120028843_PDE29_01&02_R2_011404
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
Vol. 29, Nos. 1 & 2, pp. 43–70, 2004
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Regularity of the Solution of Some Unilateral
Boundary Value Problems in Polygonal and
Polyhedral Domains
W. Chikouche,1 D. Mercier,2 and S. Nicaise2*
1
Département de Mathematiques, Centre Universitaire de Jijel, Jijel, Algeria
Université de Valenciennes et du Hainaut Cambrésis, MACS, Institut des
Sciences et Techniques de Valenciennes, Valenciennes, France
2
ABSTRACT
We consider some unilateral boundary value problems in polygonal and
polyhedral domains with unilateral transmission conditions. Regularity results
in terms of weighted Sobolev spaces are obtained using a penalization technique,
similar regularity results for the penalized problems and by showing uniform
estimates with respect to the penalization parameter.
Key Words: Unilateral problems; Regularity results.
AMS Subject Classification: 35J85; 35J25; 35B65; 49N60; 35J60.
I. INTRODUCTION
We investigate two and three dimensional transmission problems for the
Laplace operators in polygonal or polyhedral domains. Some unilateral boundary
and transmission conditions of Signorini’s type are imposed. We may expect
∗ Correspondence: S. Nicaise, Université de Valenciennes, MACS, ISTV, Le Mont
Houy, 59313 Valenciennes Cedex 9, France; Fax: (33) 03275-11940; E-mail: snicaise@
uni-valenciennes.fr.
43
DOI: 10.1081/PDE-120028843
Copyright © 2004 by Marcel Dekker, Inc.
0360-5302 (Print); 1532-4133 (Online)
www.dekker.com
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Chikouche, Mercier, and Nicaise
that the solution presents a singular behaviour near the corners and edges of
the domains, especially where the interface intersects the boundary (since it is
already the case for linear problems (Dauge, 1988; Grisvard, 1985; Kondrat’ev,
1967; Leguillon and Sanchez-Palencia, 1991; Nicaise, 1993)). Different authors have
studied such unilateral boundary value problems but without transmission (i.e.,
pure Laplace equation) either by finding sufficient conditions on the domains
which guarantee the H 2 regularity of the solution (Brézis, 1972; Caccioppoli, 1963;
Grisvard, 1975–1976; Grisvard and Iooss, 1976), or by restricting to Signorini’s
boundary conditions (Moussaoui, 1992). In Mercier and Nicaise (2002), regularity
results in fractional Sobolev spaces for the solution of unilateral boundary value
problems with transmision is obtained using a perturbation technique combined
with a lifting argument. Here we characterize the regularity of the solution in terms
of weighted Sobolev spaces. This characterization is of great interest for numerical
applications as shown in the linear theory (Apel, 1999), where some refined meshes
are used to compensate the singular behaviour of the solution. Some applications
to unilateral problems combining standard methods (from Hlavček et al., 1988 for
instance) and the refinement mesh method will be considered in a forthcoming
paper.
Our main method is inspired from the method of Caccioppoli (1963), Grisvard
(1975–1976) and Grisvard and Iooss (1976), which consists in four steps: first we
penalize the problem, secondly we estimate the second order derivatives of the
solution of the penalized problem using some integrations by parts, thirdly we
estimate uniformly the boundary terms using the monotonicity assumption and
finally we pass to the limit using some compactness arguments. This method is
here adapted to weighted Sobolev spaces (even anisotropic ones in 3D) and to
unilateral boundary and transmission conditions. For purely unilateral boundary
conditions near the interface this method fails since we can no more estimate
some boundary terms, therefore in this case we use a method based on a change
of variables (Moussaoui, 1992) allowing to pass from the regular case to the
singular one.
The paper is organized as follows: In Sec. II we prove Poincaré’s inequality
in weighted Sobolev spaces related to our transmission problem we have in
mind. Our method is based on a Fourier analysis and extends a former
result from Bailet (1996) obtained in the homogeneous case. Section III is
devoted to the presentation of the problem. We restrict ourselves to the two
dimensional case with mixed boundary condition near the interface in Sec. IV.
There, adapting the method of Caccioppoli to weighted Sobolev norms and
using Poincaré’s inequality mentioned above we obtain the regularity results
of the solution in weighted Sobolev spaces. In Sec. V we extend this kind of
results to three dimensional domains with a singular edge. As in the linear case
we first show the optimal regularity of the solution in the edge direction. In
a second step using this regularity and 2D results we conclude the singular
behaviour in the direction perpendicular to the edge. This leads to anisotropic
regularity of the solution. Finally in Sec. VI we come back to two dimensional
domains but with unilateral boundary conditions near the interface. In this case
since Poincaré’s inequality cannot be invoked we adapt a method based on a
change of variables (Moussaoui, 1992) allowing to pass from the regular case to the
singular one.
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II. POINCARÉ’S INEQUALITY
In this section, we describe a Poincaré’s inequality in weighted Sobolev spaces
defined in a sector C defined by
C = r r > 0 0 < < for a fixed ∈0 2
that is supposed to be divided into the two sectors C1 ∪ C2
defined by
C1 = r r > 0 0 < < 1 C2 = r r > 0 1 < < of respective opening 1 and 2 = − 1 . For any ∈ , the related weighted
Sobolev space are defined by
W1 C = u ∈ C r −1 u r i u ∈ L2 C i = 1 2
which is a Hilbert space equipped with the inner product
2 −2
u vW1 C =
pi r r ux2 + ux2 dx
i=12
Ci
for two positive constants p1 p2 .
The space L2 0 is now equipped with the inner product
1
u v = p1
uv d + p2
uv d
0
1
We further introduce the operator from L2 0 into itself defined by
D = ∈ H 1 0 1 ∈ H 2 0 1 2 ∈ H 2 1 and satisfying
p1 1 1 = p2 2 1 1 0 = 0 2 = 0 i = −pi i i = 1 2 ∀ ∈ D
This operator is a positive selfadjoint operator (see for instance Nicaise, 1993) with
a compact resolvant. We then denote by k2 k=1 its set of eigenvalues in increasing
order which are simple in this case and are roots of (see for instance Kozlov et al.,
2001; Leguillon and Sanchez-Palencia, 1991; Mercier, 2001; Nicaise and Sändig,
1994)
p1 sin1 sin2 − p2 cos1 cos2 = 0
(1)
Let k k=1 be the set of associated eigenvectors which forms an orthornomal basis
of L2 0 (with the above inner product).
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Chikouche, Mercier, and Nicaise
Now we are ready to formulate Poincaré’s inequality in W1 C (compare with
Proposition 2.1 of Bailet, 1996):
Lemma II.1. If < 0 then for all u ∈ W1 C with a compact support and fulfilling the
Dirichlet boundary condition
ur = 0 ∀r > 0
(2)
one has
C
ux2 r 2−2 dx ≤
2
1
ui 2 r 2 dx
2
+ 1 i=12 Ci
(3)
Proof. We write u ∈ W1 C with a compact support and satisfying (2) in the basis
k k=1 described above to get
ur · =
uk rk k=1
u
r · =
uk rk r
k=1
Therefore by the fact that the space
V = ∈ H 1 0 = 0
equipped with the inner product
= p1
1
0
d + p2
1
d
is such that V = D1/2 (where the operator was defined above), and by
Parseval’s identity, we obtain
p1
1
0
ur 2 d + p2
1
ur 2 d =
k=1
uk r2 2
2
p1
u1 r d + p2
u2 r d
0
1
2
2 1 u
1 u
1
r + 2 1 r d
= p1
r
r 0
2
2 u
1 u1
1
r + 2 r d
+p2
r
r
1
=
1
uk r2 +
k=1
k2
u r2 r2 k
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After integration in r this last identity implies in particular
i=12
pi
Ci
ui r 2 r 2 r dr d ≥
k=1 0
+12
uk r2 r 2 r dr
k=1 0
uk r2 r 2−2 r dr
(4)
By the assumption on u, the function uk belongs to L2+1/2 + , where we
recall (see for instance Grisvard, 1987, p. 28) that L2 + is the set of measurable
functions v defined on + such that
v
2L2 + =
0
vtt 2 dt < As in Proposition 2.1 of Bailet (1996) the assumption < 0 implies that uk 0 = 0.
Therefore we can write
r
uk s ds
r −1 uk r = r −1
0
With the notation from Grisvard (1985, p. 28) the above identity means that
r −1 uk r = Huk r
and by Hardy’s inequality we obtain that
0
r 2−2 uk r2 r dr ≤
1 2 2
r uk r r dr
2 0
We conclude by inserting this estimate into (4).
III. SIGNORINI TRANSMISSION PROBLEMS WITH
MIXED BOUNDARY CONDITIONS
Let us fix a bounded domain of n n = 2 or 3, with a Lipschitz-boundary .
We suppose that is decomposed into two nonoverlapping subdomains (only for
the sake of simplicity) 1 and 2 with an interface satisfying
=
1 ∪ 2 1 ∩ 2 = ∅ 1 ∩ 2 = We assume that the boundaries i of i i = 1 2 are also Lipschitz-continuous.
We further assume that is sudvided into two parts D and S with S ⊂
∩ 1 , the first part corresponds to the part of the boundary where we will
impose Dirichlet boundary conditions while on the second one we will fix Signorini
boundary conditions.
For a function u defined in we denote by ui its restriction to i .
Let us now fix two positive constants pi i = 1 2 and two maximal monotone
graphs of 2 S and , such that 0 ∈ S 0 and 0 ∈ 0 (see Brézis, 1971).
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Chikouche, Mercier, and Nicaise
We further suppose that S is different from the graph corresponding to the
Dirichlet boundary condition (namely 0 = and t = ∅ if t = 0). The
Signorini-transmission problem we have in mind is the following one:


pi −ui + ui = fi



u1 = u2






u
u
− p1 1 − p2 2 ∈ u1 1
1




u


−p1 1 ∈ S u1 


1


u=0
in i i = 1 2
on on (5)
on S on D where f ∈ L2 and u1 /1 means the outward normal derivative of u1 on the
boundary of 1 . The weak formulation of that problem is quite standard, let us
describe it for the sake of completeness: Define the variational space HD1 by
HD1 = v ∈ H 1 v = 0 on D and the bilinear form a on HD1 by
au v =
2
i=1
pi
i
ui vi + ui vi dx
(6)
Since S (resp. ) is a maximal monotone graph of 2 , there exists a convex
lower semicontinuous (l.s.c.) function jS (resp. j ) from in − +, such that
S (resp. ) is the subdifferential of jS (resp. j . We associate two l.s.c. mappings
S and defined respectively on L2 S and L2 by

 j ud if j u ∈ L1 S
S
S
S u = S
+
else,
and

 j ud if j u ∈ L1 u = +
else.
The weak formulation of problem (5) consists in finding a solution u ∈ V of
au v − u + S uS − S vS + u − v (7)
≥ fv − u dx ∀v ∈ HD1 The mapping u ∈ H 1 → S uS + u being convex l.s.c. on H 1 ,
the existence and uniqueness of a solution of problem (7) are deduced from general
results on nonlinear monotone operators (see for instance Browder, 1965 or Brézis,
1972, Theorem I.7).
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In order to obtain regularity results for the solution u of (7) we approximate it
by the family of problems

in i i = 1 2

pi −ui + ui = fi



=
u
on u

1
2





u1
u2
− p1
− p2
= u1 on (8)
1
1




u1


−p1
= S u1 on S 


1


on D u = 0
where > 0 will tend to 0 and (resp. S ) is the Yosida approximation of (resp. S ) given by = −1 Id − Id + −1 and is a nondecreasing function,
uniformly Lipschitz continuous with a Lipschitz constant equal to −1 (see for
instance Brézis, 1971). The weak formulation of (8) is:
S u1 v1 ds
au v + u1 v1 ds +
=
S
fv dx ∀v ∈ HD1 (9)
Following the proof of Theorem I.8 of Brézis (1972), we show that this last
problem has a unique solution u ∈ HD1 , which fulfils
u 1 f 0 (10)
where here and below the notation a b means that there exists a positive constant
C independent of a b and such that a ≤ Cb.
IV. 2D SIGNORINI TRANSMISSION PROBLEMS WITH
MIXED BOUNDARY CONDITIONS
Now we assume is a bounded domain of 2 and the boundary of the
subdomains i i = 1 2 is formed by open straight line segments ij j = 1 Ni ,
with Ni ∈ , enumerated clockwise such that
= 11 = 21 = =
Ni
ij i=12 j=2
In this section, we are interested into the influence between nonlinear
transmission conditions and mixed boundary conditions, therefore we will assume
that S = 12 , while D is the rest of the boundary of . We denote by Pi i = 1 N1 + N2 − 2 the vertices of where
i = 1 N1 − 1
Pi = 1i ∩ 1i+1 Pi = 2i−N1 +1 ∩ 2i−N1 +2 i = N1 N1 + N2 − 2
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Chikouche, Mercier, and Nicaise
We refer to Fig. 1 for an illustration. Without loss of generality we may assume that
P1 is situated at the origin.
To show some regularities of second order derivatives of u solution of (8)
near the origin we localize the above problem by using a radial cut-off function
≡ r ∈ + such that ≡ 1 in a neighbourhood of 0, ≡ 0 outside another
neighbourhood of 0 that we may suppose to be nonincreasing and satisfying
0 ≤ ≤ 1. Setting u = u (for the sake of shorthness we drop the dependance on
and write this right-hand side u since for the moment is fixed), we see that
u ∈ HD1 is a weak solution of

−pi ui = Fi
in i i = 1 2





on u1 = u2






u
u
− p1 1 − p2 2 = u1 on (11)
1
1




u


−p1 1 = S u1 on S 


1


u=0
on D where Fi = fi + 2pi · ui + pi − ui , which belongs to L2 Ci with the
estimate
F 0C f 0 (12)
according to (10).
In (11) the Neumann boundary data and transmission data are in H 1/2 thanks
to the next lemma:
Lemma IV.1. If is a uniformly Lipschitz continuous function such that 0 = 0.
Then for all u ∈ H 1/2 one has
u ∈ H 1/2 Figure 1. The domain .
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Proof. The uniformly Lipschitz continuity of x − y ≤ Bx − y ∀x y ∈ (13)
for some B > 0 and the property 0 = 0 imply that
x ≤ Bx ∀x ∈ Consequently we directly get
u
0 ≤ B
u
0 In the same way the estimate (13) directly yields
×
ux − uy2
ux − uy2
2
dx
dy
≤
B
dx dy
2
x − y
x − y2
×
and the conclusion follows.
Now adapting the arguments from Grisvard (1975–1976) and Grisvard and
Iooss (1976), we shall show the
Theorem IV.2.
For any 1 > > 1 − 1 and any i j k = 1 2 one has
r 2jk ui ∈ L2 i (14)
Furthermore one has
r 2jk ui 0Ci f 0 (15)
i=12 jk=12
Proof. The solution u ∈ HD1 of (11) may be seen as a solution of a nonhomogeneous transmission problem in with interior data F ∈ L2 , Neumann
boundary data −S u1 ∈ H 1/2 N and transmission data − u1 ∈ H 1/2 (see Lemma IV.1). Consequently by Theorem 2.26 of Nicaise (1993) (see also
Leguillon and Sanchez-Palencia, 1991; Lemrabet, 1977; Mercier, 2001; Nicaise and
Sändig, 1994), u admits the next decomposition
ui = uRi +
j ∈∩01
cj Sj i i = 1 2
where uRi ∈ H 2 i i = 1 2 is the regular part of u, cj are constant and Sj are the
so-called singularities of the transmission problem (9) given by
Sj r = r j j when j is the eigenvector associated with j2 described in Sec. II. The above
decomposition leads to the regularity (14) since we easily check that each term
satisfies this regularity.
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Note that the above regularities of u and Hardy’s inequality imply that ui
belongs to H2 Ci (see for instance Grisvard, 1985, p. 28), the weighted Sobolev
space of Kondratév’s type defined by
H2 Ci = v ∈ Ci r +−2 D v ∈ L2 Ci ∀ ≤ 2
which is a Hilbert space with its natural inner product.
To check the estimate (15) we start from the square of the left-hand side of (15)
and use some integrations by parts. These last ones being justified by the density of
C0 Ci = v ∈ C Ci vx = 0 for x < r and x > R
for some 0 < r < R < into H2 Ci (see e.g., Dauge, 1988). In other words we first fix a sequence un ∈ C0
Ci satisfying
n
ui → ui in H2 Ci as n → Then applying Green’s formula we have
2 2 n 2 n
n
r 2 ui 2 dx =
r jk ui jk ui dx
Ci
jk=12 Ci
=−
jk=12 Ci
n
n
j r 2 2jk ui k ui
dx+
Ci
r
2
n
ui ·
n
ui
i
ds
By Leibniz’s rule we may write
2 3 n n
n
n
n
r 2 ui 2 dx = −
j r 2 2jk ui k ui dx −
r jjk ui k ui dx
Ci
jk=12 Ci
+
n
Ci
r 2 ui · jk=12 Ci
n
ui
i
ds
Applying Green’s formula to the second term of the above right-hand side and
Leibniz’s rule, we get
n
r 2 ui 2 dx
Ci
=−
−
=−
+
n
jk=12 Ci
Ci
r
2
n
n ui
ui
i
Ci
r
Ci
r
2
n
ui
n
·
n
ui
i
n
jk=12 Ci
n
ui
j r 2 2jk ui k ui dx +
2
ds +
n
jk=12 Ci
n
j r 2 2jk ui k ui dx +
−
·
k=12 Ci
n
n ui
ui
i
n
2jj ui k r 2 k ui dx
n
ui
i
n
ds
ui k r 2 dx +
ds
n
Ci
r 2 ui 2 dx
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Using the basis i i we see that
v
v
2 v v
2 v v
− v
v · =
− 2
i
i
i i i
i i
Since v/i = ±v/r, we obtain
v
v
2 v v 2 v v
− v
v · − 2
=
i
i
i r r
r i
Inserting this identity into the above one, we arrive at
n
r 2 ui 2 dx
Ci
=−
n
jk=12 Ci
+
Ci
r
2
n
j r 2 2jk ui k ui dx +
n
n
n
k=12 Ci
n
n
2 ui ui
2 ui ui
−
i r r
r 2 i
ui k r 2 dx +
n
Ci
r 2 ui 2 dx
ds
Passing to the limit in this identity we get
r 2 ui 2 dx
Ci
=−
+
jk=12 Ci
Ci
r
2
j r 2 2jk ui k ui dx +
k=12 Ci
2 ui ui
2 u u
− 2i i
i r r
r i
ds
ui k r 2 dx +
Ci
r 2 ui 2 dx
(16)
Here the boundary term should be understood as a duality bracket since by a
standard trace theorem we have
ui ui
∈ H1/2 i r
and therefore
2 ui 2 ui
∈ H−1/2 i r r 2
1/2
(see Appendix A of Dauge, 1988). Since
where H−1/2 is the dual of H−
1/2
1/2
, the above
the property ui /i ∈ H is equivalent to r 2 ui /i ∈ H−
considerations give a meaning to the duality pair
2
ui 2 ui
r
r 2
i
The first term is justified in the same way.
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Now in order to take into account the boundary and transmission conditions
in (11) we multiply the identity (16) by pi and take the sum on i = 1 2. This yields
pi
Ci
i=12
r 2 ui 2 dx = I1 + I2 + IF + I + IS (17)
where we have set
I1 = −
i=12
I2 =
pi
i=12
IF =
pi
jk=12 Ci
k=12 Ci
pi
i=12
Ci
j r 2 2jk ui k ui dx
ui k r 2 k ui dx
r 2 ui 2 dx
2 ui ui
2 u u
− 2i i ds
i r r
r i
i=12
2 u1 u1
2 u1 u1
2
ds
−
IS =
r p1
1 r r
r 2 1
S
I =
r 2
pi
since u2 = 0 on D and therefore u2 /r = 0 on D .
Since 2 = −1 on by the transmission conditions in (11) we see that
I =
2 u1
u
r 2 − u1 1 +
u
ds
r
r
r 2 1
By integration by parts (justified by density arguments as above) and Leibniz’s rule,
we get
I = −2J + K (18)
where we have set
u
u1 1 ds
r
r
u
K = −2 r 2−1 1 u1 ds
r
J =
r 2
Reminding the definition of u = u and using again Leibniz’s rule, we may
write
u1 J = r 2 u1 + u1 ds
r
r
u1 2
= r u1 u1 + u1 ds
r
r r
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Integrating by parts in the first term we get
2 2 u1 − u1 r u1 + r u1 ds
J =
r
r r
Using Leibniz’s rule we obtain
u1 2
J =
u1 − 2r 2−1 u1 u1
r 2 2 r
−r 2 u1 u1 ds
Since the first and the second terms of the above right-hand side are nonnegative
(due to the monotonicity of and S ) we obtain
J ≥ − r 2 u1 u1 ds
Inserting this estimate in the identity (18) we obtain
I ≤ 2 r 2 u1 u1 ds + K (19)
Using the boundary condition on S in (11) and the same arguments we
prove that
IS ≤ 2 r 2 S u1 u1 ds + KS (20)
S
where we have set
u
KS = −2 r 2−1 1 S u1 ds
r
S
In view to the identity (17) we have to estimate I + IS . By the estimates (19)
and (20) we remark that
I + IS ≤ J + K
(21)
where we have set
J = 2 r 2 u1 u1 ds + 2 r 2 S u1 u1 ds
S
K = K + KS First by the properties uu ≥ 0 and S uu ≥ 0, for any u ∈ (monotonicity), we remark that
J ≤ C0
u1 u1 ds +
S u1 u1 ds S
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where C0 = max0≤r≤R r 2 r, with R > 0 large enough such that the support of
is included into B0 R. Now using the variational formulation (9) with v = u
we get
pi
i=12
=
i
ui 2 + ui 2 dx +
u1 u1 ds +
S
S u1 u1 ds
fu dx
Consequently we have
u1 u1 ds +
S
S u1 u1 ds ≤
fu dx
and by Cauchy–Schwarz’s inequality and the estimate (10), we obtain
u1 u1 ds +
S
S u1 u1 ds f 20 This estimate in the above one leads to
J f 20 (22)
The quantity K cannot be estimated directly but need to be associated with the
internal term I1 . Indeed using the definition of I1 K KS and the boundary and
transmission conditions satisfied by u we see that
ui · ui dx
r
Ci
i=12
u1
u2
u
u
2−1 u1
+2 r
− p2
ds + 2 r 2−1 1 p1 1 ds
p1
r
1
1
r
1
S
I1 + K = −2
pi
r 2−1
(23)
Green’s formula yields (again justified by density arguments as above)
u u
u
u
r 2−1 i i ds =
ui r 2−1 i + ui · r 2−1 i dx
r i
r
r
Ci
Ci
Multiplying this identity by pi and adding the result we obtain
2−1 ui
2−1 ui
+ ui · r
dx
pi ui r
r
r
Ci
i=12
u1
u2
u
u
2−1 u1
= r
− p2
r 2−1 1 p1 1 ds
ds +
p1
r
1
1
r
1
S
This identity in (23) leads to
I1 + K = I3 + I4 + I5 (24)
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where we set
ui
dx
r
Ci
i=12
2−2 ui 2
I4 = 22 − 1
pi r
dx
r
Ci
i=12
2−1
ui
ui dx
I5 = 2
ui · − ui ·
pi r
r
r
Ci
i=12
I3 = 2
pi
ui r 2−1
It then remains to estimate I2 IF I3 I4 I5 . We start with I2 : By Cauchy–
Schwarz’s inequality and the fact that pi ui = Fi we have
I2 ≤ 2
r F 0C r −1 u
0C F 0C r −1 u
0C By the estimate (12) and Young’s inequality (2ab ≤ !a2 + !−1 b2 , for all
a b ! > 0) we obtain
I2 f 20 + C
r −1 u
20C (25)
In the same manner we have
I3 f 20 + r −1 u
20C (26)
Similarly we get
IF f 20 (27)
On the other hand we direcly have
I4 r −1 u
20C (28)
Furthermore using polar coordinates we see that
ui · ui
r
ui = r −3
− ui ·
r
ui
2
and consequently we arrive at
I5 r −1 u
20C (29)
These estimates show that we need to estimate r −1 u
0C with respect to
f 0 . For that purpose we set = 2 − 2 and remark that Green’s formula yields
−
Ci
ui ui r dx =
Ci
ui · ui r dx −
Ci
ui u r ds
i i
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Multiplying this identity by pi , summing on i = 1 2 and taking into account the
boundary and transmission conditions in (11) we obtain
pi ui ui r dx =
pi ui · ui r dx + 2 u1 u1 r ds
−
Ci
i=12
Ci
i=12
+
S
2 u1 S u1 r ds
By the monotonicity of and S , we obtain
−
i=12
pi
Ci
ui ui r dx ≥
i=12
pi
Ci
ui · ui r dx
Now by Leibniz’s rule we may write
ui · ui r dx =
ui 2 r dx +
ui ui · r dx
Ci
Ci
Ci
1
=
ui 2 r dx +
ui 2 · r dx
2 Ci
Ci
Applying Green’s formula in the second term of this right-hand side and remarking
that
r = 0 on Ci i
we get
Ci
1
u 2 r dx
2 Ci i
Ci
2 ui 2 r dx −
u 2 r −2 dx
=
2 Ci i
Ci
ui · ui r dx =
ui 2 r dx −
Inserting this identity into the above estimate we have proved that
2 pi
ui 2 r dx −
ui 2 r −2 dx ≤ −
pi ui ui r dx
2
C
C
Ci
i
i
i=12
i=12
Using Poincaré’s type inequality (3) from Lemma II.1 we arrive at
2
2
+ 12 −
pi ui 2 r −2 dx ≤ −
pi ui ui r dx
4
2
Ci
Ci
i=12
i=12
Now we remark that the factor 12 − 2 /4 is positive since 1 > > 1 − 1 .
Now by Cauchy–Schwarz’s inequality we obtain
pi ui ui r dx ≤ r 2 +1 F 0C r 2 −1 u
0C F 0C r 2 −1 u
0C −
i=12
Ci
(30)
(31)
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since /2 + 1 ≥ 0. By the estimate (12) we obtain
−
pi ui ui r dx f 0 r 2 −1 u
0C 59
(32)
Ci
i=12
This estimate in (31) yields
pi ui 2 r −2 dx f 20 i=12
Ci
and consequently (32) becomes
pi ui ui r dx f 20 −
i=12
Ci
Using these last two estimates in (30) we have proved that
ui 2 r dx f 20 (33)
Ci
In summary the estimates (21), (22), (24), the estimates (25) to (29) and the
estimate (33) into (17) lead to the estimate (15).
A similar regularity result near the corner P2 common to S and 13 can
be obtained by taking p1 = p2 and = 0 so that the first singular exponent is
"2 = /2, where is the interior opening of 1 at P2 . For the corner PN1
we also have a similar regularity result except that here we have Dirichlet
boundary conditions on the boundary segments and therefore the associated
singular exponents are different from those described in Sec. II but may be
characterized as the zeros of
p2 cos2 sin1 + p1 cos1 sin2 = 0
where 1 (resp. 2 ) is the interior opening of 1 (resp. 2 ) at PN1 . Let "N1 be the
first positive root of that equation.
In a neighbourhood of the other corners Pl l = 3 N1 + N2 − 2 l = N1 u is
solution of a standard Dirichlet problem whose regularity is well known (see Dauge,
1988; Grisvard, 1985; Kondrat’ev, 1967). Using these references we get that
rl l 2jk l ui ∈ L2 i ∀j k = 1 2
where l > 1 − "l "l = /l l being the interior opening of i at Pl with i = 1
i and l is a fixed cut-off function equal to 1 near Pl and with
or 2 such that Pl ∈ a sufficiently small support. Note that we further have
rl l 2jk l ui 0i f 0 jk=12
since the boundary value problem solved by l ui does not depend on and using
the estimate (10).
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Similarly to the other corners, for the corner P1 , we denote by "1 = 1 the first
positive singular exponent. Using a partition of unity we obtain the
Theorem IV.3. For any i j k = 1 2, one has
#i 2jk ui ∈ L2 i (34)
where
#1 =
N1
rj j j=1
#2 = r1 1
N1 +N2 −2
rj j j=N1
rj is the distance to Pj and 1 > j > 1 − "j , where "j were described above.
Furthermore one has
#i 2jk ui 0i f 0 (35)
i=12 jk=12
Theorem IV.4. Let u be the solution of (5). Then for any i j k = 1 2, one has
#i 2jk ui ∈ L2 i (36)
Proof. The estimates (10) and (35) mean that ui belongs to the Hilbert space
H 2 i defined by
H 2 i = v ∈ H 1 i #i D v ∈ L2 i ∀ ∈ 2 = 2 i = 1 2
where #i was defined above; this space being equipped with the norm
21
2
2
v
2
#i D v
0i i = v
1i +
=2
Since j ∈ 0 1 from Lemma 8.4.1.2 of Grisvard (1987), the embedding
of H 2 i into H 1 i is compact. Consequently we deduce with the uniform
estimates (10) and (35) that there exist a subsequence n and a function wi ∈
H 2 i such that
lim un i = wi in H 2 i weakly i = 1 2
n →0
and
lim un i = wi in H 1 i strongly i = 1 2
n →0
As all un are in H 1 , the limit w also belongs to H 1 . Hence we may pass
to the limit in (8) and following the arguments of Theorem I.8 of Brézis (1972), the
limit w is the solution u of our problem (5).
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V. 3D SIGNORINI TRANSMISSION PROBLEMS
WITH MIXED BOUNDARY CONDITIONS
In this section, we take a three dimensional prismatic domain = 2 × 0 1,
with 2 as in the previous section (here all 2D objects are specify by
the superscript2 ). The interface 2 of 2 induces a 3D interface = 2 × 0 1
for so that is still subdivided into two subdomains i i = 1 2. The boundary
2
2
of is still divided into two parts: S = S × 0 1 with S as in Sec. IV and
2
D is the rest of the boundary, i.e., D = D × 0 1 ∪ 2 × 0 ∪ 2 × 1
(see Fig. 2).
In this section, we want to describe the regularity of the solution u of (5). We
shall show that u has optimal regularity in the edge direction and has a singular
behaviour in the direction perpendicular to the edge. Similar results were obtained
for linear problems in Apel and Nicaise (1996) and Grisvard (1987).
For that purpose u is approximated by the sequence of u solution of (8). We
first show that u has optimal regularity in the edge direction:
Theorem V.1. For any i = 1 2, one has
2j3 ui ∈ L2 i ∀j = 1 2 3
(37)
Furthermore one has
3
2j3 ui 0i f 0 (38)
i=12 j=1
Proof. The solution u ∈ HD1 of (8) may be seen as a solution of a nonhomogeneous transmission problem in with interior data f ∈ L2 , Neumann
boundary data −S u1 ∈ H 1/2 S and transmission data − u1 ∈ H 1/2 Figure 2. The domain .
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(thanks to Lemma IV.1). Consequently by the arguments of Sec. II of Apel and
Nicaise (1996) we obtain the regularity (37). As before the difficulty is to obtain the
uniform dependence in .
In a first step we assume that f ∈ H01 . We multiply the first identity of (8)
by 23 ui and integrate the result on i . This gives
pi −ui + ui 23 ui dx =
pi
fi 23 ui dx
i
i=12
i=12
i
In this left-hand side by integration by part in x3 , which is allowed since
−ui + ui belongs to H 1 i , we get
−
pi
3 −ui + ui 3 ui dx +
pi
−ui + ui i3 3 ui ds
i
i=12
=
pi
i=12
i=12
i
i
fi 23 ui dx
We remark that the boundary terms are zero since on the lateral faces i3 = 0 and
on the top and bottom faces pi −ui + ui = fi = 0. Therefore the above identity
reduces to
pi
3 ui 3 ui dx =
pi fi 23 ui + 3 ui 2 dx
(39)
i=12
i
i=12
i
Now again the assumption on f implies that ui ∈ H 1 i and therefore the vector
field v = 23j ui 3j=1 belongs to
Hdiv i = v ∈ L2 i 3 div v ∈ L2 i By Green’s formula (see for instance the identity (I.2.17) of Girault and Raviart,
1986)
div v + v · dx = v · i
valid for v ∈ Hdiv i and in H 1 i , one obtains (by taking = 3 ui which
belongs to H 1 i thanks to (37)):
pi
3 ui 3 ui dx
i=12
=−
i
i=12
pi
3
i j=1
23j ui 2
ui
dx +
pi 3
3 ui i
i=12
Inserting this identity into (39) we have obtained
3
ui
pi
23j ui 2 dx −
pi 3
3 ui
i
i j=1
i=12
i=12
=−
pi fi 23 ui + 3 ui 2 dx
i=12
i
(40)
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Using the transmission conditions in (8) we see that
ui
ui
3 ui = −
3 u1
pi 3
pi 3
−
i
i
i=12
i=12
= 3 u1 3 u1 = u1 3 u1 3 u1 ≥ 0
≥ 0. This property in (40) leads finally to
since pi
i=12
3
i j=1
23j ui 2 dx ≤ −
pi
i=12
i
fi 23 ui dx
(41)
By Cauchy–Schwarz’s and Young’s inequalities we obtain
pi
i=12
3
i j=1
23j ui 2 dx ≤
1 1 pi fi 20i +
p 2 u 2 2 i=12
2 i=12 i 3 i 0i
which is equivalent to
i=12
pi
3
i j=1
23j ui 2 dx ≤
pi fi 20i (42)
i=12
and is nothing else than (38).
Since the estimate (42) depends only on the L2 -norm of f by the density of
1
H0 into L2 we get the result for any f in L2 .
This result and Theorem IV.3 allow to get the
Theorem V.2. For any i = 1 2 one has
#i 2jk ui ∈ L2 i ∀j k = 1 2
with the estimate
#i 2jk ui 0i f 0 (43)
(44)
i=12 jk=12
Proof. For almost all x3 ∈ 0 1, u · x3 may be seen as the solution of the 2D
problem: (a justifier)

2
pi −2 ui + ui = fi + pi 23 ui
in i i = 1 2





on 2 
u1 = u2




u1
u2
− p1
− p2
= u1 on 2 (45)
1
1




u1
2


−p1
= S u1 on S 


1


2
u = 0
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2
By Theorem IV.3, we get the regularities #i 2jk ui · x3 ∈ L2 i , and the estimate
(owing to (35)).
#i 2jk ui · x3 02 f 02 +
i=12 jk=12
i
23 ui 02 i=12
i
for almost all x3 ∈ 0 1.
We conclude by integrating the square of this estimate in x3 and using the
estimate (38).
Theorem V.3. Let u be the solution of (5). Then for any i = 1 2, one has
2j3 ui ∈ L2 i ∀j = 1 2 3
(46)
#i 2jk ui ∈ L2 i ∀j k = 1 2
(47)
Proof. Similar arguments as in the proof of Theorem IV.4.
VI. 2D SIGNORINI TRANSMISSION PROBLEMS WITH
SIGNORINI BOUNDARY CONDITIONS
Here we take a 2D polygonal domain as in Sec. IV. But we want to consider
the case when the interface meets the boundary inside the Signorini boundary
part, therefore, for the sake of simplicity, we suppose that S is the boundary of (see Fig. 1). We start by the description of the regularity near the origin which is
assumed to be a point of and of S . In this case the singular exponents of the
associated transmission problem are the zeros of
p2 cos1 sin2 + p1 cos2 sin1 = 0
(48)
Denote by 1 the smallest positive singular exponent.
Since in this case Poincaré’s inequality from Sec. II is no more applicable we use
another argument inspired from Proposition III.2.2 of Moussaoui (1992). Namely
we first show the H 2 regularity when the singular exponents of the transmission
problem are larger than 1. In the singular case, by a tricky change of variables we go
back to the first case and by inverse change of variables obtain the same regularity
result than in Theorem IV.4. We will see that this method is only valid for some
special maximal graphs S and , but including the standard case of Signorini.
We start with the regular case:
Theorem VI.1. Let u be the solution of (5). Assume that 1 > 1. Then for any i j
k = 1 2 one has
2jk ui ∈ L2 i where is the cut-off function introduced in Sec. IV.
(49)
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Proof. As in Sec. IV, we approximate u by a sequence of u ∈ HD1 solution
of (8). Denote for a moment u = u which is a solution of (11). By the
assumption 1 > 1, ui belongs to H 2 Ci , i = 1 2. Now we follow the arguments of
Theorem IV.2 but with = 0 (the case p1 = p2 and = 0 is exactly the case treated
by Grisvard (1975–1976) and Grisvard and Iooss (1976)). Therefore the identity (17)
holds but with I1 = I2 = 0, while the estimate (21) also holds with K = 0. Since the
estimates (22) and (27) are still valid we actually have the estimate
pi 2 ui 2 dx f 20 i=12
Ci
Passing to the limit in yields the result.
Note that in the above proof we have not used Poincaré’s inequality since, with
the notation from Theorem IV.2 I3 = I4 = I5 = 0, and the estimation of that terms
only required Poincaré’s inequality.
In the singular case we suppose that the domain is reduced to the truncated
sector C defined by
C = r 0 < r < R 0 < < for a fixed R > 0 and take S as the boundary of C. We further restrict ourselves
to the case when S and satisfies
yS x = S x ∀x ∈ y > 0
(50)
y x = x ∀x ∈ y > 0
(51)
Such a graph is of the form (see Fig. 3)


∅
if x < 0





 − 0 if x = 0
S x = 0
if 0 < x < S 


0 + if x = S 


∅
if x > S for some S ∈ 0 + (the case S = + corresponds to the Signorini condition).
Let K be the convex set defined by
K = v ∈ H 1 C 0 ≤ vS ≤ S 0 ≤ v1 ≤ The weak formulation (7) of (5) is here equivalent to find a solution u ∈ K of
au v − u ≥ fv − u dx ∀v ∈ K
(52)
C
where a is the bilinear form defined as in (6) but replacing the domain by the
domain C.
Theorem VI.2. Under the above restrictions on , S and , the solution u of (5)
with D = ∅, satisfies
r 2jk ui ∈ L2 i ∀i j k = 1 2
when > 1 − 1 (1 being described in the beginning of this section).
(53)
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Chikouche, Mercier, and Nicaise
Figure 3. The graph S .
Proof. As in Proposition III.2.2 of Moussaoui (1992) we make the change of
variables # = r and = with some > 0. This change of variables transforms
the sector C into the sector
C = # 0 < # < R 0 < < We denote by S the boundary of C and = # 1 0 < # < R . Setting
1
U# = u# , U belongs to the convex K defined by
K = v ∈ H 1 C 0 ≤ vS ≤ S v1 = v2 0 ≤ v1 ≤ From the relations
u
−1 U
= # r
#
1 u
1 U
−1
= #
r # we directly see that the weak formulation (52) becomes : U ∈ K is solution of
2
pi
i=1
≥
Ci
C
UV − Udx +
2
i=1
1
f# V − U−2 #
pi
Ci
21−
2
Ui Vi − Ui −2 #
dx ∀V ∈ K 21−
2
dx
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67
which is the weak formulation of


−pi Ui = Fi



U1 = U2


 U1
U2
− p2
∈ U1 − p1


1
1




U

−pi i ∈ S U1 i
in Ci i = 1 2
on on (54)
on S where F ∈ L2 C is given by
Fi # = −2 #
21−
1
1
fi # − p i ui # Remark further that by change of variables we easily check that
F 0C f 0C + u
0C (55)
For this new problem the singular exponents from (48) are here the zeros of
p2 cos 1 sin 2 + p1 cos 2 sin 1 = 0
As i = i , we deduce that
=
where is a zero of (48). Choosing now such that
1
>1
problem (54) is in the setting of Theorem VI.1. From this theorem we deduce that
(in Cartesian coordinates)
2jk Ui ∈ L2 Ci ∀i j k = 1 2
(56)
This obviously implies that
#$ 2jk Ui ∈ L2 Ci ∀i j k = 1 2
(57)
with $ = − 1 + / which will be positive if is chosen such that > 1 − (which is always possible). Let us fix a radial cut-off function such that # = 1
for 0 < # < R and # ≡ 0 for R < #, for some R < R < R . We now write
#−1 k Ui # = −#−1
#
Ui s ds
# k
120028843_PDE29_01&02_R2_011404
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Chikouche, Mercier, and Nicaise
By (57) and the H 1 regularity of U , for almost all the function /#
k Ui · belongs to L2$+1/2 + (see Sec. II). With the notation from
Grisvard (1985, p. 28), the above identity means that
−1
Ui · #
# k Ui # = L
# k
and by Hardy’s inequality we obtain that
U
·
#−1 k Ui # L2$+1/2 + ≤ $−1 i
# k
L2$+1/2 + since $ > 0. Integrating the square of this identity on we get
#$−1 k Ui ∈ L2 Ci ∀i k = 1 2
From the definition of $ this regularity means that
#
−1
k Ui ∈ L2 Ci ∀i k = 1 2
By inverse change of variables we obtain
2−2
r 2−2 ui 2 r dr d ≤
# Ui 2 # d# d < Ci
Ci
(58)
(59)
thanks to (58). Similarly the expressions
2
2 ui
−2 Ui
2 2−2 Ui
+
=
−
1r
r
r 2
#
#2
1 2 ui
1 2 Ui
= 2 r 2−2
r r
# #
2
1 2 u i
2 2−2 1 Ui
=
r
r 2 2
#2 2
and (58) lead to
2 2 2 2 1 ui 1 2 u i 2
2 ui r 2 +
+ 2 2 r dr d < r
r r r Ci
(60)
The regularities (59) and (60) and the estimate
2 u 1 2 u 1 2 u 1
2
+ u
jk u 2 + +
r
r r r 2 2 r
lead to the conclusion.
Remark VI.3. Clearly the above result and the arguments of Sec. V allow to
describe the edge regularity of a 3D transmission problem with Signorini boundary
condition along the edge.
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Apel, Th., Nicaise, S. (1998). The finite element method with anisotropic mesh
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Grisvard, P., Iooss, G. (1976). Problème aux Limites Unilatéral dans des Domaines
non Réguliers, Publications des Séminaires de Mathématiques de l’Université
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conical or angular points. Trans. Moscow Math. Soc. 16:209–292.
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Mercier, D., Nicaise, S. (2002). Minimal regularity of the solution of some boundary
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02–08. Submitted to Math. Nachrichten.
Moussaoui, M. (1992). Régularité des solutions d’un problème mêlé DirichletSignorini dans un domaine polygonal plan. Comm. PDE 17:805–826.
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Nicaise, S., Sändig, A.-M. (1994). General interface problems I, II. Math. Meth.
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Received July 2002
Accepted May 2003
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120028843
Regularity of the Solution of Some Unilateral Boundary Value Problem in
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