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T itle On strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations

A uthor(s ) Giga,Y oshikazu; Ohnuma,Masaki

C itation Hokkaido University Preprint S eries in Mathematics, 719: 1-24

Is s ue D ate 2005

D O I 10.14943/83870

D oc UR L http://hdl.handle.net/2115/69528

T ype bulletin (article)

F ile Information pre719.pdf

On strong comparison principle

for semicontinuous viscosity solutions

of some nonlinear elliptic equations

Yoshikazu Giga and Masaki Ohnuma

Abstract

The strong comparison principle for semicontinuous viscosity so-lutions of some nonlinear elliptic equations are considered. For linear elliptic equations it is well known that the strong comparison prin-ciple is equivalent to the strong maximum prinprin-ciple. However, for nonlinear equations the strong maximum principle may not imply the strong comparison principle. We establish a strong comparison princi-ple for some nonlinaer elliptic equations including the minimal surface equation.

1 Introduction

We are concerned with an elliptic equation of the form (1:1) F(Du(x)D

2

u(x)) = 0 in where is a domain in IRn. The function

u: !IR is unknown and F is a given function. Here DuandD

2

udenote, respectively, the gradient of uand the Hessian ofuin variablesx. The functionF : IR

n SI

n

!IR is continuous, where SIn

denotes the space of all real nn symmetric matrices.

Our goal is to establish the strong comparison principle for viscosity so-lutions of (1.1). By the strong comparison principle we mean the principle that a subsolution u agrees with a supersolution v in if u v in and

u(x 0) =

v(x

0) at some point x

0

2 . A typical example of F = F(pX) we consider here is of the form

F(pX) =;trace

I;

pp 1 +jpj 2

X

so that (1.1) becomes (1:2) ;

p

1 +jDuj 2div

Du p

1 +jDuj 2

!

= 0 in : The equation (1.2) is called the (graph) minimal surface equation.

We shall establish the strong comparison principle for some elliptic equa-tions including the graph minimal surface equation. A solution we consider here is a viscosity solution which may not be continuous. The idea of our proof of the strong comparison principle reects that of the classical strong maximum principle to uniformly linear elliptic equations (cf. PW, GT]). By the strong maximum principle we mean the principle that a subsolution u equals a constant M if u M in and u(x

0) =

M at some point x 0

2 . Evidently the strong comparison principle implies the strong maximum prin-ciple provided that a constant is a solution. However, as we shall see later (Remark 2.6) the converse may not hold. To prove the strong maximum prin-ciple one only need to study the relation of a subsolution and its maximum value. However, to prove the strong comparison principle we have to study the relation between a subsolution and a supersolution of the equation. So we are forced to choose a test function by doubling variables. Since we use an auxiliary function as in the proof of the classical strong maximum principle for linear elliptic equations (cf. PW, GT]), our test function (xy) loses symmetry of variables between x 2 and y 2 . We need some eorts for estimates of matrices concerned with D

2(

xy). We also establish the Hopf boundary lemma. Its proof is very similar to that of the strong copmarison principle.

For generalF(pX) we need some ellipticity. If we remove elliptic condi-tion completely we have a counterexample to the strong comparison principle.

Forj

du

dxj= 1 on (;11) there are two solutions and those are not coincide on (;11) (See Remark 2.6). Moreover, we need some Lipschitz condition onp. When we lose the Lipschitz condition, we have a couterexample. In fact, for ;u;jDujm = 0 inB(0R) with 0< m <1 there is a nonconstant solution which attains its maximum at center 0 and the value is zero on the boundary of the ball B(0R). In this case even the strong maximum principle does not hold.

It is well known that for linear elliptic equations the strong comparison principle is equivalent to the strong maximum principle since linear com-binations of solutions are still solutions. The strong maximum principle of classical solutions for linear elliptic equations has been well studied (cf. PW, GT]). There are a few results on the strong maximum principle for weak solu-tion (distribusolu-tion sense) of quasilinear possibly degenerate equasolu-tions (see e.g. V, PS, GT]). For viscosity solutions Kawohl and Kutev KK] prove the strong maximum principle under continuity condition for subsolutions or supersolu-tions. Later, Bardi and Da Lio BD] improve this result without continuity assumption for solutions and they establish the strong maximum principle for a large class including the graph minimal surface equation and even for degenerate elliptic equations, for example, for thep-Laplacian equation with

p > 1. For a level set equation of the minimal surface equation a special form of a strong maximum principle for level sets of solutions was established by GOS]. Our choice of test function is close to that of KK]. In BD] to prove the strong maximum principle they just consider the relation of a subsolution

u(x) and its maximum. So they do not have to consider the test function (xy) we use they need not to estimate the second derivative D2(xy). For viscosity solutions Trudinger T] proved the strong comparison princi-ple for uniformly elliptic equations with Lipshitz cotinuity assumptions on subsolutions and supersolutions. He only state results in T, Remark 3.2] without the proof.

This paper is organized as follows. In section 2 we recall a notion of

ity solutions through denition of viscosity subsolutions and supersolutions to elliptic dierential equations. Then we list assumptions on F =F(pX). We give some comments to our assumptions. In section 3 we establish the strong comparison principle of viscosity solutions. In section 4 we prove the Hopf boundary lemma for viscosity solutions. In section 5 we give a key lemma to prove that results apply to uniform elliptic equations including (1.1).

After this work was completed, we were informed of a recent work of Ishii I] who proved the strong comparison principle for semicontinous viscosity solutions of uniformly elliptic equations. His proof is very similar to ours.

2 Denition of viscosity solutions

and assumptions on

Let be a domain in IRn. We consider a elliptic equation of form (2:1) F(Du(x)D

2

u(x)) = 0 in : Here Du and D

2

u denote, respectively, the gradient of uand the Hessian of u in variablesx.

Now we recall a denition of viscosity solutions of (2.1). We list the basic assumptions on F =F(pX).

(F1) F : IR

n SI

n

!IR is continuous where SIn denotes the space of all real

nn symmetric matrices. We will use the following notations

USC() =fupper semicontinuous functions u: !IRg LSC() =flower semicontinuous functionsu: !IRg:

De nition 2.1 Letu: !IR.

(i) A function u 2 USC() is a viscosity subsolution of (2.1), if for all '2 C

2() and local maximum points ^

xof u;'we have

F(D'(^x)D 2

'(^x))0:

(ii) A function u 2 LSC() is a viscosity supersolution of (2.1), if for all '2C

2() and local minimum points ^

x of u;' we have

F(D'(^x)D 2

'(^x))0:

(iii) If uis a viscosity subsolution and a viscosity supersolution of (2.1), then we call u a viscosity solution of (2.1).

We recall one of equivalent denition of viscosity sub- and supersolutions of (2.1) (cf. CIL]).

De nition 2.2 Letu: !IR.

(i) A function u2USC() is a viscosity subsolution of (2.1),

F(pX)0 for all (pX)2J 2+

u(x) x2: (ii) A function u2LSC() is a viscosity supersolution of (2.1),

F(pX)0 for all (pX)2J 2;

u(x) x2: Here J

2+ denotes the elliptic super 2-jet in , i.e., J

2+ is the set of (

pX)2 IRn

SI n

that satisfy

u(y)u(x) +hpy;xi+ 12hX(y;x)y;xi+o(jy;xj 2) as y!x in

where hi denotes the Euclidean inner product. Similarly, J

2; denotes the elliptic sub 2-jet in , i.e., J

2; is the set of (

pX)2IR n

SI

n that satisfy

u(y)u(x) +hpy;xi+ 12hX(y;x)y;xi+o(jy;xj 2) as y!x in :

Note that J

2+ and J

2; have the relation J

2;

u=;J 2+(

;u).

We next describe of a class of equations for which we shall establish a strong comparison principle. We shall introduce a notion called coercive.

De nition 2.3 We say that a function f : IRSI n

!IR is coercive if for each M >0 there exists a function =

M : 0

1)!IR satisfying (i) is continuous on 01) and lim

!+1

() = +1

(ii) f(pS)b(N) for allS 2SI

n,

b>0,N >0 and p2IR

n satisfying

S bI t

S;bN jpjM for some 2S

n;1. Here

I denotes the identity matrix, is a row vector, t

is the transposed vector of and S

n;1 denotes the set of unit vectors in IR n

. The function is called a bound for f.

We shall assume a kind of ellipticity and a Lipschitz continuity of deriva-tive variables p for F =F(pX).

(F2) There exists a coercive function f such that

F(pX);F(p;Y)f(pX+Y) for allp2IR

n and for all

XY 2SI n.

(F3) Let M and K be positive. There exists a positive constant L

MK such that

jF(qX);F(~qX)jL MK

jq;qj~ for all qq~2IR

n satisfying

jqjjqj~ M and for all X 2 SI

n satisfying

jjXjj K, wherejjXjj denotes the operator norm ofX as a self-adjoint operator on IRn.

We shall see that the uniform ellipticity implies (F2). Let us recall a denition of uniformly elliptic equations. Let M be positive. If there exists constant 0<

M

M such that (2:2)

M trace

Y F(pX;Y);F(pX)

M trace Y for all p 2 IR

n satisfying

jpj M, XY 2 SI n and

Y 0, then we call F = F(pX) is uniformly elliptic. It turns out that (F2) is fullled if F = F(pX) is uniformly elliptic (Proposition 2.5). Let

j (1

j n) be the set of eigenvalues of X including multiplicity. Let e

j be eigenvectors of

j. We may assume that fe

j g

n

j=1 is an orthogonal basis of IR n

. Thus we have a spectral decomposition

X = n X

j=1

j e

j e

j : We dene the plus part X

+ and minus part X

; by

X + :=

n X

j=1 (

j)+ e

j e

j

X

; := n X

j=1 (

j); e

j e

j

where (

j)+ := max f0

j

g and (

j); := min f0

j g.

Proposition 2.4 LetF be uniformly elliptic. Then we have

F(pX);F(p;Y);

M trace (

X+Y) +

;

M trace (

X+Y) ;

:

Proof. Let

Z = n X

j=1

j e

j e

j

be a spectral decomposition of Z =X+Y. We may assume that 1

2

` 0 and n n;1 `+1

<0. We calculate

F(pX);F(p;Y) =F(pX);F(pX;Z) = F(pX);F(pX;

n X j=1 j e j e j) = F(pX);F(pX;

1 e

1 e

1) +F(pX;

1 e

1 e

1)

;F(pX;( 1 e 1 e 1+ 2 e 2 e 2)) ...

+F(pX; `;1 X j=1 j e j e j)

;F(pX; ` X j=1 j e j e j)

+F(pX; ` X j=1 j e j e j)

;F(pX; `+1 X j=1 j e j e j) ...

+F(pX; n;1 X j=1 j e j e j)

;F(pX; n X j=1 j e j e j) and apply (2.2) to get

; M 1 ; M 2 ;; M `+ M j `+1

j++ M j n j =; M( 1+ + `) ; M( `+1+ + n) =; M trace Z + ; M trace Z ; 2 As we prove later (section 5) ;

M trace (

X+Y) +

;

M trace (

X+Y) ; is a coercive function for uniformly elliptic equations. Thus by Proposition 2.4 we have

Proposition 2.5

Remark 2.6

(F2) completely. In fact the strong comparison principle fails for a rst order equation j

du dx

solutionsu1(x) =x+1 andu2(x) =

;jxj+1. We observe that u 1(x)

u 2(x) on (;11) and u

1(x) u2(x) on (

;10). However, u

1(x) > u2(x) on (01). This means that the strong comparison principle is not fullled.

(ii) One would like to weaken the Lipschitz condition of F(pX) in p. For example, we consider

jF(qX);F(~qX)jLM Kjq;q~j

m

for some m (0 < m < 1). However, for such F we have a counterexample (cf. BDD]). Let 0 < m <1R >0,

F(pX) = ;trace X;jpj

m _=B(0R)

IR

n_:

For this F equation (2.1) becomes (2:3) ;u;jDuj

m _{= 0 in B(0R):}

In BD] there is a comment to (2.3). For (2.3) the strong minimum principle holds, however the strong maximum principle does not hold. In fact, u(x) =

C(Rk;jxjk) with k = (2;m)=(1;m) C = k

;1(n +k ;2)

1=(m;1) is a non constant solution to (2.3) (cf. BDD]). This means for (2.3) the strong comparison principle does not hold. So we cannot remove the Lipschitz continuity assumption completely. If we would like to weaken the assumption (F3), we have to consider another way.

Remark 2.7

(2:4) ; p

1 +jDuj 2div

Du

p

1 +jDuj 2

!

= 0 in :

For this equation F =F(pX) is given by (2:5) F(pX) =;trace

I;

pp 1 +jpj 2

X

:

This F = F(pX) is uniformly elliptic. Indeed, for (2.5) elliptic constants are taken by

M = 1

=(1 +M 2)

M = 1. An extended equation of (2.4) is the following.

(2:6)

;trace A(Du)

I;

DuDu 1 +jDuj

2

D 2

u

I;

DuDu 1 +jDuj

2

= 0 in where A(p)2SI

n

satises A(p)0 for allp2IR n

. We shall assume that for each M > 0 there exists a constant C = C(M) > 0 such that A(p) CI for all p2 IR

n

satisfying jpjM. We also assume a lower bound such that there existsc>0 satisfying cI A(p) for allp2IR

n. For (2.6)

F =F(pX) is given by

(2:7) F(pX) =;tracefA(p)R p

XR p

g R p :=

I;

pp 1 +jpj 2

:

This F = F(pX) is also uniformly elliptic. Elliptic constants are taken by

M =

c=(1 +M 2)2

M = C.

3 Strong comparison principle

Let be a domain in IRn. We consider a elliptic equation of the form (3:1) F(Du(x)D

2

u(x)) = 0 in :

Our main theorem is an extension of the strong comparison theorem to viscosity subsolutions and supersolutions to (3.1).

Theorem 3.1

F satises (F1){(F3). Letu2 USC() andv 2LSC() be, respectively, viscosity sub-and supersolutions of (3.1). Assume that uv in and that there exists a point x

0

2 such that u(x 0) =

v(x

0). Then

u v in .

If v is a constant function in and a constant function is a viscosity solution then Theorem 3.1 gives a strong maximum principle.

We shall prove Theorem 3.1 in several steps. Our proof reects that of the maximum principle to uniformly elliptic equations in classical sense. Choice of an auxiliary function and some domains in near the point x

0 are very similar to the classical work PW, GT].

Leta2, R>0,

B 0 := (

aR) x 0

2@B 0

B 1 :=

B(x 0

R

2 ) where B(aR) denotes the open ball in IR

n of radius

R centered at a. Let for >0 and x2

z(x) :=e ;R

2 ;e

;jx;aj 2

: By denition one observes that

(3:2)

;1<z(x)<0 in B 0

z(x) = 0 on @B 0

0<z(x)<1 outside B

0 :

Let w(xy) be a function on . We set for (xy)2 and" >0 (xy) := "z(x) + jx;yj

2 (xy) :=w(xy);(xy):

For proof of Theorem 3.1 we have to study maximum points of (xy) on B

1 B

1 and their values. First we shall consider the value of (

xx) for x2@B

1.

Proposition3.2 LetB 0

B 1 and

z(x) as stated above. There exists" 0

>0 such that if 0<" <"

0 then

w(xx);"z(x)<0 on @B 1

for all >0 provided that w is upper semicontinuous on , w(xx)0 for allx2 and

w(xx)<0 if x2B 0

nfx 0

g w(x

0 x

0) = 0 :

Proof. We will divide the boundary of B

1 into two pieces:

C 1 :=

@B 1

\B 0

C 2 :=

@B 1

nB 0 clearly @B

1 is a disjoint union of C

1 and C

2. Since

w(xx)<0 on a compact set C

1, there exists a constant

` > 0 that satises w(xx) ;` on C 1 by upper semicontinuity of w. By (3.2) z(x) 0 on C

1. We shall take "

0 >0 such that

;`;" 0

z(x)<0 on C 1

: By (3.2) z(x)>0 on C

2. We easily see that for any " >0

w(xx);"z(x)<0 on C 2

: Thus we observe that if 0<"<"

0,

w(xx);"z(x)<0 on @B 1 for all >0. 2

We next study properties of maximum points of (xy) on B 1

B 1.

Proposition 3.3 Suppose thatw be upper semicontinuous on and that

w(xx)<0 if x 2B 0

nfx 0

g w(x

0 x

0) = 0 : Let B

0 B

1 and as stated above and let "

0 be as in Proposition 3.2. Let (xy) attain its maximum at (x

y

) 2 B

1 B

1 for all 0

<" < "

0. Then jx

;y

j ! 0 as ! +1 this convergence is uniform in 0 < " < " 0 and >0.

In particular, there exists a point ^x 2 B

1 such that x

y

! x^ as ! +1 by taking a subsequence.

Proof. We easily see (x

y )

0 for all 0 < " < " 0 and

> 0, since (x y ) (x 0 x

0) = 0. We observe that

w(x

y )

;"z(x ) jx ;y j 2 : By boundedness ofB

1 and upper semicontinuity of

w, there exists a positive constant M such that

w(x

y )

;"z(x )

M for all (x y ) 2B 1 B 1 : We now observe that

0 jx ;y j 2 M

which yields jx

;y

j!0 as !+1. 2

Proposition 3.4 Assume the same hypotheses of Proposition 3.3. Then there exists

0

>0 such that if >

0 then attains its maximum over B

1 at an interior point (x

y ) 2B 1 B

1 for all 0

<"<" 0 and

>0.

Proof. We will show ^x 2 B

1. Suppose that ^ x 2 @B

1. By denition of and (x

y

)

0 we have

w(x

y )

;"z(x ) (x y ) 0: Letting !+1 by taking a subsequence we observe that

w(^xx^);"z(^x)0

which contradicts to Proposition 3.2. Thus if >0 is su ciently large say >

0, then x y 2B 1. 2

For the proof of Theorem 3.1 we will use a maximum principle for semi-continuous functions due to Crandall and Ishii CIL]. In particular, we shall study several properties on matrices which are useful to calculate matrices appeared in their theory.

Let

d(x) := 2"e ;jx;aj

2

B :=d(x)(I;2(x;a)(x;a)):

Lemma 3.5

>0 there exists 0

>0 such that if >

0, then

(i) B+B

2

2d(x)I

(ii) t

(B+B 2)

;d(x)jj 2

N

1 for all

x2B 1

whereis an outward normal vector on@B

0at x

0 2@B

0such that =x

0 ;a.

Proof. (i) By direct calculation we have

B+B 2

= d(x)I;2(x;a)(x;a)

+d(x)fI ;4(x;a)(x;a) + 4 2

jx;aj 2(

x;a)(x;a)g] d(x)I+d(x)fI + 4

2

jx;aj 2

(x;a)(x;a)g]: We set

M :=d(x)fI+ 4 2

jx;aj 2(

x;a)(x;a)g: We note that there exists

1

>0 such that if > 1 then

M I. Since

M ;I = (d(x);1)I + 4 2

d(x)jx;aj 2(

x;a)(x;a) t

p(M ;I)p= (d(x);1)jpj 2+ 4

2

d(x)jx;aj 2

hpx;ai 2 for all p2IR

n : Since R

2

jx;aj 3 2

R, we have

d(x)!0 and 2

Now we observe that there exists 1

>0 so that (d(x);1)jpj

2+ 4

2

d(x)jx;aj 2

hpx;ai 2

0 if >

1. Therefore, we have if >

1 then

M I. By 1 we see that if >

1, then

B+B 2

2d(x)I for all x2B 1

: (ii) By direct calculation and the Schwarz inequality

t

(B+B 2)

=d(x)jj

2

;2h x;ai 2 +d(x)(jj

2

;4h x;ai 2

+ 4 2

jx;aj 2

h x;ai 2

)]

d(x)jj 2

;2h x;ai 2 +d(x)(jj

2+ 4

2 jj

2

jx;aj 4)] =d(x)jj

2

1;2h jj

x;ai 2 +d(x)(1 + 4

2

jx;aj 4)]

: Note that h x;ai > 0 for all x 2 B

1. For all N

1

> 0 there exists 2

>0 such that if >

2 then 1;2h

jj

x;ai 2+

d(x)(1 + 4 2

jx;aj 4)

;N 1 for all x 2 B

1. Thus for all N

1

> 0 there exists 2

> 0 such that if > 2 then

t

(B +B 2)

;d(x)jj 2

N

1 for all

x2B 1

2 Now we are in a position to prove Theorem 3.1.

Proof of Theorem 3.1. We will argue by contradiction. We set w(xy) = u(x);v(y) so that wis upper semicontinuous on . Suppose that there would exist a pointx

1

2 such thatu(x 1)

<v(x

1). By a standard argument there would exist an open ball B

0 with B 0 andx 0 0 2@B

0 that satises u<v in B

0 nfx

0 0

g u(x

0 0) =

v(x 0 0)

We shall replace x 0 0 with

x 0 since

u(x 0) =

v(x

0). We set B

0 =

B(aR) and B

1 = B(x

0

R

2) so that B

1

. Now we see that all conclusions of Proposition 3.2{3.4 would hold for = w; on B

1 B

1 for su ciently small " and su ciently large . Proposition 3.4 says that attains its maximum over B

1 B

1 at ( x y ) 2 B 1 B

1 for su ciently small

" > 0 and su ciently large >0. In particular,

u(x);v(y)u(x )

;v(y

) + (

xy);(x

y )

: Expanding at (x

y

) we get x( x y ) y( x y ) A 2J

2+( u(x

) ;v(y

)) with

A=D 2( x y ) = xx( x y

) yx( x y ) xy( x y

) yy( x y ) where x =

D

x, xx = D

2

xx is an

nn matrix and so on. We shall apply the elliptic version of Crandall{Ishii's Lemma CIL, Theorem3.2]. We see that for all positive , there exists XY 2SI

n such that (i)

(x( x

y

)

X)2J 2+

u(x )

(y(

x

y )

Y)2J 2+(

;v(y )) (, (; y( x y )

;Y)2J 2;

v(y )) (ii) (MI) ; 1 +jjAjj I 2n X O O Y

A+A 2

: Here J

2+ and J

2;, respectively, denote closure of J

2+ and J

2; (cf. CIL]). By direct calculation

x =

"Dz(x) + 2(x;y) = 2"e ;jx;aj

2

(x;a) + 2(x;y) y =

;2(x;y) xx =

"D 2

z(x) + 2 I = 2"e ;jx;aj

2

(I;2(x;a)(x;a)) + 2 I xy = yx =

;2 I yy = 2

By denition ofdandB (see the paragraph just before Lemma 3.5) we obtain the identity at x=x

A =

B+ 2 I ;2 I ;2 I 2 I

: From (MI) we observe that

X+Y B +B 2

: Let(x) =d(x)(x;a) and letp

= 2

(x;y) so that x =

(x)+p . Since u is a viscosity subsolution of (3.1), we have

(3:3) F((x

) +p

X)0: Since v is a viscosity supersolution of (3.1), we have

(3:4) F(p

;Y)0: Subtracting (3.4) from (3.3), we get

(3:5) F((x

) +p

X);F(p

;Y) 0: By (F3) we see that

F((x

) +p

X);F(p

X) ;L MK

j(x

)j =;L

MK d(x

)jx

;aj: By (F2) and Lemma 3.5 we observe that

F(p

X);F(p

;Y)f(p

X+Y)2d(x

)( N

1 2 ) for all N

1

>0 by taking su ciently large. From (3.5) and R 2jx

;aj 3R we see

02d(x

)(N 2)

;L MK

d(x

)32R

where N 1 = 2

N

2. Since

d(x)>0 we have 02(N

2) ;L

MK 3 2R : Letting N

2

! +1 yields (N 2)

! +1. This means that there exists N 0 such that if N

2 >N

0 then

L MK

3

2R<2(N 2)

:

We get a contradiction. Now we have completed the proof of Theorem 3.1. 2

Remark 3.6

space variable x of the form (3:6) F(Du(x)D

2

u(x));'(x) = 0 in

provided ' 2 C(). The proof is almost same as that of Theorem 3.1. By the same procedure we have

02d(x

)(N 2)

;L MK

d(x

)32R;'(x ) +

'(y )

:

From Proposition 3.2 letting ! +1 by taking a subsequence we observe that

02d(^x)(N 2)

;L MK

d(^x)32R : for some ^x 2 B

1. Since

d(^x)> 0 we get a contradiction again. A typical example of (3.6) is

;div

Du p

1 +jDuj 2

!

='(x) in

which is called the curvature equation.

4 The Hopf boundary Lemma

In this section we establish the Hopf boundary Lemma. We consider a equa-tion of the form

(4:1) F(DuD 2

u) = 0 in IR n

Theorem 4.1

(F2) and (F3). Letu2USC(fx 0

g) andv 2LSC(fx 0

g) be a viscosity subsolution and a supersolution of (4.1), respectively.

Assume that

uv in fx 0

g there exists a ballB

0

and a point x 0

2@B

0 such that

u<v in "B 0

nfx 0

g and u(x

0) = v(x

0). Then for anyw2IR

n satisfying

hwi<0, (4:2) limsup

s#0

(u;v)(x 0+

sw);(u;v)(x 0) s

chwi

with some c > 0 independent of w and , where denotes the outward normal of the boundary @B at x

0.

Proof. LetB 0 =

B(aR) and let z be the same function as in (3.2). To show (4.2) if su ces to prove

(4:3) (u;v;"z)(x)0 in Z

for su ciently small">0 (0<"<1) and a domainZwhich is neighborhood of x

0 and is contained in B

0. If we have (4.3), we can see (u;v;"z)(x

1)

(u;v;"z)(x

0) for all x

For small s>0 we setx 1 =

x 0+

sw. Now we observe that (u;v)(x

0+

sw);(u;v)(x 0) s

"z(x

0 +

sw);"z(x 0) s

: Since h wi<0, we get

lim s#0

sup (u;v)(x 0+

sw);(u;v)(x 0) s

"hDz(x 0)

wi = 2"e

;R 2

h wi<0: Thus we obtain (4.2).

It remains to prove (4.3). We argue by contradiction. LetB 1 =

B(x 0

R 2) and Z =B

0 \B

1. Suppose that for all

"(0<"<1) there would exist ~x2Z" such that

(u;v;"z)(~x) = max Z

(u;v;"z) = "

>0: On the boundary@Z there exits "

0

>0 such that if" 2(0"

0) then (4:4) (u;v;"z)(x)0 on @Z:

We see that ~x2Z and

max Z

(u;v;"z) = "

: Now we set

(xy) = "z(x) + jx;yj 2

where >0. We dene

(xy) =u(x);v(y);(xy):

Let attain its maximum at ("xy") 2 Z" Z" for all " 2 (0" 0) and

> 0, i.e.,

max ZZ

(xy) = ("xy"): We easily see that ("xy")>0 since

(4:5) max ZZ

(xy)max Z

(u;v;")(x) = "

We observe that

M u("x);v("y);"z("x)> jx";yj" 2

0 and there exists ^x2Z" such that

"

xy"!x^ as !+1

by taking a subsequence. Note that ^x2Z. Suppose that ^x2@Z. By (4.5)

u("x);v("y);"z("x)u("x);v("y);"z("x); jx";yj" 2

" >0 Letting ! +1 by taking a subsequence we have (u ;v ;"z)(^x) > 0 that contradicts (4.4). Thus if > 0 is su ciently large say >

0, then "

xy"2 Z. Since u(x);v(y) u("x);v("y) + (xy);("x y"), we argue in the same way as in the proof of Theorem 3.1 with x

= " x y

= "

y to get a contradiction. 2

Remark 4.2

0 if u and v are dierentiable at x=x

0. For linear elliptic equations the Hopf boundary Lemma implies the strong maximum principle. For some nonlin-ear degenerate elliptic equations a version of the Hopf boundary Lemma is established by BD, Theorem 1] to prove the strong maximum principle for semicontinuous viscosity solutions. In their situation v is taken a constant.

The proof of Theorem 4.1 is essentialy the same as that of Theorem 3.1. However, u and v may not satises the equation (4.1) at x = x

0. So we should discuss separately the place where w; takes maximum values.

5 Key lemma for uniformly elliptic equations

We shall prove a key lemma to prove that a uniformly elliptic operator fullls (F2). It su ces to verify that;trace(X+Y)

+

;trace(X+Y)

; appeared in Propsition 2.4 is a coercive function. Here is a key lemma.

Lemma 5.1

n satisfy

S bI (5.1)

t

S;bN for some 2S n;1

(5.2)

where S

n;1 denotes the set of unit vector in IR

n. Then we have traceS

++

traceS ;

(n;1)b; N

n b:

Proof. We may assume that S is a diagonal matrix. Let i (1

i n) be eigenvalues of S. From (5.1) we see

i

b for all i. From (5.2) there exists number ` that satises

`

;bN=n. We may assume that

1 2 j 0> j+1 n;1 n :

From (5.2) at least one eigenvalue is negative. We do not worry about the case all eigenvalues are negative. By the denition of S

+ and S

; we see that traceS + = j X k=1 k traceS ; = n X k=j+1 k : Then we obtain

traceS ++ traceS ; = j X k=1 k+ n X k=j+1 k6=` k+ ` : By (5.1) and (5.2) we see that

j X k=1 b+ n X k=j+1 k6=` b; Nb n

(n;1)b; Nb

n : 2

Remark 5.2

uni-formly elliptic equations coercive function f and a function which is a bound for f are following for each M >0 if jpjM then

f(pS) =;

Mtrace S + ; Mtrace S ;

(N) =; M(

Acknowledgements

This work is partly supported by the Grant-in-Aid fund of COE "Math-ematics of Nonlinear Structure via Singularities". The work of the rst au-thor is partly supported by the Grant-in-Aid for Scientic Research, No. 14204011, the Japan Society of the Promotion of Science. Much of the work of the rst author was done when he was a faculty of the Mathematics De-partment of Hokkaido University. The second author was partly supported by JSPS Grant-in-Aid for Young Scientists (B) through grant No. 15740106. The authors were grateful to Dr. Kazuyuki Yama-uchi for his fruitful discus-sion.

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Authors:

Yoshikazu Giga

Graduate School of Mathematical Sciences University of Tokyo

Komaba 3-8-1 Meguro, Tokyo 153-8914

Japan

Masaki Ohnuma

Department of Mathematical and Natural Sciences The University of Tokushima

Tokushima 770-8502 Japan