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Comptes Rendus
Mécanique
Yurij A. Alkhutov and Gregory A. Chechkin
The Meyer’s estimate of solutions to Zaremba problem for second-order
elliptic equations in divergent form
Volume 349, issue 2 (2021), p. 299-304
<https://doi.org/10.5802/crmeca.87>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
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Comptes Rendus
Mécanique
2021, 349, n 2, p. 299-304
https://doi.org/10.5802/crmeca.87
Short paper / Note
The Meyer’s estimate of solutions to Zaremba
problem for second-order elliptic equations in
divergent form
L’estimation de Meyer pour les solutions au problème de
Zaremba pour les équations elliptiques du second ordre
sous forme divergente
Yurij A. Alkhutova and Gregory A. Chechkin∗, b, c , d
a A.G. and N.G. Stoletov Vladimir State University, Gor’kogo St., 87, Vladimir, 600000,
Russia
b M.V. Lomonosov Moscow State University, Leninskie Gory, 1, Moscow, 119991,
Russia
c Institute of Mathematics with Computing Center - Subdivision of the Ufa Federal
Research Center of Russian Academy of Science, Chernyshevskogo st., 112, Ufa,
450008, Russia
d Institute of Mathematics and Mathematical Modeling, Pushkin st. 125, Almaty,
050010, Kazakhstan
E-mails: yurij-alkhutov@yandex.ru (Yu. A. Alkhutov), chechkin@mech.math.msu.su
(G. A. Chechkin)
Dedicated to our wives.
Blessed is the man that hath a wise wife. . .
(Sirach 25:11, translated from the Orthodox Bible)
Abstract. In this paper we obtain an estimate for the increased integrability of the gradient of the solution
to the Zaremba problem for divergent elliptic operator in a bounded domain with nontrivial capacity of the
Dirichlet boundary conditions.
Résumé. Dans cet article, nous obtenons une estimation de l’intégrabilité accrue du gradient de la solution
du problème de Zaremba pour un opérateur elliptique divergent dans un domaine borné avec une capacité
non triviale des conditions aux limites de Dirichlet.
∗Corresponding author.
ISSN (electronic) : 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/
300 Yurij A. Alkhutov and Gregory A. Chechkin
Keywords. Meyers estimates, Mixed problem, Embedding theorems, Capacity, Rapidly alternating type of
boundary conditions.
Mots-clés. Estimations de Meyers, Problème mixte, Théorèmes d’intégration, Capacité, Type de conditions
aux limites à alternance rapide.
2020 Mathematics Subject Classification. 76A15, 35K61, 35A01, 35A02.
Manuscript received 21st April 2021, revised 12th May 2021, accepted 17th May 2021.
1. Introduction
In this paper we estimate solutions to the Zaremba problem for elliptic equations in bounded
Lipschitz domain D ∈Rn , where n > 1, of the form
L u := div(a(x)∇u) (1)
with uniformly elliptic measurable and symmetric matrix a(x) = {ai j (x)}, i.e. ai j = a j i and
α−1|ξ|2 ≤
n∑
i , j=1
ai j (x)ξiξ j ≤α|ξ|2 for almost all x ∈ D and for all ξ ∈Rn . (2)
Below we assume that the set F ⊂ ∂D is closed (possibly disconnected and having nontrivial
microstructure, n −2 < dimF ≤ n −1) and denote G = ∂D \ F . Consider the Zaremba problem
L u = div f in D, u = 0 on F, ∂u
∂ν
= 0 on G , (3)
where ∂u/∂ν is an outer conormal derivative of u, and the components of the vector-function
f = ( f1, . . . , fn) are functions from L2(D). We define a solution to problem (3) in the following
way. Denote by W 12 (D,F ) the completion of the set of infinitely differentiable in the closure of D
functions, vanishing in the vicinity of F , by the norm
∥ u ∥W 12 (D,F )=
(∫
D
u2 dx +
∫
D
|∇u|2 dx
)1/2
.
The function u ∈W 12 (D,F ) is a solution to problem (3) if the following integral identity holds∫
D
a∇u ·∇ϕdx =
∫
D
f ·∇ϕdx (4)
for all functions ϕ ∈W 12 (D,F ).
Note that for the Laplace equation in the case of sufficiently smooth boundary of the domain
D and the boundary of the Neumann data G , classical solvability of problem (3) is proved in [1] by
the potential theory methods. One of the first papers on properties of solutions to the Zaremba
problem for nondivergent elliptic equations with regular coefficients is [2]. In this paper, in
particular, the author discovered that at the junction of the Dirichlet and Neumann data the
smoothness of solutions is lost.
In the case of homogeneous Dirichlet problem for (3) with right-hand side f ∈ Lp (D), where
p > 2, the increased integrability of the gradient of solutions to divergent uniformly elliptic
equations with measurable coefficients on the plane follows from the results of [3]. Later, the
same problem in multidimensional domain with sufficiently regular boundary was considered
in [4].
Our interest is in the increased integrability of the gradient of a solution to problem (3).
The condition on the structure of the support of the Dirichlet data F plays the key role. For
C. R. Mécanique — 2021, 349, n 2, 299-304
Yurij A. Alkhutov and Gregory A. Chechkin 301
the formulation of the result we need the notion of capacity. Let us define the capacity Cp (K ),
1 < p < n, for the compact set K ⊂Rn by the identity
Cp (K ) = inf
{∫
Rn
|∇ϕ|p dx : ϕ ∈C∞0 (Rn),ϕ≥ 1 on K
}
.
Here and throughout, B x0r is the open ball with radius r centered in the point x0, and mesn−1(E)
is the (n −1)-measure of the set E ⊂ ∂D . Taking p = 2n/(n +2) as n > 2 and p = 3/2 as n = 2 we
assume one of the following conditions to be satisfied: for an arbitrary point x0 ∈ F as r ≤ r0 either
the inequality
Cp (F ∩B x0r ) ≥ c0r n−p , (5)
or the inequality
mesn−1(F ∩B x0r ) ≥ c0r n−1 (6)
holds true, where the positive constant c0 does not depend on x0 and r . The condition (6) is
stronger then (5), but it is more visual. Note that under one of these conditions the function
v ∈W 12 (D,F ) satisfies the Friedrichs inequality∫
D
v2 dx ≤C
∫
D
|∇v |2 dx, (7)
which leads to the unique solvability of the problem (3) because of the Lax–Milgram theorem [5].
If the condition (6) holds, the inequality (7) is well known, since mesn−1(F ) > 0, and for the
condition (5) this inequality follows, for instance, from the results of the monograph [6]. Let us
explain this point in more detail. Denote by Qd the open cube with edge length d and faces
parallel to the coordinate axes. Assume that the Lipschitz domain D has the diameter d and
D ⊂ Qd . Let us define the capacity Cp (K ,Q2d ) of the compact K ⊂ Qd with respect to the cube
Q2d , by the formula
Cp (K ,Q2d ) = inf
{∫
Q2d
|∇ϕ|p dx : ϕ ∈C∞0 (Q2d ),ϕ≥ 1 on K
}
. (8)
The theorem from [6, Section 14.1.2] and the comments to the results of the Chapter 14 about
Lipschitz domains lead, in particular, to the inequality∫
D
v2 dx ≤ C (n,D)d
n
C2(F,Q2d )
∫
D
|∇v |2 dx, (9)
for functions v ∈ W 12 (D,F ). Then we use the condition (5). First of all note that for 1 < p < 2 the
definition of the capacity Cp (K ,Q2d ) and the Hölder inequality gives the estimate
Cp (K ,Q2d ) ≤ |Q2d |(2−p)/2C p/22 (K ,Q2d ), (10)
where |Q2d | is the n-dimensional measure of the cube Q2d . Now we use the fact for 1 < p < n (see
Proposition 4 from [7]) there exists a positive constant γ(n, p) ≥ 1 such that
Cp (K ) ≤Cp (K ,Qd ) ≤ γCp (K ). (11)
The condition (5) for 1 < p < 2 leads to Cp (F ) > 0. Consequently C2(F,Q2d ) > 0 because of (10)
and (11). Using (9), we get the Friedrichs inequality (7).
2. Main result
To formulate the main result we give the definition of Lipschitz domain D in more detail.
A domain D will be called a Lipschitz domain, if for any point x0 ∈ ∂D there exists an open cube
Q centered in x0, faces of which are parallel to the coordinate axes, the length of the cube edges
are independent of x0, and in some Cartesian coordinate system with origin in x0 the set Q ∩∂D
is a graph of the Lipschitz function xn = g (x1, . . . , xn−1) with the Lipschitz constant independent
C. R. Mécanique — 2021, 349, n 2, 299-304
302 Yurij A. Alkhutov and Gregory A. Chechkin
of x0. Denote the length of the edge of such cubes by 2R0, and the Lipschitz constant of the
respective functions g by L. For definiteness, we assume that the set Q ∩D is located above the
graph of the function g , and the constant r0 from the conditions (5) and (6) is less than or equal
to the constant R0.
Theorem 1. If f ∈ L2+δ0 (D) with δ0 > 0, then there exist positive constants δ(n,δ0) < δ0 and C
such that for the solution to problem (3) the estimate∫
D
|∇u|2+δdx ≤C
∫
D
| f |2+δdx (12)
holds, where C depends only onδ0, the dimension n, the ellipticity constantα from (2), the constant
c0 from (5) and (6), and also the constants L and R0 from the definition of the Lipschitz domain D.
3. Proof
The proof of the Theorem is based on the internal estimates of increased summation and on the
estimates of increased summation in the vicinity of the boundary for the gradient of solutions to
problem (3). First, we establish the estimate for the gradient of solutions to problem (3) in the
vicinity of the boundary of the domain D . Here we use the technique of the straightening of the
boundary ∂D . Assuming QR0 = {x : |xi | < R0, i = 1, . . . ,n}, we consider an arbitrary point x0 ∈ ∂D
and such local Cartesian system with origin in x0, that the part of the boundary ∂D lying in the
cube QR0 , is defined in this coordinate system by the equation xn = g (x ′), where x ′ = (x1, . . . , xn−1),
and g is a Lipschitz function with the Lipschitz constant L. Moreover the domain DR0 =QR0 ∩D
is situated on the set of points xn > g (x ′). Next, in QR0 we pass to the new coordinate system by
nondegenerate transformation
y ′ = x ′, yn = xn − g (x ′). (13)
It is clear that the part of the boundary QR0 ∩∂D transforms to a hyperplane
PR0 = {y : |yi | < R0, i = 1, . . . ,n −1, yn = 0}.
Also it is easy to see that the image of the domain QR0 contains the cube
KR0 = {y : |yi | < (1+
p
n −1L)−1R0, i = 1, . . . ,n}. (14)
In addition in the semi-cube K +R0 = KR0 ∩ {y : yn > 0} which is contained in the image of the
domain D ∩QR0 , problem (3) has the form
L̃ u = div f̃ in K +R0 , u = 0 on F̃R0 ,
∂u
∂ν̃
= 0 on G̃R0 . (15)
We keep the same notation for its solution. Here
L̃ u := div(b(y)∇u) (16)
is the elliptic operator with symmetric matrix b(y) = {bi j (y)} satisfying
β−1|ξ|2 ≤
n∑
i , j=1
bi j (x)ξiξ j ≤β|ξ|2 for almost all y ∈ K +R0 and for all ξ ∈Rn ,
where the constant β depends only on the constantα from (2) and the Lipschitz constant L of the
function g . The right-hand side of the equation has the form
f̃ (y) = ( f̃1(y), . . . , f̃n(y)), where f̃i (y) = fi (y ′, yn + g (y ′)) for i = 1, . . . ,n −1,
f̃n(y) =
n−1∑
i=1
∂g (y ′)
∂yi
fi (y
′, yn + g (y ′))+ fn(y ′, yn + g (y ′)).
(17)
C. R. Mécanique — 2021, 349, n 2, 299-304
Yurij A. Alkhutov and Gregory A. Chechkin 303
The sets F̃R0 and G̃R0 satisfy F̃R0 = F̃ ∩PR0 ∩KR0 and G̃R0 = G̃ ∩PR0 ∩KR0 , where F̃ , G̃ are the
images of the sets F ∩QR0 and G ∩QR0 respectively, and ∂u/∂ν̃ is an outer conormal derivative of
the function u, connected with the operator (16).
Consider the even continuation of the function u, satisfying (15), with respect to the hyper-
plane {y : yn = 0}. We keep the same notation for the continued function. This function satisfies
the following relation:
L̃1u = divh in KR0 \ F̃R0 , u = 0 on F̃R0 . (18)
Here
L̃1u := div(c(y)∇u),
the positive definite matrix c = {ci j (y)} is such that the elements c j n(y) = c j n(y) for j 6= n are
odd continuations of the elements b j n(y) from (16), and all others elements ci j (y) are even
continuations of bi j (y). The vector-functions h = (h1, . . . ,hn) in (18) are defined in an analogous
way: the components hi (y) for i = 1, . . . ,n − 1 are even continuations of the components f̃i (y)
from (15), and hn(y) is odd continuation of f̃n(y). It is clear that the solution to problem (18) is a
function u ∈W 12 (KR0 ), satisfying the integral identity (see (4))∫
KR0
c(y)∇u ·∇ϕdy =
∫
KR0
h ·∇ϕdy (19)
for all test-functions ϕ ∈ W 12 (KR0 ,FR0 ). Here W 12 (KR0 ,FR0 ) is the closure of the set of infinitely
differentiable functions in the closure of KR0 , vanishing in the vicinity of ∂KR0 and FR0 , by the
norm
∥ u ∥W 12 (KR0 ,FR0 )=
(∫
KR0
u2 dx +
∫
KR0
|∇u|2 dx
)1/2
.
Denote by Q y0R the open cube centered in y0, the edge length equal to 2R, and faces parallel to the
coordinate axes. Below we assume that
y0 ∈ KR0/2 \∂KR0/2, where R ≤ 12 dist(y0,∂KR0/2).
and denote ∫
−
Q
y0
R
f dx = 1|Q y0R |
∫
Q
y0
R
f dx,
where |Q y0R | is the n-dimensional measure of the cube Q
y0
R .
By means of the conditions (5) or (6), the appropriate choice of the test-function in (19),
the embedding theorem (see [6, Section 14.1.2] and the estimate of the Proposition 4 from [6,
Section 13.1.1]), and also the Poincaré–Sobolev inequality with p from (5), we obtain(∫
−
Q
y0
R
|∇u|2 dy
)1/2
≤C (n,α,c0,L)
((∫
−
Q
y0
2R
|∇u|p dy
)1/p
+
(∫
−
Q
y0
2R
|h|2 dy
)1/2)
.
From this estimate (holds true for all considered cubes Q y0R ) and the generalized Gehring Lemma
(see [8], [9], and also [10, Chapter 7]), bearing in mind the edge length of the cube KR0 (see (14)),
we get under the condition h ∈ L2+δ0 (KR0 ) with δ0 > 0, the estimate∫
KR0/4
|∇u|2+δdy ≤C (n,α,δ0,c0,L,R0)
∫
KR0/2
|h|2+δdy
with positive constant δ= δ(n,δ0). This estimate can be rewritten because of the evenness of the
function u with respect to the hyperplane {y : yn = 0}, in the form (see (15))∫
K +R0/4
|∇u|2+δdy ≤C (n,α,δ0,c0,L,R0)
∫
K +R0/2
| f̃ |2+δdy. (20)
C. R. Mécanique — 2021, 349, n 2, 299-304
304 Yurij A. Alkhutov and Gregory A. Chechkin
We make the inverse transformation to (13). It is easy to see that the preimage of the semi-cube
K +R0/2 is contained in the set DR0 , and the preimage of the semi-cube K
+
R0/4
contains the set DθR0 ,
where θ = θ(n,L) > 0. Keeping in mind (17), by means of (20) we have∫
DθR0
|∇u|2+δdx ≤C (n,α,δ0,c0,L,R0)
∫
DR0
| f |2+δdx.
Passing to the Cartesian system of coordinates with the origin in x0 ∈ ∂D , which we used from the
very beginning of the reasoning, we get∫
D∩Qx0
θR0
|∇u|2+δdx ≤C (n,α,δ0,c0,L,R0)
∫
D∩Qx0R0
| f |2+δdx.
Since x0 ∈ ∂D is an arbitrary boundary point and the boundary ∂D is compact, one can find such
finite cover of ∂D , that the closed set
Dθ1R0 = {x ∈ D : dist(x,∂D) ≤ θ1R0}, θ1 = θ1(n,L) > 0
is contained in the union of the sets D ∩Qxi
θR0
, where xi ∈ ∂D . Then, summing the inequalities∫
D∩Qxi
θR0
|∇u|2+δdx ≤C (n,α,δ0,c0,L,R0)
∫
D∩QxiR0
| f |2+δdx,
we came to the estimate∫
Dθ1R0
|∇u|2+δdx ≤C (n,α,δ0,c0,L,R0)
∫
D
| f |2+δdx.
The internal estimate ∫
D\Dθ1R0
|∇u|2+δdx ≤C (n,α,δ0,R0)
∫
D
| f |2+δdx
is well known and follows, for instance, from the paper [4]. As a result, combining the last two
inequalities, we came to (12).
Acknowledgments
The results of YuAA in Section 3 is supported in part by RFBR (project 19-01-00184). The results
of GAC in Sections 1 and 2 is partially supported by RSF (project 20-1120272).
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