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A counterexample to the Hopf-Oleinik lemma (elliptic case)

TL;DR: In particular, this article showed that for convex domains the Dini-type assumption is the necessary and sufficient condition which guarantees the Hopf-Oleinik type estimates for the boundary of $Omega.
Abstract: We construct a new counterexample confirming the sharpness of the Dini-type condition for the boundary of $\Omega$. In particular, we show that for convex domains the Dini-type assumption is the necessary and sufficient condition which guarantees the Hopf-Oleinik type estimates.

Summary (2 min read)

Introduction

  • It shows that for convex domains, the C 1;Dini assumption on @ is the necessary and sufficient condition providing the estimates of Hopf–Oleinik type.
  • The significant result in this field is the Hopf–Oleinik lemma, known also as the “boundary point principle”.

440 DARYA E. APUSHKINSKAYA AND ALEXANDER I. NAZAROV

  • Among these directions are the extension of the class of operators and the class of solutions, as well as the weakening of assumptions on the boundary.
  • The reduction of the assumptions on the boundary of up to C 1;Dini-regularity was realized for various elliptic operators in the papers [Widman 1967; Himčenko 1970; Lieberman 1985] (see also [Safonov 2008]).
  • The authors emphasize that in their counterexample the Dini condition fails for the supremum of F.x0/=jx0j, while in all the previous results of this kind, it fails for the infimum of F.x0/=jx0j.
  • In other words, the authors show that violating the Dini condition just in one direction causes the failure of the Hopf–Oleinik lemma.
  • The authors also adopt the convention regarding summation with respect to repeated indices.

442 DARYA E. APUSHKINSKAYA AND ALEXANDER I. NAZAROV

  • The authors say that a function satisfies the Dini condition at zero if j .r/j6 C .r/; and belongs to the class D1.
  • (b) ı1.r/ satisfies the Dini condition at zero if and only if ı.r/ satisfies the Dini condition at zero.
  • 0, the authors can easily see that the domain is contained in a dihedral wedge with the angle less than and the edge going through the origin.
  • For this case, the statement of Theorem 4.2 is proved already in [Apushkinskaya and Nazarov 2000, Theorem 4.3].

444 DARYA E. APUSHKINSKAYA AND ALEXANDER I. NAZAROV

  • Unlike the natural properties (i)–(ii), assumption (iii) is a rather “technical” one.
  • Now, taking into account properties (i) and (ii) of the space X . /, the authors immediately get a contradiction with inequalities (13).
  • An application of Lemma 2.5 finishes the proof.
  • (b) Let Qbi".x 0/Diu.x 0/ < bi.x0/Diu.x 0/ 6 0. In this case, the authors decrease all the coefficients Qbi".x0/ corresponding to the negative summands such that the sums bi"Diu and b iDiu become equal.

446 DARYA E. APUSHKINSKAYA AND ALEXANDER I. NAZAROV

  • The proof is similar in spirit to [Apushkinskaya and Ural0tseva 1995, Lemma 1].
  • Observe also that B 0.Qz/ , otherwise the authors get a contradiction with the definition of ı.r/.

454 DARYA E. APUSHKINSKAYA AND ALEXANDER I. NAZAROV

  • Let the assumptions of Theorem 4.1 hold, and suppose ı.r/D max jx0j6r F.x0/ jx0j does not satisfy the Dini condition at zero.
  • The authors would like to thank the anonymous referee for essential comments.

456 DARYA E. APUSHKINSKAYA AND ALEXANDER I. NAZAROV

  • References [Aleksandrov 1960] A. D. Aleksandrov, “Certain estimates for the Dirichlet problem”, Dokl.
  • In Russian; translated in St. Petersburg Math.

458 DARYA E. APUSHKINSKAYA AND ALEXANDER I. NAZAROV

  • Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices.
  • APDE peer review and production are managed by EditFlow® from MSP.

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ANALYSIS & PDE
msp
Volume 9 No. 2 2016
DARYA E. APUSHKINSKAYA AND ALEXANDER I. NAZAROV
A COUNTEREXAMPLE TO THE HOPF–OLEINIK LEMMA
(ELLIPTIC CASE)

ANALYSIS AND PDE
Vol. 9, No. 2, 2016
dx.doi.org/10.2140/apde.2016.9.439
msp
A COUNTEREXAMPLE TO THE HOPF–OLEINIK LEMMA (ELLIPTIC CASE)
DARYA E. APUSHKINSKAYA
AND
ALEXANDER I. NAZAROV
Dedicated to Professor M.V. Safonov
We construct a new counterexample to the Hopf–Oleinik boundary point lemma. It shows that for convex
domains, the C
1;Dini
assumption on @ is the necessary and sufficient condition providing the estimates
of Hopf–Oleinik type.
1. Introduction
The influence of the properties of a domain on the behavior of a solution is one of the most important
topics in the qualitative analysis of partial differential equations.
The significant result in this field is the Hopf–Oleinik lemma, known also as the “boundary point
principle”. This celebrated lemma states:
Lemma.
Let
u
be a nonconstant solution to a second-order homogeneous uniformly elliptic nondivergence
equation with bounded measurable coefficients, and let
u
attend its extremum at a point
x
0
located on the
boundary of a domain
R
n
. Then
.@u=@n/.x
0
/
is necessarily nonzero provided that
@
satisfies the
proper assumptions at x
0
.
This result was established in a pioneering paper of S. Zaremba [1910] for the Laplace equation in
a 3-dimensional domain
having an interior touching ball at
x
0
and generalized by G. Giraud [1932;
1933] to equations with Hölder-continuous leading coefficients and continuous lower-order coefficients in
domains belonging to the class C
1
with ˛ 2 .0; 1/.
Notice that a related assertion about the negativity on
@
of the normal derivative of the Green’s
function corresponding to the Dirichlet problem for the Laplace operator was proved much earlier for
2-dimensional smooth domains by C. Neumann [1888] (see also [Korn 1901]). The result of [Neumann
1888] was extended for operators with lower-order coefficients by L. Lichtenstein [1924]. The same
version of the boundary point principle for the Laplacian and 3-dimensional domains satisfying a more
flexible interior paraboloid condition was obtained by M. V. Keldysch and M. A. Lavrent’ev [1937].
A crucial step in studying the boundary point principle was made by E. Hopf [1952] and O. A. Ole
˘
ınik
[1952], who simultaneously and independently proved the statement for the general elliptic equations
with bounded coefficients and domains satisfying an interior ball condition at x
0
.
Later the efforts of many mathematicians were focused on the generalization of the boundary point
principle in several directions (for the details, we refer the reader to [Alvarado et al. 2011; Alvarado
MSC2010: 35J15, 35B45.
Keywords: elliptic equations, Hopf–Oleinik lemma, Dini continuity, counterexample.
439

440 DARYA E. APUSHKINSKAYA AND ALEXANDER I. NAZAROV
2011] and references therein). Among these directions are the extension of the class of operators and the
class of solutions, as well as the weakening of assumptions on the boundary.
The widening of the class of operators to singular/degenerate ones was made in the papers [Kamynin and
Him
ˇ
cenko 1975; 1977; Alvarado et al. 2011], while the uniform elliptic operators with unbounded lower-
order coefficients were studied in [Safonov 2010; Nazarov 2012] (see also [Nazarov and Uraltseva 2009]).
We mention also the publications [Tolksdorf 1983; Mikayelyan and Shahgholian 2015], where the boundary
point principle was established for a class of degenerate quasilinear operators including the
p
-Laplacian.
We note that before 2010, all the results were formulated for classical solutions, i.e.,
u 2 C
2
./
. The
class of solutions was expanded in [Safonov 2010] to strong generalized solutions with Sobolev’s second-
order derivatives. The latter requirement seems to be natural in the study of nondivergent elliptic equations.
The reduction of the assumptions on the boundary of
up to
C
1;Dini
-regularity was realized for various
elliptic operators in the papers [Widman 1967; Him
ˇ
cenko 1970; Lieberman 1985] (see also [Safonov
2008]). A weakened form of the Hopf–Oleinik lemma (the existence of a boundary point
x
1
in any
neighborhood of
x
0
and a direction
`
such that
.@u=@`/.x
1
/ ¤ 0
) was proved in [Nadirashvili 1983] for
a much wider class of domains including all Lipschitz ones. We mention also the paper [Sweers 1997],
where the behavior of superharmonic functions near the boundary of a 2-dimensional domain with corners
is described in terms of the main eigenfunction of the Dirichlet Laplacian.
The sharpness of some requirements was confirmed by corresponding counterexamples constructed in
[Widman 1967; Him
ˇ
cenko 1970; Kamynin and Him
ˇ
cenko 1975; Safonov 2008; Alvarado et al. 2011;
Nazarov 2012]. In particular, the counterexamples from [Widman 1967; Him
ˇ
cenko 1970; Safonov 2008]
show that the Hopf–Oleinik result fails for domains lying entirely in non-Dini paraboloids.
The main result of our paper is a new counterexample (see Theorem 4.2) showing the sharpness of the
Dini condition for the boundary of
. The simplest version of this counterexample can be formulated as
follows:
Counterexample.
Let
be a convex domain in
R
n
, let
@
in a neighborhood of the origin be described
by the equation
x
n
D F.x
0
/
with
F > 0
and
F.0/ D 0
, and let
u 2 W
2
n;loc
./ \C./
be a solution of the
uniformly elliptic equation
a
ij
.x/D
i
D
j
u D 0 in :
Suppose also that uj
@
vanishes at a neighborhood of the origin. If , in addition, the function
ı.r / D sup
jx
0
j6r
F.x
0
/
jx
0
j
is not Dini-continuous at zero, then .@u=@n/.0/ D 0.
Thus, it turns out that for convex domains, the Dini-continuity assumption on
ı.r /
is necessary and
sufficient for the validity of the boundary point principle. We emphasize that in our counterexample the
Dini condition fails for the supremum of
F.x
0
/=jx
0
j
, while in all the previous results of this kind, it fails
for the infimum of
F.x
0
/=jx
0
j
. In other words, we show that violating the Dini condition just in one
direction causes the failure of the Hopf–Oleinik lemma.

A COUNTEREXAMPLE TO THE HOPFOLEINIK LEMMA (ELLIPTIC CASE) 441
Notation and conventions. Throughout the paper we use the following notation:
x D .x
0
; x
n
/ D .x
1
; : : : ; x
n1
; x
n
/ is a point in R
n
.
R
n
C
D fx 2 R
n
W x
n
> 0g.
jxj; jx
0
j are the Euclidean norms in the corresponding spaces.
E
denotes the characteristic function of the set E R
n
.
is a bounded domain in R
n
with boundary @.
P
r;h
. Nx
0
/ D fx 2 R
n
W jx
0
Nx
0
j < r; 0 < x
n
< hg and P
r
. Nx
0
/ D P
r;r
. Nx
0
/.
P
r;h
D P
r;h
.0/ and P
r
D P
r
.0/.
B
r
.x
0
/ is the open ball in R
n
with center x
0
and radius r; B
r
D B
r
.0/.
For r
1
< r
2
, we define the annulus B.x
0
; r
1
; r
2
/ D B
r
2
.x
0
/ nB
r
1
.x
0
/.
v
C
D maxfv; 0g and v
D maxfv; 0g.
kk
1;
denotes the norm in L
1
./.
We adopt the convention that the indices
i
and
j
run from
1
to
n
. We also adopt the convention
regarding summation with respect to repeated indices.
D
i
denotes the operator of (weak) differentiation with respect to x
i
.
D D .D
0
; D
n
/ D .D
1
; : : : ; D
n1
; D
n
/.
L is a linear uniformly elliptic operator with measurable coefficients
Lu a
ij
.x/D
i
D
j
u Cb
i
.x/D
i
u; I
n
.a
ij
.x//
1
I
n
; (1)
where I
n
is the n n identity matrix. We define b.x/ D .b
1
.x/; : : : ; b
n
.x//.
We use the letters
C
and
N
(with or without indices) to denote various constants. To indicate that,
say, C depends on some parameters, we list them in parentheses: C./.
Definition 1.1. We say that a function W Œ0; 1 ! R
C
belongs to the class D
1
if
is increasing, .0/ D 0, and .1/ D 1;
.t /=t is summable and decreasing.
Remark 1.2.
Our assumption about the decay of
.t/=t
is not restrictive. Indeed, for any increasing
function W Œ0; 1 ! R
C
satisfying .0/ D 0 and .1/ D 1 and having summable .t/=t, we can define
Q.t / D t sup
2Œt;1
./
; t 2 .0; 1/:
It is easy to see that Q 2 D
1
, Q.t/=t decreases and .t / 6 Q.t/ for all t 2 .0; 1.
Definition 1.3. Let a function belong to the class D
1
. We define the function J
as
J
.s/ WD
s
Z
0
./
d: (2)

442 DARYA E. APUSHKINSKAYA AND ALEXANDER I. NAZAROV
Remark 1.4. The decreasing of .t/=t implies
.t/ 6 J
.t/ 8t 2 Œ0; 1: (3)
In addition, for t 6 t
0
6 1, we have
.t=t
0
/ D
.t =t
0
/
t=t
0
t =t
0
6
.t/
t
t =t
0
D
.t/
t
0
; (4)
and, similarly,
J
.t=t
0
/ 6
J
.t/
t
0
: (5)
Definition 1.5. We say that a function satisfies the Dini condition at zero if
j.r/j 6 C.r /;
and belongs to the class D
1
.
2. Preliminaries
Properties of .
Let
be a bounded domain in
R
n
. Without loss of generality, we may assume
0 2 @
.
Suppose that
is locally convex in a neighborhood of the origin. Without restriction, the latter means
that for some 0 < R
0
6 1, we have
P
R
0
\ D
˚
.x
0
; x
n
/ 2 R
n
W jx
0
j 6 R
0
; F.x
0
/ < x
n
< R
0
;
where F is a convex nonnegative function satisfying F.0/ D 0.
For r 2 .0; R
0
/, we define the functions ı D ı.r/ and ı
1
D ı
1
.r/ by the formulas
ı.r / WD max
jx
0
j6r
F.x
0
/
jx
0
j
; ı
1
.r/ WD max
jx
0
j6r
jrF.x
0
/j: (6)
Lemma 2.1. The following statements hold:
(a) ı
1
.r/ ! 0 as r ! 0 if and only if ı.r/ ! 0 as r ! 0.
(b) ı
1
.r/ satisfies the Dini condition at zero if and only if ı.r / satisfies the Dini condition at zero.
Proof. By the convexity of F, we have for any x
0
and z
0
, the estimate
F.z
0
/ > F.x
0
/ CrF.x
0
/ .z
0
x
0
/: (7)
Therefore,
jrF.x
0
/j > rF.x
0
/
x
0
jx
0
j
>
F.x
0
/
jx
0
j
;
and, consequently,
ı
1
.r/ > ı.r /: (8)
On the other hand, for any r <
1
2
R
0
, we can find a point x
0
such that
jrF.x
0
/j D ı
1
.r/:

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