
∅≤|x|≤(1+ϑ)∅
u(x)≤C inf
∅≤|x|≤(1+ϑ)∅
u(x). (5.2)
Proof.The proof is a slight modiﬁcation of the proof of Theorem 4.9. Here, we shall
emphasize the dependence of the constants onc1.Ifx0∈Ωletr0=
1
4
|x0|so that
B2r0(x0)⊂Ω. We claim that for everyp∈(0,∞) there exists positive constants
C=C(p)andϑ0depending onc1such that for allx0∈Ωand0<r≤ϑ0r0
sup
B
r/2(x0)
|u(x)|≤C
α
1
|Br(x0)|
∅
Br(x0)
|u(y)|
p
dy
ρ
1/p
. (5.3)
To prove (5.3) we ﬁxσ∈(0,4) and note that
sup
x∈Ω
r
σ
V
L
N
4−σ(Br(x))
=C1<∞
whereC1depends onc1. This fact together with the embedding property ( 4.2)
yields the Lemma 3.2 for eachBsr0(x0)⊂Btr0(x0)⊂Ω. Clearly, Lemma 4.7 and
4.8 hold true. We have to bound
sup
x∈B2r(x0)
∅
B2r(x0)
V(y)
|x−y|
N−4
dy,
uniformly with respect tox0∈Ω. Indeed, since
sup
x∈B2r(x0)
∅
B2r(x0)
V(y)
|x−y|
N−4
dy≤c

r
σ
0
V
L
N
4−σ(B2r
0
(x0))

r
σ
r
σ
0
,
then
sup
x∈B2r(x0)
∅
B2r(x0)
V(y)
|x−y|
N−4
dy≤δ,
holds, provided that 0<r≤ϑ0r0,whereϑ0=δ/(2cC1). Hence, it suﬃces to
apply those lemmas in order to obtain (5.3). Combining (5.3) with
inf
B
r/2(x0)
u(x)≥
c
|Br(x0)|
∅
Br(x0)
u(y)dy,
corresponding tou≥0and−∆u≥0, we obtain for 0<r≤ϑ0r0
sup
B
r/2(x0)
u(x)≤Cinf
B
r/2(x0)
u(x). (5.4)
Finally, the inequality (5.2) follows from (5.4) by applying a standard covering
argument in the annulusω≤|x|≤(1 +ϑ)ω. ∅
5.1. Universal estimates
The following lemma is due to Mitidieri and Pohozaev [17].
Lemma 5.4.Letq>1. Assume thatuis a weak solution of the problem
∆
2
u(x)≥|u(x)|
q
,x∈Ω⊂R
N
, (5.5)

24 G. Caristi and E. Mitidieri
then, there existC>0independent ofuandR>0such that
∅
BR
|u(x)|
q
dx≤CR
N(q−1)−4q
q−1, (5.6)
whereBR=BR(x0)⊂Ω.
Proof.Chooseψ∈C
∞
0(R
N
) radially symmetric such thatsupp(ψ)=B2(0), 0≤
ψ≤1andψ≡1inB1(0). Givenx0∈ΩandR>0 such thatB2R(x0)⊂⊂Ω,
deﬁneψR=ψ(R
−1
(x−x0)). Multiply (5.5) byψand integrate by parts. By H¨older
inequality, we get
∅
Ω
|u|
q
ψRdx≤
∅
Ω
∆
2
uψRdx≤
∅
Ω
u∆
2
ψRdx≤
∅
Ω
|u|
q
ψRdx
1
q
α
∅
Ω
|∆
2
ψ|
q
∅
ψ
q
∅
−1
dx
ρ1
q
∅
.
By a standard change of variables we obtain:
∅
BR
|u|
q
dx≤C
α
∅
B1
|∆
2
ψ|
q
∅
ψ
q
∅
−1
dx
ρ1
q
∅
R
N(q−1)−4q
q−1.
This concludes the proof. ∅
Theorem 5.5.LetN>4andΩ=R
N
.Letf:R→Rbe a given function such
that there existsq∈(1,N/(N−4))andK>1such that
|u|
q
≤f(u)≤K|u|
q
,u∈R (5.7)
Then, there existC1>0andC2>0(depending only onN,qandK)such that
ifu∈L
q
loc
(Ωis a weak solution of
∆
2
u(x)=f(u(x)),x∈Ω, (5.8)
then, the following estimates hold
|u(x)|≤C1d(x, ∂Ω)
−
4
q−1,x∈Ω, (5.9)
|∆u(x)|≤C2d(x, ∂Ω)
−
2(q+1)
q−1,x∈Ω. (5.10)
Proof.Letx0∈ΩandR0>0 be such thatB2R0(x0)⊂⊂Ω. Sinceq<
N
N−4
,we
can ﬁxσ>0 such that
q=
N
N−4+σ
<
N
N−4
.
It follows thatV≡|u|
q−1
∈L
N/(4−σ)
loc
(Ω
V∈K
N,2
loc
(Ω
Letx0∈ΩandR0be chosen as in Theorem 3.3. Takeδ0=R/8. From (5.7)
and Lemma 2.7, we get:
|u(x)|≤CR
4−N
0
Vu
L
1
(B3R
2
(x0)+CR
−N
0
u
L
1
(BR
0
4
+R
(x0))

Harnack Inequality and Biharmonic Equations 25
whereCandC1are positive constants which depend onN. From Lemma 5.4 and
H¨older inequality, we get that
|u(x)|≤C1R
4−N
0
R
N(q−1)−4q
q−1
0
+C2R
−N
0
R
N(q−1)−4q
q(q−1)
0
R
N
q
∅
0
, (5.11)
which implies that for anyx∈BR(x0)
|u(x)|≤CR
−
4
q−1.
Thus (5.9) follows by choosingR=d(x, ∂Ω).A similar argument as above can be
used to show that indeed (5.10) holds. We omit the details. ∅
Acknowledgment
We thank the referee for useful comments.
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[3] I. Bachar, H. Mˆaagli, and L. Mˆaatoug,Positive solutions of nonlinear elliptic
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2
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2002(2002), No.41, pp. 1–24.
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Vol.XLVI(1998), pp. 257–294.
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Springer Verlag, 1983.
[12] H.-Ch. Grunau, G. Sweers,Positivity for equations involving polyharmonic operators
with Dirichlet boundary conditions, Math.Ann.307(1997), 589–626.

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linear partial diﬀerential equations and inequalities, Tr. Math.Inst. Steklova, Ross.
Akad. Nauk, vol. 234, 2001.
[18] M. Nicolescu,Les Fonctions Polyharmoniques, Hermann and Cie Editeurs, Paris
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113(1965), 219–240.
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(1964), 247–302.
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tors and related topics,Math.Z.203(1990), 129–152.
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(1983), 13–18.
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Gabriella Caristi and Enzo Mitidieri
Dipartimento di Matematica e Informatica
Universit`adiTrieste
I-34127 Trieste, Italy
e-mail:[email protected]
e-mail:[email protected]

Operator Theory:
Advances and Applications, Vol. 168, 27–43
c∅2006 Birkh¨auser Verlag Basel/Switzerland
Cl´ement Interpolation and Its Role
in Adaptive Finite Element Error Control
Carsten Carstensen
In honor of the retirement of Philippe Cl´ement.
Abstract.Several approximation operators followed Philippe Cl´ement’s sem-
inal paper in 1975 and are hence known as Cl´ement-type interpolation op-
erators, weak-, or quasi-interpolation operators. Those operators map some
Sobolev spaceV⊂W
k,p
(Ω Vh⊂W
k,p
(Ω)
and generalize nodal interpolation operators wheneverW
k,p
(Ω)ﬃ⊂C
0
( Ω),
i.e., whenp≤n/kfor a bounded Lipschitz domain Ω⊂R
n
. The original
motivation wasH
2
ﬃ⊂C
0
(Ω) for higher dimensionsn≥4 and hence nodal
interpolation is not well deﬁned.
Todays main use of the approximation operators is for a reliability proof
in a posteriori error control. The survey reports on the class of Cl´ement type
interpolation operators, its use in a posteriori ﬁnite element error control and
for coarsening in adaptive mesh design.
1. Introduction: Motivation and applications
The ﬁnite element method (FEM) is the driving force and dominating tool behind
today’s computational sciences and engineering. It is common sense that the FEM
should be implemented in an adaptive mesh-reﬁning version with a posteriori error
control. This paper high-lights one mathematical key tool in the justiﬁcation of
those adaptive ﬁnite element methods (AFEMs) which dates back to Philippe
Cl´ement’s seminal paper in 1975.
The origin of his operators, today known under the description Cl´ement-type
interpolation operators, weak-, or quasi-interpolation operators in the literature,
however, was completely diﬀerent.
The author is supported by the DFG Research CenterMatheon“Mathematics for key technolo-
gies” in Berlin.

28 C. Carstensen
Indeed, many second-order elliptic boundary value problems are recast in a
weak form
a(u, v)=b(v) for allv∈V,
where (V,a) is some Hilbert space with induced norm · aandb∈V
∗
has the
Riesz representationu∈V. Given a subspaceVh, the discrete solutionuh∈Vh
satisﬁes
a(uh,vh)=b(vh) for allvh∈Vh.
The errore:=u−uh∈Vthen satisﬁes thebest-approximation property
e a=min
vh∈Vh
u−vh a.
(The proof is given at the end of Subsection 3.1 below for completeness.)
The classical estimation of the upper bound u−vh areplacesvhby the
nodal interpolation operator applied to the exact solutionu. In the applications,
one typically has
u∈V∩H
s
(Ω;R
m
)
for some Sobolev spaceH
s
(Ω;R
m
) on some domain Ω and some regularity pa-
rameters≥1. The higher regularity withs>1 is subject to regularity theory of
elliptic PDEs andsdepends on the smoothness of data and coeﬃcients as well as
on the boundary conditions and on the geometry (e.g., corners) of Ω. An optimal
regularity for 2D and convex domains orC
2
boundaries iss= 2. For smallers
and/or for higher space dimensions, there holds
u∈V∩H
s
(Ω;R
m
)⊂C( Ω;R
m
).
Thus the nodal interpolation operator, which evaluates the functionuat a ﬁnite
number of points, isnotdeﬁned well.
The Cl´ement operatorsJ(v) cures that diﬃculty in that they replace the
evaluation at discrete points by some initial local best approximation ofvin the
L
2
sense followed by the point evaluation of this local best approximation at the
discrete degrees of freedom.
This small modiﬁcation is essential in order to receive approximation and
stability properties in the sense that
v−J(v)
H
r
(Ω)≤Ch
s−r
v
H
s
(Ω)
with some mesh-size independent constantC>0 and the maximal mesh-size
h>0. The point is that the range for the exponentssandris possible with
r=0,1ands=1,2 whiles= 1 is excluded for the nodal interpolation operator
even for dimensionn≥2.
AtthetimeofCl´ement’s research, this improvement seemed to be regarded
as a marginal technicality in the a priori error control – this possible underestima-
tion is displayed in Ciarlet’s ﬁnite element book [17] where Cl´ement’s research is
summarized as an exercise (3.2.3

Cl´ement Interpolation in AFEM 29
The relevance of Cl´ement’s results was highlighted in the a posteriori er-
ror analysis of the last two decades where the aforementioned ﬁrst-order ap-
proximation and stability property ofJforr≤s= 1 had been exploited.
Amongst the most inﬂuential pioneering publications on a posteriori error con-
trol are [2, 21, 5, 19] followed by many others. The readers may ﬁnd it rewarding
to study the survey articles of [20, 7] and the books of [26, 1, 3, 4] for a ﬁrst insight
and further references.
This paper gives a very general approach to Cl´ement-type interpolation in
Section 2 and comments on its applications in the a posteriori error analysis in
Section 3 where further applications such as the error reduction in adaptive ﬁnite
element schemes and the coarsening are seen as currently open ﬁelds.
2. Cl´ement-type approximation
This section aims at a ﬁrst glance of a class of approximation operatorsJ:
W
1,p
(Ω)→W
1,p
(Ω ement
and called Cl´ement-type interpolation operators or sometimesquasi-interpolation
operators.
The most relevant properties include the localﬁrst order approximation, i.e.,
for allv∈V⊂W
1,p
(Ω Dv∈L
p
(Ω)
n
in Ω⊂R
n
and an
underlying triangulation with local mesh-sizeh∈L
∞
(Ω V h,
there holds
v−J(v)
L
p
(Ω)≤C hDv
L
p
(Ω)and h
−1
(v−J(v))
L
p
(Ω)≤C Dv
L
p
(Ω),
and theW
1,p
(Ω)stability property, i.e., forv∈V,
DJ(v)
L
p
(Ω)≤C Dv
L
p
(Ω).
A discussion on a particular operator with an additionalorthogonality property
ends this section. Although the emphasis in this section is not on the evaluation
of constants and their sharp estimation, there is a listM
1,...,M5of highlighted
relevant parameters.
Deﬁnition 2.1 (abstract assumptions).LetΩbe a bounded Lipschitz domain in
R
n
,let1<p,q<∞with1/p+1/q=1andm, n∈N={1,2,3,...}.Suppose
that(ϕ
z:z∈N)is a Lipschitz continuous partition of unity [associated toV h
below] onΩwith
ω
z:={x∈Ω:ϕ(x) =0}⊆Ω z⊂Ω.
The setsΩ
zare supposed to be open, nonvoid, and connected supersets ofω zof
volume|Ω
z|. For anyz∈NletA z⊆R
m
be nonvoid, convex, and closed and let
Π
z:R
m
→R
m
be the orthogonal projection ontoA zinR
m
(with respect to the
Euclidean metric).Let
V⊆W
1,p
(Ω;R
m
)andV h:={
ϕ
z∈N
azϕz:az∈Az}.

30 C. Carstensen
For anyz∈NdenoteVz:=V|Ωz:={v|Ωz:v∈V}⊆W
1,p
(Ωz;R
m
)and let
Hz>0andez>0. Suppose there exists a map
Jz:Vz→R
m
such that the following two hypothesis(H1)–(H2)hold for allz∈Nand for all
v∈Vzwith¯vz:=|Ωz|
−1
τ
Ωz
v(x)dx∈R
m
:
(H1) v−Πz( vz)
L
p
(Ωz)≤Hz Dv
L
p
(Ωz);
(H2) Jz(v|Ωz)−¯vz
L
p
(Ωz)≤ezHz Dv
L
p
(Ωz).
Typically, (H1) describes some Poincar´e-Friedrichs inequality withHzω
diam(Ωz)whereΩzis some local neighborhood of a node.
The subsequent list of parameters illustrates constants crucial in the analysis
of the approximation and stability estimates.
Deﬁnition 2.2 (Some constants).Under the assumptions of the previous deﬁnition
set(the piecewise constant function)
H(x):=max
z∈N
x∈Ωz
Hzforx∈Ω
and, with1/p+1/q=1, deﬁneM1,...,M5by
M1:= max
z∈N
ϕz
L
∞
(Ω),
M2:= max
x∈Ω
ϕ
z∈N
|ϕz(x)|,
M3:= max
x∈Ω
card{z∈N:x∈Ωz},
M4:= max
z∈N
(1 +ez),
M5:= sup
x∈Ω
α
ϕ
z∈N
H(x)
q
|Dϕz(x)|
q
ρ
1/q
.
Example2.3 (P1FEM in 2D). Given a regular triangulationTof the unit square
Ω=(0,1)
2
into triangles,m=1,n=2,letV:=H
1
0(Ωp=q=2,andlet
(ϕz:z∈N) denote the nodal basis functions of all the nodes with respect to the
ﬁrst-order ﬁnite element space. The boundary conditions are described by the sets
(Az:z∈N). For each nodezin the interior, writtenz∈N∩Ω,Az=R, while
Az:={0}forzon the boundary, writtenz∈N∩∂Ω, ΓD=∂Ω. LetJzdenote
the integral mean operator
Jz(v):=|ωz|
−1
∅
ωz
v(x)dxforv∈Vz⊆L
p
(Ωz)
on the patchωz=Ωz={x∈Ω:ϕ(x)>0}withez= 0 in (H2). The Dirichlet
boundary conditions are then prescribed by (v∈R
m
)
Πz(v):=
π
vifz∈N∩Ω,
0ifz∈N∩∂Ω.

Cl´ement Interpolation in AFEM 31
There holds (H1) with Poincar´e and Friedrich’s inequalities and
Hz:=
π
cP(ωz)if z∈N∩Ω,
cF(ωz,(∂Ω)∩(∂ωz)) ifz∈N∩∂Ω;
Hzis of the form global constant times diam(ωz). It is always true that
M1=M2=1=M4andM3=3.
Since Dϕz
L
∞
(Ω)≈H
−1
zthere holds
M5ω1.
Notice thatM5may be very large for small angles in the triangulation while the
constantsM1,...,M4are robust with respect to small aspect ratios.
The aforementioned example is not exactly the choice of [18] but certainly
amongst the natural choices [14, 25].
The announced approximation and stability properties are provided by the
following main theorem.
Theorem 2.4.The mapJ:V→Vh;vλ→
η
z∈N
Πz(Jz(v|Ωz))ϕzsatisﬁes
(a)∃c1>0∀v∈V, v−J(v)
L
p
(Ω)≤c1 HDv
L
p
(Ω);
(b∃c2>0∀v∈V, H
−1
(v−J(v))
L
p
(Ω)≤c2 Dv
L
p
(Ω),
(c∃c3>0∀v∈V, DJ(v)
L
p
(Ω)≤c3 Dv
L
p
(Ω).
The constantsc1andc2depend onM1,...,M4and onp, q, m, nwhilec3depends
also onM5.
Proof.Givenv∈V,letvz:=Jz(v|Ωz),z∈N, be the coeﬃcient in
J(v)=
ϕ
z∈N
Πz(vz)ϕz
and let vz:=|Ωz|
−1
τ
Ωz
v(x)dxbe the local integral mean.
The ﬁrst step of this proof establishes the estimate
v−Πz(vz)
L
p
(Ωz)≤M4Hz Dv
L
p
(Ωz). (2.1)
The assumptions (H1) and (H2) lead to
v|Ωz−Πz(vz)
L
p
(Ωz)≤Hz Dv
L
p
(Ωz),
vz−vz
L
p
(Ωz)≤ezHz Dv
L
p
(Ωz).
Since Πzis non-expansive (Lipschitz with Lip(Πz)≤1;R
m
is endowed with the
Euclidean norm used inL
p
(Ω;R
m
)) there holds
Πz(vz)−Πz(vz)
L
p
(Ωz)≤ vz−vz
L
p
(Ω).
The combination of the aforementioned estimates yields an upper bound of the
right-hand side in
v−Πz(vz)
L
p
(Ωz)≤ v−Πz(vz)
L
p
(Ωz)+ Πz(vz)−Π(vz)
L
p
(Ωz)
and, in this way, establishes (2.1) and concludes the ﬁrst step.

32 C. Carstensen
Step two establishes assertion (a) of the theorem. Since (ϕ
z)z∈Nis a partition
of unity,
η
z∈N
ϕz=1inΩ,thereholds
v−J(v)
p
L
p
(Ω)
=
ϕ
z∈N
(v−Π z(vz))ϕz
p
L
p
(Ω)
=
∈
Ω
χ
χ
χ
χ
χ
ϕ
z∈N
ϕ
1/q
z
ϕ
1/p
z
(v−Π z(vz))
χ
χ
χ
χ
χ
p
dx.
H¨olders’s inequality inη
1
and
η
z∈N
|ϕz|≤M 2lead to
v−J(v)
p
L
p
(Ω)
≤
∈
Ω
α
ϕ
z∈N
|ϕz||v−Π z(vz)|
p
ρα
ϕ
z∈N
|ϕz|
ρ
p/q
dx
≤M
p/q
2
ϕ
z∈N
∈
Ω
|ϕz||v−Π z(vz)|
p
dx
≤M
1M
p/q
2
ϕ
z∈N
v−Π z(vz)
p
L
p
(Ωz)
.
This and estimate (2.1) from the ﬁrst step show assertion (a) with
c
1:=M
1/p
1
M
1/q
2
M
1/p
3
M4.
In fact, the last step involves the overlaps by means ofM
3and the deﬁnition of
H(x):=max{H
z:∃z∈N,x∈Ω z}:
ϕ
z∈N
H
p
z
Dv
p
L
p
(Ωz)
≤
∈
Ω
ϕ
z∈N
x∈Ω
z
H
p
z
|Dv(x)|
p
dx
≤
∈
Ω
H(x)
p
(
ϕ
z∈N
x∈Ω
z
1)|Dv(x)|
p
dx
≤M
3 HDv
p
L
p
(Ω)
.
This concludes step two and the proof of assertion (a).
Step three establishes assumption (b
ments as in step two shows
H
−1
(v−J(v))
p
L
p
(Ω)
≤M 1M
p/q
2
ϕ
z∈N
H
−1
(v−Π z(vz))
p
L
p
(Ωz)
.
SinceH
z≤H(x)forx∈Ω z,z∈N, there holds
H
−1
(v−J(v))
p
L
p
(Ω)
≤M 1M
p/q
2
ϕ
z∈N
H
−p
z
v−Π z(vz)
p
L
p
(Ωz)
.
Based on (2.1), the proof of assertion (b
two. This yields (b) withc
2:=M
1/p
M
1/q
2
M
1/p
3
M4.

Cl´ement Interpolation in AFEM 33
Step four establishes assumption (c
ϕ
z∈N
ϕz=1thereholds
ϕ
z∈N
Dϕz=0
almost everywhere in Ω. With the characteristic functionχ
z∈L
∞
(Ω z,
deﬁned byχ
z|Ωz=1andχ z|
Ω\Ω z
= 0 it follows that
DJ(v)
p
L
p
(Ω)
=
ﬀ
Ω
χ
χ
χ
χ
χ
ϕ
z∈N
χz(x)H
−1
(x)(Πz(vz)−v(x))Dϕ z(x)H(x)
χ
χ
χ
χ
χ
p
dx.
H¨older’s inequality inη
1
shows, for almost everyx∈Ω,
χ
χ
χ
χ
χ
ϕ
z∈N
χz(x)H
−1
(x)(Πz(vz)−v(x))Dϕ z(x)H(x)
χ
χ
χ
χ
χ
≤
α
ϕ
z∈N
χz(x)H
−p
(x)|Π z(vz)−v(x)|
p
ρ
1/pα
ϕ
z∈N
H(x)
q
|Dϕz(x)|
q
ρ
1/q
≤M 5
α
ϕ
z∈N
χz(x)H
−p
z
|Πz(vz)−v(x)|
p
ρ
1/p
.
The combination of the two preceding estimates in this step four is followed by
(2.1) and then results in
DJ(v)
p
L
p
(Ω)
≤M
p
4
M
p
5
ϕ
z∈N
Dv
p
L
p
(Ωz)
.
The proof concludes as in step two and establishes assertion (cc
3:=
M
1/p
3
M4M5. ﬀ
In order to discuss a particular operator designed to introduce an extra or-
thogonality property, additional assumptions are necessary.
Deﬁnition 2.5 (Free nodes for Dirichlet boundary conditions).Adopt the notation
of Deﬁnition2.1and suppose, in addition, that
V
h⊂V:={v∈W
1,p
(Ω;R
m
):v=0onΓ D},
whereΓ
Dis some(possibly empty)closed part of the boundary that includes a
complete set of edges in the sense that, for eachE∈E,eitherE⊂Γ
DorE∩Γ D⊂
N.LetK:=N\Γ
Dand, for allz∈N,
A
z=
ν
R
m
ifz∈K;
{0}otherwise.
Given anyz∈KletJ(z)⊂Nsuch thatz∈J(z)and
ψ
z:=
ϕ
x∈J(z)
ϕx

34 C. Carstensen
deﬁnes a Lipschitz partition(ψz:z∈K)of unity withΩz⊇{x∈Ω:ψz(x)=0}.
Then
Jz(v):=
τ
Ωz
vψzdx
τ
Ωz
ϕzdx
deﬁnes an approximation operatorJ:V→Vh.Set
M6:= max
z∈K
ϕz
−1
L
1
(Ωz)
ψz
L
q
(Ωz) w
L
p
(Ωz)|Ωz|
1/p
.
Theorem 2.6.The operatorJ:V→Vhfrom Deﬁnition2.5satisﬁes(a)–(cfrom
Theorem2.4plus theorthogonality property
χ
χ
χ
χ
∅
Ω
f·(v−Jv)dx
χ
χ
χ
χ
≤c4 Dv
L
p
(Ω)
α
ϕ
z∈K
H
q
zmin
fz∈R
m
f−fz
q
L
q
(Ωz)
)
ρ
1/q
.
Remark2.7.A Poincar´e inequality in casef∈W
1,q
(Ω
f−fz
L
q
(Ωz)≤Cp(Ωz)Hz Df
L
q
(Ωz),
illustrates that the right-hand side in Theorem 2.6 is of the form
O( H
2
L
∞
(Ω)
) Dv
L
p
(Ω)
and then of higher order (compared with the error of ﬁrst-order FEM).
Proof of Theorem2.6.In order to employ Theorem 2.4, it remains to check the
conditions (H1)–(H2). For anyv∈Vz=V|Ωz, (H1) is either a Poincar´eor
Friedrichs inequality forz∈Korz∈K∩ΓDas in Example 2.3. Moreover,
given anyv∈Vzand
Jz(v):=
∅
Ω
vψzdx/
∅
Ω
ϕzdx∈R
m
,
there holdsJz(vz)=vzifψz≡ϕz(i.e.,J(z)={z})forvz:=|Ωz|
−1
τ
Ωz
v(x)dx.
Then, forw=v|Ωz−vz, there holds
Jz(v|Ωz)−vz
L
p
(Ωz)= Jz(w)
L
p
(Ωz)
≤ ϕz
−1
L
1
(Ωz)
ψz
L
q
(Ωz) w
L
p
(Ωz)|Ωz|
1/p
≤M6 w
L
p
(Ωz)≤M6Hz Dv
L
p
(Ωz).
Forψz≡ϕz(i.e.,J(z)={z}),Ωzincludes nodes at the boundary and∂Ωz∩ΓD
has positive surface measure. Therefore, Friedrichs inequality guarantees
vz
L
p
(Ωz)≤c5Hz Dvz
L
p
(Ωz)for allvz∈Vz.
Then (H2) follows immediately from this and
Jz(v)
L
p
(Ωz)+ vz
L
p
(Ωz)≤c6 vz
L
p
(Ωz).
The orthogonality property is based on the design ofJz(v|Ωz) and the partition of
unity (ψz:z∈K). In fact,
∅
Ωz
(v(x)ψz(x)−Jz(v|Ωz)ϕz(x))dx=0

Cl´ement Interpolation in AFEM 35
for anyz∈Kand so, for anyf
z∈R
m
, there follows
∅
Ω
f(v−Jv)dx
=
∅
Ω
f(x)
α
ϕ
z∈K
v(x)ψ z(x)−
ϕ
z∈K
Jz(v|Ωz)ϕz(x)
ρ
dx
=
ϕ
z∈K
∅
Ωz
f(x)·(v(x)ψ z(x)−J z(v|Ωz)ϕz(x))dx
=
ϕ
z∈K
∅
Ωz
(f(x)−f z)·(v(x)ψ z(x)−J z(v|Ωz)ϕz(x))dx
≤
ϕ
z∈K
∂
H
z f−f z
L
q
(Ωz)
δ∂
H
−1
z
vψz−Jz(v|Ωz)ϕz
L
p
(Ωz)
δ
≤
α
ϕ
z∈K
H
q
z
f−f z
q
L
q
(Ωz)
ρ
1/qα
ϕ
z∈K
H
−p
z
vψz−Jz(v|Ωz)ϕz
p
L
p
(Ωz)
ρ
1/p
.
Forψ
z≡ϕz(i.e.,J(z)={z}) it follows
v(ψ
z−ϕz)
L
p
(Ωz)+ (v−J z(v|Ωz))ϕz
L
p
(Ωz)≤c7Hz Dv
L
p
(Ωz)
as above. Forψ z ≡ϕz, a Friedrichs inequality shows
v
L
p
(Ω)≤c8Hz Dv
L
p
(Ωz)
and completes the proof. The remaining details are similar to the arguments of
Theorem 2.4 and hence omitted. ∅
3. A posteriori residual control
This section reviews explicit residual-based error estimators in a very abstract
form. The subsequent benchmark example motivates this analysis of the dual norm
of a linear functional.
3.1. Model example
For the benchmark example suppose that an underlying PDE yields to some bi-
linear formaon some Sobolev spaceVsuch that (V,a)isHilbertspaceandthe
right-hand sidebis a bounded linear functional, writtenb∈V
∗
. The Riesz repre-
sentationuofbis called weak solution and is characterized byu∈Vwith
a(u, v)=b(v) for allv∈V.
The ﬁnite element solutionu
hbelongs to some ﬁnite-dimensional subspaceV hof
V, the ﬁnite element space, and is characterized byu
h∈Vhwith
a(u
h,vh)=b(v h) for allv h∈Vh.
In other words,u
his the Riesz representation of the restricted right-hand sideb| Vh
in the discrete Hilbert space (V h,a|Vh×Vh
).

36 C. Carstensen
The goal of a posteriori error control is to represent the error
e:=u−uh∈V
in the energy norm · a=
κ
a(·,·). One particular property is the Galerkin
orthogonality
a(e, vh) = 0 for allvh∈Vh.
This followed by a Cauchy inequality yields, for allvh∈Vh,that
e
2
a
=a(e, u−vh)≤ e a u−vh a.
This implies the best-approximation property
e a=min
vh∈Vh
u−vh a
claimed in the introduction.
3.2. Error and residual
Givena,b,anduh, the residual
R:=b−a(uh,·)∈V
∗
is a bounded linear functional inV, writtenR∈V
∗
. Its dual norm is
R V
∗:= sup
v∈V\{0}
R(v)/ v a=sup
v∈V\{0}
a(e, v)/ v a= e a<∞. (3.1)
The ﬁrst equality in (3.1) is a deﬁnition, the second equality immediately follows
from
R=a(e,·).
Notice that the Galerkin orthogonality immediately translates into
Vh⊂kernR.
A Cauchy inequality with respect to the scalar productaresults in
R V
∗≤ e a
and hence in one of two inequalities of the last identity R V
∗= e ain (3.1).
The remaining converse inequalities follows in fact withv=eyields ﬁnally the
equality; the proof of (3.1) is ﬁnished.
The identity (3.1) means that the error (estimation) in the energy normis
equivalentto the (computation of the) dual norm of the given residual. Further-
more, it is even ofcomparable computational eﬀortto compute an optimalv=e
in (3.1) or to computee. The proof of (3.1) yields even a stability estimate: Given
anyv∈Vwith v a= 1, the relative error ofR(v) as an approximation to e a
equals
e a−R(v)
e a
=1/2 v−e/ e a
2
a. (3.2)

Cl´ement Interpolation in AFEM 37
In fact, given anyv∈Vwith v a= 1, the identity (3.2) follows from
1−a(e/ e a,v)=
1
2
a(e/ e a,e/ e a)−a(e/ e a,v)+
1
2
a(v, v)
=
1
2
v−e/ e a
2
a
. ∅
The interpretation of the error estimate (3.2) is as follows. The maximizingv
in (3.1) (i.e.,v∈Vwith maximalR(v) subject to v a≤1) is unique and
equalse/ e a. As a consequence, the computation of the maximizingvin (3.1)
is equivalent to and indeed equally expensive as the computation of the unknown
e/ e aand so essentially of the exact solutionu.
Therefore, a posteriori error analysis aims to compute lower and upper bounds
of R V
∗instead of its exact value. However, the use of any good guess of the
exact solution from some postprocessing with extrapolation or superconvergence
phenomenon is welcome (and, e.g., may enterv).
3.3. Residual representation formula
Without stating examples in explicit form, it is stressed that linear and nonlinear
boundary value problems of second order elliptic PDEs typically lead (after some
partial integration) to some explicit representation of the residualR:=b−a(uh,·).
LetTdenote the underlying regular triangulation of the domain and letEdenote
a set of edges forn= 2. Then, the data and the discrete solutionuhlead to given
quantitiesrT∈R
m
, the explicit volume residual, andrE∈R
m
, the jump residual,
for any element domainTand any edgeEsuch that theresidual representation
formula
R(v)=
ϕ
T∈T
∅
T
rT·vdx−
ϕ
E∈E
∅
E
rE·vds (3.3)
holds for allv∈V.
The goal of a posteriori error control is to establish guaranteed lower and
upper bounds of the (energy norm of the) error and hence of the (dual norm of
the) residual.
Any choice ofv∈V\{0}immediately leads to a lower error bound via
η:=|R(v)/ v a|≤ e a.
Indeed, edge and element-oriented bubble functions are studied forvin [26] to
prove eﬃciency.
More challenging appears reliability, i.e., the design of upper error bounds
which require extra conditions. In fact, the Galerkin orthogonalityVh⊂kernRis
supposed to hold for the data in (3.3).
3.4. Explicit residual-based error estimators
Given the explicit volume and jump residualsrTandrEin (3.3), one deﬁnes the
explicit residual-based estimator
η
2
R
:=
ϕ
T∈T
h
2
T
rT
2
L
2
(T)
+
ϕ
E∈EΩ
hE rE
2
L
2
(E)
, (3.4)

38 C. Carstensen
which is reliable in the sense that
e
a≤Cη R. (3.5)
The proof of (3.5) follows from the Cl´ement-type interpolation operators and their
properties. In fact, given anyv∈Vand anyv
h∈Vhsetw:=v−v h. Then the
residual representation formula withV
h⊂kernRplus Cauchy inequalities lead to
R(v)=R(w)=
ϕ
T∈T
∈
T
rT·wdx−
ϕ
E∈EΩ
∈
E
rE·wds
≤
ϕ
T∈T
∂
h
T rT
L
2
(T)
δ∂
h
−1
T
w
L
2
(T)
δ
+
ϕ
E∈EΩ
θ
h
1/2
E
rE
L
2
(E)
βθ
h
−1/2
E
w
L
2
(E)

≤
α
ϕ
T∈T
h
2
T
rT
2
L
2
(T)
ρ
1/2α
ϕ
T∈T
h
−2
T
w
2
L
2
(T)
ρ
1/2
+
α
ϕ
E∈EΩ
hE rE
2
L
2
(E)
ρ
1/2α
ϕ
E∈EΩ
h
−1
E
w
2
L
2
(E)
ρ
1/2
.
The well-established trace inequality for each elementTof diameterh
Tand its
boundary∂Tshows
h
−1
T
w
2
L
2
(∂T)
≤CT
θ
h
−2
T
w
2
L
2
(T)
+ Dw
2
L
2
(T)

with some size-independent constantC
T(which solely depends on the shape of
the element domain). This allows an estimate of
α
ϕ
E∈EΩ
h
−1
E
w
2
L
2
(E)
ρ
1/2
≤C
ϕ
T∈T
θ
h
−2
T
w
2
L
2
(T)
+ Dw
2
L
2
(T)

.
The localized ﬁrst-order approximation and stability property of the Cl´ement-
interpolation shows
ϕ
T∈T
θ
h
−2
T
w
2
L
2
(T)
+ Dw
2
L
2
(T)

≤C
ϕ
T∈T
Dv
2
L
2
(T)
.
Altogether, it follows that
R(v)≤Cη
R|v|
H
1
(Ω).
This proves reliability in the sense of (3.5) ∈
3.5. Error reduction and discrete residual control
Diﬀerent norms of the residual play an important role in the design of convergent
adaptive meshes. SupposeV
h≡Vis the ﬁnite element space with a discrete
solutionu
h≡uand its residualR≡R :=b−a(u ,·). The design task in
adaptive ﬁnite element methods is to ﬁnd a superspaceV
+1⊃Vassociated with

Cl´ement Interpolation in AFEM 39
some ﬁner mesh such that the discrete solutionu
+1inV+1⊂Vis strictly more
accurate. To make this precise suppose that 0<ζ<1and0≤δsatisfyerror
reductionin the sense that
u−u
+1
2
a
≤ζ u−u
2
a
+δ. (3.6)
The interpretation is that the error is reduced by a factor at mostζ
1/2
<1upto
the termsδwhich describe small eﬀects. The error reduction leads to discuss the
equivalentdiscrete residual control
(1−ζ) R
2
V
∗≤ R
2
V
∗
θ+1+δ. (3.7)
Therein, R
V
∗
θ+1denotes the dual norm of the restricted functionalR| Vθ+1
,namely
R
V
∗
θ+1:= sup
v∈V θ+1\{0}
R(v)/ v a.
The point is that (3.6)⇔(3.7). The proof follows from (3.1) which leads to
R
V
∗= u−u aand R V
∗
θ+1= u +1−u a.
This plus the Galerkin orthogonality and the Pythagoras theorem
u−u

2
a
= u−u +1
2
a
+ u +1−u
2
a
show that (3.7) reads
(1−ζ) u−u

2
a
≤ u−u
2
a
− u−u +1
2
a
+δ
and this is (3.6). Altogether, (3.6)⇔(3.7) and one needs to study thediscrete
residual control.
One standard technique is the bulk criterion for the explicit error estimator
η
Rfollowed by local discrete eﬃciency. The latter involves proper mesh-reﬁnement
rules and hence this is not included in this paper.
3.6. Remarks
This subsection lists a few short comments on various tasks in the a posteriori
error analysis.
3.6.1. Dominating edge contributions.For ﬁrst-order ﬁnite element methods in
simple situations of the Laplace, Lam´e problem, or the primal formulation of
elastoplasticity, the volume termr
T=fcan be substituted by the higher-order
term of oscillations, i.e.,
e
2
a
≤C
θ
osc(f)
2
+
ϕ
E∈EΩ
hE rE
2
L
2
(E)

. (3.8)
Therein, for each nodez∈Nwith nodal basis functionϕ
zand patchω z:={x∈
Ω:ϕ
z(x) =0}of diameterh zand the source termf∈L
2
(Ω)
m
with integral mean
f
z:=|ω z|
−1
τ
ωz
f(x)dx∈R
m
the oscillations offare deﬁned by
osc(f):=
θϕ
z∈N
h
2
z
f−f z
2
L
2
(ωz)

1/2
.

40 C. Carstensen
Notice forf∈H
1
(Ω)
m
and the mesh-sizeh
T
∈P0(T) there holds
osc(f)≤C h
2
T
Df
L
2
(Ω)
and so osc(f) is of quadratic and hence of higher order. We refer to [10, 9, 16], [22],
[6], and [24] for further details on and proofs of (3.8). The proof in [16] is based on
Theorem 2.6. In fact, the arguments of Subsection 3.4 are combined withr
T=f
for the lowest-order ﬁnite element method at hand. Then, Theorem 2.6 shows
∅
Ω
fwdx≤c 4 Dv
L
2
(Ω)osc(f)
and leads to the announced higher-order term.
Dominating edge contributions lead to eﬃciency of averaging schemes [10, 11].
3.6.2. Reliability constants.The global constants in the approximation and sta-
bility properties of Cl´ement-type operators enter the reliability constants. The
estimates of [14] illustrate some numbers as local eigenvalue problems. Based on
numerical calculations, there is proof that, for all meshes with right-isosceles tri-
angles and the Laplace equation with right-hand sidef∈L
2
(Ω
reliability in the sense of
D(u−u
h)
L
2
(Ω)≤
α
ϕ
T∈T
h
2
T
f
2
L
2
(T)
ρ
1/2
+
α
ϕ
E∈E
hE
∅
E
[∂uh/∂νE]
2
ds
ρ
1/2
with an undisplayed constant Crel<1 in the upper bound [15].
The aspect of upper bounds with constants and corresponding overestimation
is empirically discussed in [13].
3.6.3. Coarsening.One actual challenging detail is coarsening. Reason number
one is that coarsening cannot be avoided for time evolving problems to capture
moving singularities. Reason number two is the design of algorithms which are
guaranteed to convergence in optimal complexity [8].
Any coarsening excludes the Pythagoras theorem (which assumesV
⊂V+1)
and so perturbations need to be controlled which leads to the estimation of
min
vθ+1∈Vθ+1
v+1−u a.
Ongoing research investigates the eﬀective estimation of this term.
3.6.4. Other norms and goal functionals.The previous sections concern estima-
tions of the error in the energy norm. Other norms are certainly of some interest
as well as the error with respect to a certain goal functional. The later is some
given bounded and linear functionalη:V→Rwith respect to which one aims to
monitor the error. That is, one wants to ﬁnd computable lower and upper bounds
for the (unknown) quantity
|η(u)−η(u
h)|=|η(e)|.

Cl´ement Interpolation in AFEM 41
Typical examples of goal functionals are described byL
2
functions, e.g.,
η(v)=
ﬀ
Ω
ζvdx∀v∈V
for a givenζ∈L
2
(Ω) or as contour integrals. To bound or approximateJ(e)one
considers thedual problem
a(v, z)=η(v)∀v∈V (3.9)
with exact solutionz∈V(guaranteed by the Lax-Milgram lemma) and the dis-
crete solutionz
h∈Vhof
a(v
h,zh)=η(v h)∀v h∈Vh.
Setf:=z−z
h. Based on the Galerkin orthogonalitya(e, z h)=0oneinfers
η(e)=a(e, z)=a(e, z−z
h)=a(e, f). (3.10)
Cauchy inequalities lead to the a posteriori estimate
|η(e)|≤ e
a f a≤ηuηz.
Indeed, utilizing the primal and dual residualR
uandR zinV
∗
, deﬁned by
R
u:=b−a(u h,·)andR z:=η−a(·,z h),
computable upper error bounds for e
V≤ηuand f V≤ηzcan be found by
the arguments of the energy error estimators [1, 3]. Indeed, the parallelogram rule
shows
2η(e)=2a(e, f)= e+f
2
a
− e
2
a
− f
2
a
.
This right-hand side can be written in terms of residuals, in the spirit of (3.1),
namely e
a= R u V
∗, f a= R z V
∗,and
e+f
a= R u+z V
∗
for
R
u+z:=b+η−a(u h+zh,·)=R u+Rz∈V
∗
.
Therefore, the estimation ofη(e) is reduced to the computation of lower and upper
error bounds for the three residualsR
u,Rz,andR u+zwith respect to the energy
norm. This illustrates that the energy error estimation techniques of the previous
sections may be employed for goal-oriented error control [1, 3].
Alternatively, Rannacher et al. investigated the weighted residual technique
where the Cl´ement interpolation operator is employed as well. The reader is re-
ferred to [4].

42 C. Carstensen
References
[1] Ainsworth, M. and Oden, J.T. (2000).A posteriori error estimation in ﬁnite element
analysis.Wiley-Interscience [John Wiley & Sons], New York. xx+240.
[2] Babuˇska I. and Miller A. (1987)A feedback ﬁnite element method with a posteriori
error estimation. I. The ﬁnite element method and some properties of the a posteriori
estimator.Comp. Methods Appl. Mech. Engrg.,61, 1, 1–40.
[3] Babuˇska, I. and Strouboulis, T. (2001).The ﬁnite element method and its reliability.
The Clarendon Press Oxford University Press, New York, xii+802.
[4] Bangerth, W. and Rannacher, R. (2003).Adaptive ﬁnite element methods for diﬀer-
ential equations.(Lectures in Mathematics ETH Z¨urich) Birkh¨auser Verlag, Basel,
viii+207.
[5] Bank, R.E. and Weiser, A. (1985).Some a posteriori error estimators for elliptic
partial diﬀerential equations.Math. Comp.,44, 170, 283–301.
[6] Becker, R. and Rannacher, R. (1996).A feed-back approach to error control in ﬁnite
element methods: basic analysis and examples.East-West J. Numer. Math,4,4,
237–264.
[7] Becker, R. and Rannacher, R. (2001).An optimal control approach to a posteriori
error estimation in ﬁnite element methods.Acta Numerica, Cambridge University
Press, 2001, 1–102.
[8] Binev, P., Dahmen, W. and DeVore, R. (2004).Adaptive ﬁnite element methods with
convergence rates.Numer. Math.,97, 2, 219–268.
[9] Carstensen, C. (1999).Quasi-interpolation and a posteriori error analysis in ﬁnite
element method.M2AN Math. Model. Numer. Anal.,33, 6, 1187–1202.
[10] Carstensen, C. (2004).Some remarks on the history and future of averaging tech-
niques in a posteriori ﬁnite element error analysis.ZAMM Z. Angew. Math. Mech.,
84, 1, 3–21.
[11] Carstensen C.All ﬁrst-order averaging techniques for a posteriori ﬁnite element
error control on unstructured grids are eﬃcient and reliable. Math. Comp.73(2004)
1153-1165.
[12] Carstensen, C. (2006). On the Convergence of Adaptive FEM for Convex Minimiza-
tion Problems (in preparation).
[13] Carstensen, C., Bartels, S. and Klose, R. (2001).An experimental survey of a pos-
teriori Courant ﬁnite element error control for the Poisson equation.Adv. Comput.
Math.,15, 1-4, 79–106.
[14] Carstensen, C. and Funken, S.A.Constants in Cl´ement-interpolation error and resid-
ual based a posteriori estimates in ﬁnite element methods.East-West J. Numer.
Math.8(3) (2000) 153–175.
[15] Carstensen, C. and Funken, S.A.Fully reliable localised error control in the FEM.
SIAM J. Sci. Comp.21(4) (2000) 1465–1484.
[16] Carstensen, C. and Verf¨urth, R. (1999).Edge residuals dominate a posteriori error
estimates for low order ﬁnite element methods.SIAM J. Numer. Anal.,36, 5, 1571–
1587.
[17] Ciarlet, P.G.The Finite Element Method for Elliptic Problems.North-Holland, 1978.
(reprinted in the Classics in Applied Mathematics Series, SIAM, 2002)

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and he indicated Bernard who seemed to me to have a greater
confusion than the discovery gave a cause for.
“Bernard has been good enough,” said I. “You discover two Scots,
Father Hamilton, in a somewhat sentimental situation. The lady did
me the honour to be interested in my little travels, and I did my best
to keep her informed.”
He turned away as he had been shot, hiding his face, but I saw
from his neck that he had grown as white as parchment.
“What in the world have I done?” thinks I, and concluded that he
was angry for my taking the liberty to use the dismissed servant as a
go-between. In a moment or two he turned about again, eying me
closely, and at last he put his hand upon my shoulder as a
schoolmaster might do upon a boy's.
“My good Paul,” said he, “how old are you?”
“Twenty-one come Martinmas,” I said.
“Expiscate! elucidate! 'Come Martinmas,'” says he, “and what does
that mean? But no matter—twenty-one says my barbarian; sure 'tis
a right young age, a very baby of an age, an age in frocks if one
that has it has lived the best of his life with sheep and bullocks.”
“Sir,” I said, indignant, “I was in very honest company among the
same sheep and bullocks.”
“Hush!” said he, and put up his hand, eying me with compassion
and kindness. “If thou only knew it, lad, thou art due me a civil
attention at the very least. Sure there is no harm in my mentioning
that thou art mighty ingenuous for thy years. 'Tis the quality I would
be the last to find fault with, but sometimes it has its
inconveniences. And Bernard”—he turned to the Swiss who was still
greatly disturbed—“Bernard is a somewhat older gentleman. Perhaps
he will say—our good Bernard—if he was the person I have to thank
for taking the sting out of the wasp, for extracting the bullet from
my pistol? Ah! I see he is the veritable person. Adorable Bernard, let
that stand to his credit!”
Then Bernard fell trembling like a saugh tree, and protested he did
but what he was told.

“And a good thing, too,” said the priest, still very pale but with no
displeasure. “And a good thing too, else poor Buhot, that I have
seen an infinity of headachy dawns with, had been beyond any
interest in cards or prisoners. For that I shall forgive you the rest
that I can guess at. Take Monsieur Grog's letter where you have
taken the rest, and be gone.”
The Swiss went out much crestfallen from an interview that was
beyond my comprehension.
When he was gone Father Hamilton fell into a profound
meditation, walking up and down his room muttering to himself.
“Faith, I never had such a problem presented to me before,” said
he, stopping his walk; “I know not whether to laugh or swear. I feel
that I have been made a fool of, and yet nothing better could have
happened. And so my Croque-mort, my good Monsieur Propriety,
has been writing the lady? I should not wonder if he thought she
loved him.”
“Nothing so bold,” I cried. “You might without impropriety have
seen every one of my letters, and seen in them no more than a
seaman's log.”
“A seaman's log!” said he, smiling faintly and rubbing his massive
chin; “nothing would give the lady more delight, I am sure. A
seaman's log! And I might have seen them without impropriety,
might I? That I'll swear was what her ladyship took very good care
to obviate. Come now, did she not caution thee against telling me of
this correspondence?”
I confessed it was so; that the lady naturally feared she might be
made the subject of light talk, and I had promised that in that
respect she should suffer nothing for her kindly interest in a
countryman.
The priest laughed consumedly at this.
“Interest in her countryman!” said he. “Oh, lad, wilt be the death
of me for thy unexpected spots of innocence.”
“And as to that,” I said, “you must have had a sort of
correspondence with her yourself.”

“I!” said he. “Comment!”
“To be quite frank with you,” said I, “it has been the cause of
some vexatious thoughts to me that the letter I carried to the Prince
was directed in Miss Walkinshaw's hand of write, and as Buhot
informed me, it was the same letter that was to wile his Royal
Highness to his fate in the Rue des Reservoirs.” Father Hamilton
groaned, as he did at any time the terrible affair was mentioned.
“It is true, Paul, quite true,” said he, “but the letter was a forgery.
I'll give the lady the credit to say she never had a hand in it.”
“I am glad to hear that, for it removes some perplexities that have
troubled me for a while back.”
“Ah,” said he, “and your perplexities and mine are not over even
now, poor Paul. This Bernard is like to be the ruin of me yet. For
you, however, I have no fear, but it is another matter with the poor
old fool from Dixmunde.”
His voice broke, he displayed thus and otherwise so troubled a
mind and so great a reluctance to let me know the cause of it that I
thought it well to leave him for a while and let him recover his old
manner.
To that end I put on my coat and hat and went out rather earlier
than usual for my evening walk.

I
CHAPTER XXVIII
THE MAN WITH THE TARTAN WAISTCOAT
t was the first of May. But for Father Hamilton's birds, and some
scanty signs of it in the small garden, the lengthened day and
the kindlier air of the evenings, I might never have known what
season it was out of the almanac, for all seasons were much the
same, no doubt, in the Isle of the City where the priest and I
sequestered. 'Twas ever the shade of the tenements there; the
towers of the churches never greened nor budded; I would have
waited long, in truth, for the scent of the lilac and the chatter of the
rook among these melancholy temples.
Till that night I had never ventured farther from the gloomy
vicinity of the hospital than I thought I could safely retrace without
the necessity of asking any one the way; but this night, more
courageous, or perhaps more careless than usual, I crossed the
bridge of Notre Dame and found myself in something like the Paris
of the priest's rhapsodies and the same all thrilling with the passion
of the summer. It was not flower nor tree, though these were not
wanting, but the spirit in the air—young girls laughing in the by-
going with merriest eyes, windows wide open letting out the sounds
of songs, the pavements like a river with zesty life of Highland hills
when the frosts above are broken and the overhanging boughs have
been flattering it all the way in the valleys.
I was fair infected. My step, that had been unco' dull and heavy, I
fear, and going to the time of dirges on the Isle, went to a different
tune; my being rhymed and sang. I had got the length of the Rue de
Richelieu and humming to myself in the friendliest key, with the
good-natured people pressing about me, when of a sudden it began
to rain. There was no close in the neighbourhood where I could

shelter from the elements, but in front of me was the door of a
tavern called the Tête du Duc de Burgoyne shining with invitation,
and in I went.
A fat wife sat at a counter; a pot-boy, with a cry of “V'ià!” that was
like a sheep's complaining, served two ancient citizens in skull-caps
that played the game of dominoes, and he came to me with my
humble order of a litre of ordinary and a piece of bread for the good
of the house.
Outside the rain pelted, and the folks upon the pavement ran, and
by-and-by the tavern-room filled up with shelterers like myself and
kept the pot-boy busy. Among the last to enter was a group of five
that took a seat at another corner of the room than that where I sat
my lone at a little table. At first I scarcely noticed them until I heard
a word of Scots. I think the man that used it spoke of “gully-knives,”
but at least the phrase was the broadest lallands, and went about
my heart.
I put down my piece of bread and looked across the room in
wonder to see that three of the men were gazing intently at myself.
The fourth was hid by those in front of him; the fifth that had
spoken had a tartan waistcoat and eyes that were like a gled's,
though they were not on me. In spite of that, 'twas plain that of me
he spoke, and that I was the object of some speculation among
them.
No one that has not been lonely in a foreign town, and hungered
for communion with those that know his native tongue, can guess
how much I longed for speech with this compatriot that in dress and
eye and accent brought back the place of my nativity in one wild
surge of memory. Every bawbee in my pocket would not have been
too much to pay for such a privilege, but it might not be unless the
overtures came from the persons in the corner.
Very deliberately, though all in a commotion within, I ate my piece
and drank my wine before the stare of the three men, and at last, on
the whisper of one of them, another produced a box of dice.

“No, no!” said the man with the tartan waistcoat hurriedly, with a
glance from the tail of his eye at me, but they persisted in their
purpose and began to throw. My countryman in tartan got the last
chance, of which he seemed reluctant to avail himself till the one
unseen said: “Vous avez le de'', Kilbride.”
Kilbride! the name was the call of whaups at home upon the
moors!
He laughed, shook, and tossed carelessly, and then the laugh was
all with them, for whatever they had played for he had seemingly
lost and the dice were now put by.
He rose somewhat confused, looked dubiously across at me with a
reddening face, and then came over with his hat in his hand.
“Pardon, Monsieur,” he began; then checked the French, and said:
“Have I a countryman here?”
“It is like enough,” said I, with a bow and looking at his tartan. “I
am from Scotland myself.”
He smiled at that with a look of some relief and took a vacant
chair on the other side of my small table.
“I have come better speed with my impudence,” said he in the
Hielan' accent, “than I expected or deserved. My name's Kilbride—
MacKellar of Kilbride—and I am here with another Highland
gentleman of the name of Grant and two or three French friends we
picked up at the door of the play-house. Are you come off the
Highlands, if I make take the liberty?”
“My name is lowland,” said I, “and I hail from the shire of
Renfrew.”
“Ah,” said he, with a vanity that was laughable. “What a pity! I
wish you had been Gaelic, but of course you cannot help it being
otherwise, and indeed there are many estimable persons in the
lowlands.”
“And a great wheen of Highland gentlemen very glad to join them
there too,” said I, resenting the implication.

“Of course, of course,” said he heartily. “There is no occasion for
offence.”
“Confound the offence, Mr. MacKellar!” said I. “Do you not think I
am just too glad at this minute to hear a Scottish tongue and see a
tartan waistcoat? Heilan' or Lowlan', we are all the same” when our
feet are off the heather.
“Not exactly,” he corrected, “but still and on we understand each
other. You must be thinking it gey droll, sir, that a band of strangers
in a common tavern would have the boldness to stare at you like my
friends there, and toss a dice about you in front of your face, but
that is the difference between us. If I had been in your place I
would have thrown the jug across at them, but here I am not better
nor the rest, because the dice fell to me, and I was one that must
decide the wadger.”
“Oh, and was I the object of a wadger?” said I, wondering what
we were coming to.
“Indeed, and that you were,” said he shamefacedly, “and I'm
affronted to tell it. But when Grant saw you first he swore you were
a countryman, and there was some difference of opinion.”
“And what, may I ask, did Kilbride side with?”
“Oh,” said he promptly, “I had never a doubt about that. I knew
you were Scots, but what beat me was to say whether you were
Hielan' or Lowlan'.” “And how, if it's a fair question, did you come to
the conclusion that I was a countryman of any sort?” said I.
He laughed softly, and “Man,” said he, “I could never make any
mistake about that, whatever of it. There's many a bird that's like
the woodcock, but the woodcock will aye be kennin' which is which,
as the other man said. Thae bones were never built on bread and
wine. It's a French coat you have there, and a cockit hat (by your
leave), but to my view you were as plainly from Scotland as if you
had a blue bonnet on your head and a sprig of heather in your
lapels. And here am I giving you the strange cow's welcome (as the
other man said), and that is all inquiry and no information. You must
just be excusing our bit foolish wadger, and if the proposal would

come favourably from myself, that is of a notable family, though at
present under a sort of cloud, as the other fellow said, I would be
proud to have you share in the bottle of wine that was dependent
upon Grant's impudent wadger. I can pass my word for my friends
there that they are all gentry like ourselves—of the very best, in
troth, though not over-nice in putting this task on myself.”
I would have liked brawly to spend an hour out any company than
my own, but the indulgence was manifestly one involving the danger
of discovery; it was, as I told myself, the greatest folly to be sitting
in a tavern at all, so MacKellar's manner immediately grew cold
when he saw a swithering in my countenance.
“Of course,” said he, reddening and rising, “of course, every
gentleman has his own affairs, and I would be the last to make a
song of it if you have any dubiety about my friends and me. I'll allow
the thing looks very like a gambler's contrivance.”
“No, no, Mr. MacKellar,” said I hurriedly, unwilling to let us part like
that, “I'm swithering here just because I'm like yoursel' of it and
under a cloud of my own.”
“Dod! Is that so?” said he quite cheerfully again, and clapping
down, “then I'm all the better pleased that the thing that made the
roebuck swim the loch—and that's necessity—as the other man said,
should have driven me over here to precognosce you. But when you
say you are under a cloud, that is to make another way of it
altogether, and I will not be asking you over, for there is a gentleman
there among the five of us who might be making trouble of it.”
“Have you a brother in Glasgow College?” says I suddenly, putting
a question that had been in my mind ever since he had mentioned
his name.
“Indeed, and I have that,” said he quickly, “but now he is following
the law in Edinburgh, where I am in the hopes it will be paying him
better than ever it paid me that has lost two fine old castles and the
best part of a parish by the same. You'll not be sitting there and
telling me surely that you know my young brother Alasdair?”

“Man! him and me lodged together in Lucky Grant's, in Crombie's
Land in the High Street, for two Sessions,” said I.
“What!” said MacKellar. “And you'll be the lad that snow-balled the
bylie, and your name will be Greig?”
As he said it he bent to look under the table, then drew up
suddenly with a startled face and a whisper of a whistle on his lips.
“My goodness!” said he, in a cautious tone, “and that beats all.
You'll be the lad that broke jyle with the priest that shot at Buhot,
and there you are, you amadain, like a gull with your red brogues on
you, crying 'come and catch me' in two languages. I'm telling you to
keep thae feet of yours under this table till we're out of here, if it
should be the morn's morning. No—that's too long, for by the morn's
morning Buhot's men will be at the Hôtel Dieu, and the end of the
story will be little talk and the sound of blows, as the other man
said.”
Every now and then as he spoke he would look over his shoulder
with a quick glance at his friends—a very anxious man, but no more
anxious than Paul Greig.
“Mercy on us!” said I, “do you tell me you ken all that?”
“I ken a lot more than that,” said he, “but that's the latest of my
budget, and I'm giving it to you for the sake of the shoes and my
brother Alasdair, that is a writer in Edinburgh. There's not two
Scotchmen drinking a bowl in Paris town this night that does not ken
your description, and it's kent by them at the other table there—
where better?—but because you have that coat on you that was
surely made for you when you were in better health, as the other
man said, and because your long trams of legs and red shoes are
under the table there's none of them suspects you. And now that
I'm thinking of it, I would not go near the hospital place again.”
“Oh! but the priest's there,” said I, “and it would never do for me
to be leaving him there without a warning.”
“A warning!” said MacKellar with contempt. “I'm astonished to
hear you, Mr. Greig. The filthy brock that he is!”

“If you're one of the Prince's party,” said I, “and it has every look
of it, or, indeed, whether you are or not, I'll allow you have some
cause to blame Father Hamilton, but as for me, I'm bound to him
because we have been in some troubles together.”
“What's all this about 'bound to him'?” said MacKellar with a kind
of sneer. “The dog that's tethered with a black pudding needs no
pity, as the other man said, and I would leave this fellow to shift for
himself.”
“Thank you,” said I, “but I'll not be doing that.”
“Well, well,” said he, “it's your business, and let me tell you that
you're nothing but a fool to be tangled up with the creature. That's
Kilbride's advice to you. Let me tell you this more of it, that they're
not troubling themselves much about you at all now that you have
given them the information.”
“Information!” I said with a start. “What do you mean by that?”
He prepared to join his friends, with a smile of some slyness, and
gave me no satisfaction on the point.
“You'll maybe ken best yourself,” said he, “and I'm thinking your
name will have to be Robertson and yourself a decent Englishman
for my friends on the other side of the room there. Between here
and yonder I'll have to be making up a bonny lie or two that will put
them off the scent of you.”
A bonny lie or two seemed to serve the purpose, for their interest
in me appeared to go no further, and by-and-by, when it was
obvious that there would be no remission of the rain, they rose to
go.
The last that went out of the door turned on the threshold and
looked at me with a smile of recognition and amusement.
It was Buhot!

W
CHAPTER XXIX
WHEREIN THE PRIEST LEAVES ME, AND I
MAKE AN INLAND VOYAGE
hat this marvel betokened was altogether beyond my
comprehension, but the five men were no sooner gone
than I clapped on my hat and drew up the collar of my coat
and ran like fury through the plashing streets for the place
that was our temporary home. It must have been an intuition of the
raised that guided me; my way was made without reflection on it, at
pure hazard, and yet I landed through a multitude of winding and
bewildering streets upon the Isle of the City and in front of the Hôtel
Dieu in a much shorter time than it had taken me to get from there
to the Duke of Burgundy's Head.
I banged past the doorkeeper, jumped upstairs to the clergyman's
quarters, threw open the door and—found Father Hamilton was
gone!
About the matter there could be no manner of dubiety, for he had
left a letter directed to myself upon the drawers-head.
“My Good Paul (said the epistle, that I have kept till now as a
memorial of my adventure): When you return you will discover from
this that I have taken leave a l'anglaise, and I fancy I can see my
secretary looking like the arms of Bourges (though that is an unkind
imputation). 'Tis fated, seemingly, that there shall be no rest for the
sole of the foot of poor Father Hamilton. I had no sooner got to like
a loose collar, and an unbuttoned vest, and the seclusion of a cell,
than I must be plucked out; and now when my birds—the darlings!
—are on the very point of hatching I must make adieux. Oh! la belle
équipée! M. Buhot knows where I am—that's certain, so I must
remove myself, and this time I do not propose to burden M. Paul

Greig with my company, for it will be a miracle if they fail to find me.
As for my dear Croque-mort, he can have the glass coach and
Jacques and Bernard, and doubtless the best he can do with them is
to take all to Dunkerque and leave them there. I myself, I go sans
trompette, and no inquiries will discover to him where I go.”
As a postscript he added, “And 'twas only a sailor's log, dear lad!
My poor young Paul!” When I read the letter I was puzzled
tremendously, and at first I felt inclined to blame the priest for a
scurvy flitting to rid himself of my society, but a little deliberation
convinced me that no such ignoble consideration was at the bottom
of his flight. If I read his epistle aright the step he took was in my
own interest, though how it could be so there was no surmising. In
any case he was gone; his friend in the hospital told me he had set
out behind myself, and taken a candle with him and given a farewell
visit to his birds, and almost cried about them and about myself, and
then departed for good to conceal himself, in some other part of the
city, probably, but exactly where his friend had no way of guessing.
And it was a further evidence of the priest's good feeling to myself
(if such were needed) that he had left a sum of a hundred livres for
me towards the costs of my future movements.
I left the Hôtel Dieu at midnight to wander very melancholy about
the streets for a time, and finally came out upon the river's bank,
where some small vessels hung at a wooden quay. I saw them in
moonlight (for now the rain was gone), and there rose in me such a
feeling as I had often experienced as a lad in another parish than
the Mearns, to see the road that led from strangeness past my
mother's door. The river seemed a pathway out of mystery and
discontent to the open sea, and the open sea was the same that
beat about the shores of Britain, and my thought took flight there
and then to Britain, but stopped for a space, like a wearied bird,
upon the town Dunkerque. There is one who reads this who will
judge kindly, and pardon when I say that I felt a sort of tenderness
for the lady there, who was not only my one friend in France, so far
as I could guess, but, next to my mother, the only woman who knew
my shame and still retained regard for me. And thinking about

Scotland and about Dunkerque, and seeing that watery highway to
them both, I was seized with a great repugnance for the city I stood
in, and felt that I must take my feet from there at once. Father
Hamilton was lost to me: that was certain. I could no more have
found him in this tanglement of streets and strange faces than I
could have found a needle in a haystack, and I felt disinclined to
make the trial. Nor was I prepared to avail myself of his offer of the
coach and horses, for to go travelling again in them would be to
court Bicêtre anew.
There was a group of busses or barges at the quay, as I have said,
all huddled together as it were animals seeking warmth, with their
bows nuzzling each other, and on one of them there were
preparations being made for her departure. A cargo of empty casks
was piled up in her, lights were being hung up at her bow and stern,
and one of her crew was ashore in the very act of casting off her
ropes. At a flash it occurred to me that I had here the safest and the
speediest means of flight.
I ran at once to the edge of the quay and clumsily propounded a
question as to where the barge was bound for.
“Rouen or thereabouts,” said the master.
I asked if I could have a passage, and chinked my money in my
pocket.
My French might have been but middling, but Lewis d'Or talks in a
language all can understand.
Ten minutes later we were in the fairway of the river running
down through the city which, in that last look I was ever fated to
have of it, seemed to brood on either hand of us like bordering hills,
and at morning we were at a place by name Triel.
Of all the rivers I have seen I must think the Seine the finest. It
runs in loops like my native Forth, sometimes in great, wide
stretches that have the semblance of moorland lochs. In that fine
weather, with a sun that was most genial, the country round about
us basked and smiled. We moved upon the fairest waters, by magic
gardens, and the borders of enchanted little towns. Now it would be

a meadow sloping backward from the bank, where reeds were
nodding, to the horizon; now an orchard standing upon grass that
was the rarest green, then a village with rusty roofs and spires and
the continual chime of bells, with women washing upon stones or
men silent upon wherries fishing. Every link of the river opened up a
fresher wonder; if not some poplared isle that had the invitation to a
childish escapade, 'twould be another town, or the garden of a
château, maybe, with ladies walking stately on the lawns, perhaps
alone, perhaps with cavaliers about them as if they moved in some
odd woodland minuet. I can mind of songs that came from open
windows, sung in women's voices; of girls that stood drawing water
and smiled on us as we passed, at home in our craft of fortune, and
still the lucky roamers seeing the world so pleasantly without the
trouble of moving a step from our galley fire.
Sometimes in the middle of the days we would stop at a red-
faced, ancient inn, with bowers whose tables almost had their feet
dipped in the river, and there would eat a meal and linger on a pot
of wine while our barge fell asleep at her tether and dreamt of the
open sea. About us in these inns came the kind country-people and
talked of trivial things for the mere sake of talking, because the
weather was sweet and God so gracious; homely sounds would waft
from the byres and from the barns—the laugh of bairns, the whistle
of boys, the low of cattle.
At night we moored wherever we might be, and once I mind of a
place called Andelys, selvedged with chalky cliffs and lorded over by
a castle called Gaillard, that had in every aspect of it something of
the clash of weapons and of trumpet-cry. The sky shone blue
through its gaping gables and its crumbling windows like so many
eyes; the birds that wheeled all round it seemed to taunt it for its
inability. The old wars over, the deep fosse silent, the strong men
gone—and there at its foot the thriving town so loud with sounds of
peaceful trade! Whoever has been young, and has the eye for what
is beautiful and great and stately, must have felt in such a scene that
craving for companionship that tickles like a laugh within the heart—
that longing for some one to feel with him, and understand, and look

upon with silence. In my case 'twas two women I would have there
with me just to look upon this Gaillard and the town below it.
Then the bending, gliding river again, the willow and the aspen
edges, the hazy orchards and the emerald swards; hamlets, towns,
farm-steadings, châteaux, kirks, and mills; the flying mallard, the
leaping perch, the silver dawns, the starry nights, the ripple of the
water in my dreams, and at last the city of Rouen. My ship of fortune
went no further on.
I slept a night in an inn upon the quay, and early the next
morning, having bought a pair of boots to save my red shoes, I took
the road over a hill that left Rouen and all its steeples, reeking at the
bottom of a bowl. I walked all day, through woods and meadows
and trim small towns and orchards, and late in the gloaming came
upon the port of Havre de Grace.
The sea was sounding there, and the smell of it was like a
salutation. I went out at night from my inn, and fairly joyed in its
propinquity, and was so keen on it that I was at the quay before it
was well daylight. The harbour was full of vessels. It was not long
ere I got word of one that was in trim for Dunkerque, to which I
took a passage, and by favour of congenial weather came upon the
afternoon of the second day.
Dunkerque was more busy with soldiers than ever, all the arms of
France seemed to be collected there, and ships of war and flat-
bottomed boats innumerable were in the harbour.
At the first go-off I made for the lodgings I had parted from so
unceremoniously on the morning of that noisy glass coach.
The house, as I have said before, was over a baker's shop, and
was reached by a common outer stair that rose from a court-yard
behind. Though internally the domicile was well enough, indeed had
a sort of old-fashioned gentility, and was kept by a woman whose
man had been a colonel of dragoons, but now was a tippling
pensioner upon the king, and his own wife's labours, it was,
externally, somewhat mean, the place a solid merchant of our own
country might inhabit, but scarce the place wherein to look for royal

blood. What was my astonishment, then, when, as I climbed the
stair, I came face to face with the Prince!
I felt the stair swing off below me and half distrusted my senses,
but I had the presence of mind to take my hat off.
“Bon jour, Monsieur, said he, with a slight hiccough, and I saw
that he was flushed and meant to pass with an evasion. There and
then a daft notion to explain myself and my relations with the priest
who had planned his assassination came to me, and I stopped and
spoke.
“Your Royal Highness—-” I began, and at that he grew purple.
“Cest un drôle de corps!” said he, and, always speaking in French,
said he again:
“You make an error, Monsieur; I have not the honour of Monsieur's
acquaintance,” and looked at me with a bold eye and a
disconcerting.
“Greig,” I blurted, a perfect lout, and surely as blind as a mole that
never saw his desire, “I had the honour to meet your Royal Highness
at Versailles.”
“My Royal Highness!” said he, this time in English. “I think
Monsieur mistakes himself.” And then, when he saw how crestfallen I
was, he smiled and hiccoughed again. “You are going to call on our
good Clancarty,” said he. “In that case please tell him to translate to
you the proverb, Oui phis sait plus se tait.”
“There is no necessity, Monsieur,” I answered promptly. “Now that
I look closer I see I was mistaken. The person I did you the honour
to take you for was one in whose opinion (if he took the trouble to
think of me at all) I should have liked to re-establish myself, that
was all.”
In spite of his dissipation there was something noble in his
manner—a style of the shoulders and the hands, a poise of the head
that I might practise for years and come no closer on than any nowt
upon my father's fields. It was that which I remember best of our
engagement on the stair, and that at the last of it he put out his
hand to bid me good-day.

“My name,” says he, “is Monsieur Albany so long as I am in
Dunkerque. À bon entendeur salut! I hope we may meet again,
Monsieur Greig.” He looked down at the black boots I had bought
me in Rouen. “If I might take the liberty to suggest it,” said he,
smiling, “I should abide by the others. I have never seen their
wearer wanting wit, esprit, and prudence—which are qualities that at
this moment I desire above all in those that count themselves my
friends.”
And with that he was gone. I watched him descend the remainder
of the stair with much deliberation, and did not move a step myself
until the tip of his scabbard had gone round the corner of the close.

C
CHAPTER XXX
A GUID CONCEIT OF MYSELF LEADS ME FAR
ASTRAY
lancarty and Thurot were playing cards, so intent upon that
recreation that I was in the middle of the floor before they
realised who it was the servant had ushered in.
“Mon Dieu! Monsieur Blanc-bec! Il n'y a pas de petit chez
soi!” cried Thurot, dropping his hand, and they jumped to their feet
to greet me.
“I'll be hanged if you want assurance, child,” said Clancarty,
surveying me from head to foot as if I were some curiosity. “Here's
your exploits ringing about the world, and not wholly to your credit,
and you must walk into the very place where they will find the
smallest admiration.”
“Not meaning the lodging of Captain Thurot,” said I. “Whatever
my reputation may be with the world, I make bold to think he and
you will believe me better than I may seem at the first glance.”
“The first glance!” cried his lordship. “Gad, the first glance
suggests that Bicêtre agreed with our Scotsman. Sure, they must
have fed you on oatmeal. I'd give a hatful of louis d'or to see Father
Hamilton, for if he throve so marvellously in the flesh as his
secretary he must look like the side of St. Eloi. One obviously grows
fat on regicide—fatter than a few poor devils I know do upon
devotion to princes.”
Thurot's face assured me that I was as welcome there as ever I
had been. He chid Clancarty for his badinage, and told me he was
certain all along that the first place I should make for after my flight

from Bicêtre (of which all the world knew) would be Dunkerque.
“And a good thing too, M. Greig,” said he.
“Not so good,” says I, “but what I must meet on your stair the
very man-”
“Stop!” he cried, and put his finger on his lip. “In these parts we
know only a certain M. Albany, who is, my faith! a good friend of
your own if you only knew it.”
“I scarcely see how that can be,” said I. “If any man has a cause
to dislike me it is his Roy—”
“M. Albany,” corrected Thurot.
“It is M. Albany, for whom, it seems, I was the decoy in a business
that makes me sick to think on. I would expect no more than that he
had gone out there to send the officers upon my heels, and for me
to be sitting here may be simple suicide.”
Clancarty laughed. “Tis the way of youth,” said he, “to attach far
too much importance to itself. Take our word for it, M. Greig, all
France is not scurrying round looking for the nephew of Andrew
Greig. Faith, and I wonder at you, my dear Thurot, that has an
Occasion here—a veritable Occasion—and never so much as says
bottle. Stap me if I have a friend come to me from a dungeon
without wishing him joy in a glass of burgundy!”
The burgundy was forthcoming, and his lordship made the most of
it, while Captain Thurot was at pains to assure me that my position
was by no means so bad as I considered it. In truth, he said, the
police had their own reasons for congratulating themselves on my
going out of their way. They knew very well, as M. Albany did, that I
had been the catspaw of the priest, who was himself no better than
that same, and for that reason as likely to escape further
molestation as I was myself.
Thurot spoke with authority, and hinted that he had the word of
M. Albany himself for what he said. I scarcely knew which pleased
me best—that I should be free myself or that the priest should have
a certain security in his concealment.

I told them of Buhot, and how oddly he had shown his
complacence to his escaped prisoner in the tavern of the Duke of
Burgundy's Head. At that they laughed.
“Buhot!” cried his lordship. “My faith! Ned must have been tickled
to see his escaped prisoner in such a cosy cachette as the Duke's
Head, where he and I, and Andy Greig—ay! and this same priest—
tossed many a glass, Ciel! the affair runs like a play. All it wants to
make this the most delightful of farces is that you should have
Father Hamilton outside the door to come in at a whistle. Art sure
the fat old man is not in your waistcoat pocket? Anyhow, here's his
good health....”
=== MISSING PAGES (274-288) ===

CHAPTER XXXI.
THE BARD OF LOVE WHO WROTE WITH OLD
MATERIALS

W
CHAPTER XXXII.
THE DUEL IN THE AUBERGE GARDEN
hoever it was that moved at the instigation of Madame on
my behalf, he put speed into the business, for the very next
day I was told my sous-lieutenancy was waiting at the
headquarters of the regiment. A severance that seemed
almost impossible to me before I learned from the lady's own lips
that her heart was elsewhere engaged was now a thing to long for
eagerly, and I felt that the sooner I was out of Dunkerque and
employed about something more important than the tying of my hair
and the teasing of my heart with thinking, the better for myself.
Teasing my heart, I say, because Miss Walkinshaw had her own
reasons for refusing to see me any more, and do what I might I
could never manage to come face to face with her. Perhaps on the
whole it was as well, for what in the world I was to say to the lady,
supposing I were privileged, it beats me now to fancy. Anyhow, the
opportunity never came my way, though, for the few days that
elapsed before I departed from Dunkerque, I spent hours in the Rue
de la Boucherie sipping sirops on the terrace of the Café Coignet
opposite her lodging, or at night on the old game of humming
ancient love-songs to her high and distant window. All I got for my
pains were brief and tantalising glimpses of her shadow on the
curtains; an attenuate kind of bliss it must be owned, and yet
counted by Master Red-Shoes (who suffered from nostalgia, not
from love, if he had had the sense to know it) a very delirium of
delight.
One night there was an odd thing came to pass. But, first of all, I
must tell that more than once of an evening, as I would be in the
street and staring across at Miss Walkinshaw's windows, I saw his
Royal Highness in the neighbourhood. His cloak might be

voluminous, his hat dragged down upon the very nose of him, but
still the step was unmistakable. If there had been the smallest doubt
of it, there came one evening when he passed me so close in the
light of an oil lamp that I saw the very blotches on his countenance.
What was more, he saw and recognised me, though he passed
without any other sign than the flash of an eye and a halfstep of
hesitation.

“H'm,” thinks I, “here's Monsieur Albany looking as if he might, like
myself, be trying to content himself with the mere shadows of

things.”
He saw me more than once, and at last there came a night when
a fellow in drink came staving down the street on the side I was on
and jostled me in the by-going without a word of apology.
“Pardonnez, Monsieur!” said I in irony, with my hat off to give him
a hint at his manners.
He lurched a second time against me and put up his hand to catch
my chin, as if I were a wench, “Mon Dieu! Monsieur Blanc-bec, 'tis
time you were home,” said he in French, and stuttered some ribaldry
that made me smack his face with an open hand.
“I saw his Royal Highness in the neighbourhood—”
At once he sobered with suspicious suddenness if I had had the
sense to reflect upon it, and gave me his name and direction as one
George Bonnat, of the Marine. “Monsieur will do me the honour of a
meeting behind the Auberge Cassard after petit dejeuner to-
morrow,” said he, and named a friend. It was the first time I was
ever challenged. It should have rung in the skull of me like an alarm,
but I cannot recall at this date that my heart beat a stroke the faster,
or that the invitation vexed me more than if it had been one to the
share of a bottle of wine. “It seems a pretty ceremony about a
cursed impertinence on the part of a man in liquor,” I said, “but I'm
ready to meet you either before or after petit déjeuner, as it best
suits you, and my name's Greig, by your leave.”
“Very well, Monsieur Greig,” said he; “except that you stupidly
impede the pavement and talk French like a Spanish cow (comme
une vache espagnole), you seem a gentleman of much
accommodation. Eight o'clock then, behind the auberge,” and off
went Sir Ruffler, singularly straight and business-like, with a
profound congé for the unfortunate wretch he planned to thrust a
spit through in the morning.
I went home at once, to find Thurot and Clancarty at lansquenet.
They were as elate at my story as if I had been asked to dine with
Louis.

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