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 HOPF–OLEINIK 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 maxfv; 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/: