N-functions and the weak comparison principle

JOURNAL

36, 257-269

OF DIFFERENTIAL

EQUATIONS

N-Functions

and the

Department

of

Mathematics,

Weak

(1980)

Comparison

VICTOR

L. SHAPIRO

University

of California,

Received

July 30, 1979

Principle*

Riverside,

California

92521

1. INTRODUCTION A(t) is called an N-function if A(t) = si 01(s)ds for 0 < t < 00 where LYis a right continuous nondecreasing function on [0, co) with a(t) > 0 for t > 0, a(0) = 0, and Km,,, a(t) = +ox. (See [l, p. 2281 or [4, p. 61.) In this paper, we shall deal only with N-functions A which are twice differentiable on (0, 03) with A”(t) > 0 for t > 0; so, in particular, dA/dt = A’ = a: is strictly increasing and continuous for 0 < t < 00. Defining CC = 01-l as in [l, p. 2291 and setting A-(t) = si e(s) ds, we see that A- is also an N-function. We shall refer to A- as the complementary N-function to A. We shall be concerned with the comparison principle for the divergence structure operator of the form Qu = div A’(1 Du 1) 1Du 1-l Du + F(x, u)

(1.1)

for functions u defined in a bounded open set Sz where as usual Du = D,u) and D,u = au/&v, _ In particular, we shall be interested in (DA.., establishing a result along the following lines: Qv ,( Qu

in

52,

u
then u < v in 8. In the sequel, F in (1.1) will be a finite-valued with the property that

on

aII

function

(1.2) defined on 52 x R

F[x, u(x)] is Lebesgue measurable on Q whenever

u is

(1.3)

Also, we shall suppose that

I w% 41 < c4l s I) I s I-l + f(X), * This research was partially Grant MCS 76-02163.

supported

by the National

Science Foundation

(1.4) under

257 0022-0396/80/050257-13$02.00/O

Copyright 0 1980by AcademicPress, Inc. All rights of reproductionin any fox-mreserved.

258

VICTOR

L.

SHAPIRO

where c is a positive constant and f is a nonnegative measurable function with A-(f) in Ll(sZ). If s is zero the first term on the \right-hand side of the inequality in (1.4) is understood to be zero. In the sequel, we shall also assume that A satisfies a da-condition near infinity (see [l, p. 2521) i.e., there exist a constant K and a t, > 0 such that A(2t) < M(t) for t, < t < co. We shall say u is in LA(Q) if jc, A(1 u 1) dx < co. We shall say u is in FVA(SJ) if first u is in W&,!(Q), and second both u and Du are in LA@). WiyA(Q) will mean there is a sequence (&)F with & in Cam(a) such that jsaA(1 u - & I) and

- Ml) +0 s41Ddu a

as j+,

for i = l,..., n.

Next, we recall for the reader Young’s inequality, [I, p. 2291, i.e.,

st < A(t) + A-(s)

O
(1.5)

o
(1.6)

and the following inequality [I, p. 2301, A-[/l(t)

t-l] < A(t),

Also, we see from [l, p. 2321 that if A satisfies a As-condition near infinity, then there is a constant c and a t, 3 0 such that A’(t) < L-A(t) t-1

(1.7)

for 2, < t < co. Since we are interested here in the weak comparison principle, we now give weak interpretations to the inequalities in (1.2). (See [3, Chaps. 8, 91 and [8] for analogous situations.) In particular, for 52 a bounded open set in [w” with u and z, in IVA(S), we shall say u < v weakly (A)

on

%J

if (u - v)+ is in W,‘*A(f2) where as usual uf = max(u, 0). Also, we shall say Qv < Qu weakly (A)

in

D

provided

Q(u, v; 4) =

s, W(Du) - G(Wl .D+ - [F(x,u) -F@, v)l+>d.x< 0 (1.8)

holds for all nonnegative 4 in C,,l(sZ) where G = (Gi ,..., G,) and G,(x) = A’(1 x I) xi 1x 1-l =o

for

x#O

for

x = 0.

(1.9)

WEAK

COMPARISON

259

PRINCIPLE

Before proceeding, we note from (1.3) through (1.7) that Q(u, v; 4) is well defined for II and z, in wlJ(Q) and I$ in C:(Q). Besides assuming that A meets a As-condition near infinity and A”(t) exists and is positive for 0 < t < co, we shall also assume that A meets one of three other possible conditions. The first two of these conditions are easy to state: A”(t)

is nonincreasing

for 0 < t < co;

(1.10)

A”(t)

is nondecreasing

for 0 < t < co.

(1.11)

It is easy to see that A(t) = t” meets (I .lO) for 1 < p 2 < p < co. Also, it is easy to see that A(t) = (1 + t)” t)“/p + p-l meets (1.10) for 1 < p < 3/2 and (1.11) for 2 < for Q < p < 2, A meets neither (1 .lO) or (1.11). To cover we introduce the following condition.

< 2 and (1.11) for log(1 + t) - (1 + p < co. However, this last range of p

A”(t) is continuous for 0 < t < co with A”(0) > 0. Furthermore, (1.12) there is a t, > 0 such that A”(t) is nonincreasing for t, < t < CO. An easy computation shows that the last-mentioned for the range 3 < p < 2. We intend to establish the following theorem:

A above does meet (I. 12)

THEOREM. Let A be an N-function which satisfies a AX-condition near infinity and is twice-dayerentiable on the open interval (0, co) with A”(t) > 0 for 0 < t < co. Let JJ be a bounded open set, and let F(x, s) be a $nite-valued function deJined on D x R which meets (1.3) and (1.4) and which is nonincreasing in s for fixed x in G. Also, let A meet one of the conditions (1. IO), (1.1 I), or (1.12). Suppose that both u and v are in WS~(J!I) and that Q(v) < Q(U) weukZy (A) in J2. Then if u < v weakly (A) on c%, it follows that

u < v almost everywhere in C’.

(1.13)

We observe that if u is continuous in a, 0 on X?, and in lPA(Q), then it follows that u < 0 weakly (A) on aa. Consequently, in view of the remarks made concerning (l.lO), (l.ll), and (1.12), we see that the above theorem constitutes both a generalization and an extension of [5, Lemma 11. (It was this lemma and its proof which in part motivated the present paper.) For other results involving the comparision principle for nonlinear elliptic partial differential equations, we refer the reader to [2, 6, 81. Other A(t) that meet the conditions in the hypothesis of the theorem are, for example, (1 + t)P[log(l + t)12 with 1 < p < 00 and (1 + t)[log(l + t)]” with 1 < 4 < 2. For a comparison principle for nonlinear elliptic partial differential equations of a somewhat different nature than presented here, we refer the reader to a result of Serrin’s, [9, p. 1861.

260

VICTOR L. SHAPIRO

2. A FUNDAMENTAL LEMMA The following

lemma will be needed for the proof of the theorem.

LEMMA. Let A be an N-function which is twice dzfferentiable on the open interval (0, 00) with A”(t) > 0 for 0 < t < CO.Let G = (G, ,..., G,) where G, is dejned by (1.9). Thm according as A meets (I. lo), (1.1 l), or (1.12), the following inequalities prevail respectively for x and y in W: [G(y)

-

G(x)] . (y -

x) 2 I y -

x I [A’(1 x I + I Y -

x I) - A’(1 x 111;

[G(y) - G(x)] . (y - 4 3 IY - xl PUIY - x1/4)1; [G(Y)

- G(x)1. (Y - 4 3

cA

1Y -

X / LAY

X I +

(2.1) (2.2)

1Y -

X 1)

- A'(1 x I + 3 I y - x I 4%

(2.3)

where c, is a positive constant depending on A. Since limt+s+ A’(t) = 0, it follows from (1.9) that Gi is continuous in UP. Consequently both the left-hand and the right-hand sides of the inequalities in (2.1), (2.2), and (2.3) can be viewed as continuous functions on UP x W. Therefore, it is easy to see that to establish the lemma, we need only show that (2.1), (2.2), and (2.3) hold for x and y fixed with x # y and 0 $+{x(t) : 0 < t < 11,

(2.4)

where

z(t) = x + t[

with

5 = y -

x.

WJ)

In order to do this, we set

f&(t) = GLW

(2.6)

for i = I,..., n and observe from the hypothesis that

of the lemma, (1.9), and (2.4)

H,(t) is continuous From (2.6), we further

and H;(t) exists for 0 < t < 1.

(2.7)

see that

Z’(t) = 41 #)I) I z(t)l-l S,lj + [A”(I +)I) - 4

+)I) I 4Wl

I 4W~i(t) 44 lj

(2.8)

WEAK

COMPARISON

261

PRINCIPLE

for 0 < t < 1, where Sij is the Kronecker-6 and the summation has been used. (This convention will continue to be used in the sequel.) From (2.8), it follows that

+ [a

441) - 4

4~

I 4wli

I 4vww

(2.9)

for 0 < t < 1. Also, we observe from (2.4) to (2.8), the fact that A’ is nondecreasing on (0, cc), and [7, 11.83, p. 3681 that [G(y) - G&)J(y,

- xi) = L1 &Hz’(t) dt.

(2.10)

Let us now suppose that A satisfies (1.10). Then it follows that A’(t) t-l 3 A”(t) for t > 0. Using this fact in conjunction with the Cauchy-Schwarz inequality, we see from (2.9) that I&(t) 3 A”([ z(t)l) / 5 I2 for 0 < t < 1. We consequently conclude from (2.10) that

Myi) - G(xi)l(~,- 4 2 I t; I2j.’0 A”(1 W) dt.

(2.11)

NOW 1z(t)/ < 1x I + t 15 / for 0 < t < 1. We obtain therefore from (1.10) and (2.11) that

[G(yJ - G(~i)l(yr - xi) 3 I 5 I s,“’ &I x I + 4 ds 3 I 5 I [A’(1 x I + I 5 I) - 41 x 01. (2-W Equation (2.5) in conjunction with (2.12) gives (2.1). In order to establish (2.2), we suppose that A meets (l.ll), i.e., A”(t) is nondecreasing on (0, ok). Consequently for t > 0, d[A’(t) t-l]/dt > 0. We have therefore that A”(t) > A’(t) t-l for t > 0 and that A’(t) t-l is nondecreasing on (0, co). But then it follows from (2.9) and (2.10) that

MY,) - GWI(Y, - 4 2 I 5 I2j-l 41 #I> I z(W dt.

(2.13)

0

Now one of the following two situations hold:

I x I d I 5 l/Z

(2.14)

I 3 I > I 1:l/2.

(2.15)

262

VICTOR

L.

SHAPIRO

If (2.14) holds, then for 2 < t < 1,

I +)I 3 3 I 5 l/4 - I x I 3 I 5 l/4. Therefore, the right-hand side of the inequality in (2.13) majorizes

and (2.2) is established in this case. Next, suppose that (2.15) holds. Then for 0 < t < $,

I x(t)1 >, I x I - t I 5 I 3 I x I - I 5 l/4 3 I * l/2 3 I 5 l/4* Therefore, the right-hand side of the inequality in (2.13) majorizes

I 5 I2L1’44A’(I 5 l/4)1 5 l-1 dt, and (2.2) is also established in this case. It remains to show that (2.3) holds under the assumption that A meets (1.12). To do this we must first specify the positive constant c, which arises in (2.3). We proceed with this matter. We first observe from (1.12) and the hypothesis of the lemma that both A’(t) t-l and A”(t) can be viewed as positive continuous functions for 0 < t < co. Also, we see that there exists t, > 0 such that A”(t) is nonincreasing for t, < t < CO.

(2.16)

Consequently, it follows that a positive constant ci exists such that A’(t) t-l > CIA”(t) for 0 < t < t, . But this fact in conjunction with (2.16) gives, after a minor computation, that a constant cp with 0 < c2 < 1 exists such that A’(t) t-l > (2.17)

c2An(t) for 0 < t < c/3. Also, we see that positive constants cs and cq exist such that for

A”(t) > cs

(2.18)

0 < t < 4t, ;

h > cJA’(t + h) - A’(t)]

for

t and h > 0.

(2.19)

We define the positive constant c,, in (2.3) as follows: C

A = min[c, , ca , cacacJ.

(2.20)

WEAK

COMPARISON

263

PRINCIPLE

Continuing with the establishing of (2.3), we observe from (2.9) and the Cauchy-Schwarz inequality that for 0 < t < 1,

CR(t) 3 min[A”(I 4t>l), A’(1 z(t)l) I .dt>l-*I I 5 12. Consequently, it follows from (2.10) and (2.17) that (2.21)

[G(Y)- ‘341 . (Y - 4 >, czI 5 I2jl-W WI] dt. 0 Now, with t, defined in (2.16), four mutually exclusive cases arise. I:

Ixj<2t*

and

I Y I G 2t* ; I Y I > 2t* ;

II :

/ X / < 2t*

and

III :

1x 1 > 2t*

and

IYI <2t*;

IV :

/ x 1 > 2t*

and

I Y I > 2t* .

We shall show that (2.3) holds in each of these cases. Suppose that I holds. Then it follows from (2.5) that I x(t)\ < 1x 1 + \ y f < 4t* for 0 < t < 1. Consequently, it follows from (2.18) (2.21) that [G(Y)

- W41 . (Y - 4 3 4c2c3 I 5 I I 5 I 4-l 3

c2c2c4

3

cA

15 I [A’(I x I + I 5 1) - A’(l x I + 3 I 4 I 4-l)

I 5 / [A’(

1x

1 +

I 4 i)

-

A’(I

x

1 +

3

1 b 1 4-1)-,,

and (2.3) is established in this case. Suppose that IT holds. Then it follows from (2.5) that for 2 < t < 1, 1z(t)1 = I(1 - t)x + ty I > 3 I y I 4-l - I x I 4-l > t, * Also, / z(t)1 < I x 1+ t I [ /. We conclude therefore from (2.16) and (2.21) that [G(Y)

- GWI * (Y - 4 2 czI 5 I2j3;,.4”(i x I + t I 5 I) dt b

c2

I 1 I [A’(1 x I + I 5 I) - 4

x I + 3 I 5 14-l)]. (2.22)

But then (2.3) follows immediately from (2.20) and the proof for Case II is complete. Suppose that III holds. Then it follows from (2.5) that for 0 < t < 4-1, 1z(t)1 > 3 1x ( 4-I - 1y 14-l 3 3t*2-* - f*2-l

3 t* *

264

VICTOR L. SHAPIRO

Also, 1z(t)] < 1x / + t 15 I. We conclude therefore from (2.16) and (2.21) that MY)

- W41 *(Y - 4 >, ~2 I 5 I2j-‘” 41 x I + t I 5 I) dt 0

3 c2 I 1 I2j’ A”(1 x I + t I 5 I) dt. 314

(2.23)

But then (2.3) follows exactly as it did in Case IT. It remains to establish (2.3) when IV holds. We subdivide IV into two further cases. IV, :

I x I > 2t* 3 I Y I > 2t* IVI, : I x I > 2t* J I Y I > 2t*

and and

It-1 <2 1x1; I51 >2lxl.

Suppose IV, holds. Then it follows from (2.5) that for 0 < t < 4--l,

I x(t)1 = I x + t5 I 3 I x I - I 5 I 4-l > I x I 2-r 3 t, * Also, I @>I < I .2”I + t I 5 I. We conclude therefore from (2.16) and (2.21) that (2.23) holds. Consequently, it follows from (2.20) that [G(Y)

- ‘WI . (y - x>3

cA

/ 5 1[A”(] x 1+ I 6 1)- A’(1x I +

3 15 / 4-71

and (2.3) is established for the Case IV, . Finally, suppose IVr, holds. Then it follows from (2.5) that for 2 < 1z(t)\

= 1t( + x I > 3 / 5 I 4-r - I x / >, I x 12-r 3

t,

t

,< 1,

.

Also, I @)I < I x I + t I 5 I. We conclude therefore from (2.16) and (2.21) that (2.22) holds. But then (2.3) follows immediately from (2.20), and the proof for Case IV, is complete. The proof of the lemma is therefore complete. 3. VARIOUS REMARKS We shall need various remarks to prove the theorem. Throughout this section, we shall assume that A is an N-function which satisfies a da-condition near infinity and is twice-differentiable on the open interval (0, r~) with A”(t) > 0 for t > 0. Also, we shall need some more facts about LA(Q) and LAY(Q), where J2 is a bounded open subset in IP. In particular, if sszA-(\ V I) dx < co, then it follows from the convexity of A- that

II VIIA-,n=inf(k>O:~A-(iVIK-l)dx~l)

WEAK

COMPARISON

265

PRINCIPLE

is finite. Likewise, if so A(] u I) dx < co, then 11u ]la,o is finite and the following generalized Holder’s inequality holds (see [I, p. 2341):

15fa 44 %4dx < 2 IIu IIA.RII v IIP,R .

(3.1)

Also, we observe from [l, p. 2361 that if u and the sequence {ui}~ are inD(SZ), then ;$I 1 A(] uj - u 1) dx = 0 implies that $ir II ui a

u llA,o = 0.

(3.2)

The first remark we establish is the following:

REMARK 1. Let w be in WtVA(Q) with Dw = 0 almost everywhere in 52. Then w = 0 almost everywhere in .Q. By assumption

there exists {+&” with (bi in C,-(G)

j;& j a 41 w -&

such that

I)dx = 0

and ,!irsj

0

for

A(ID,w-DD,+jI)dx=O

Let 7 be a polynomial. Then both A$ is easy to see from (3.1) and (3.2) that

i = l,...,n.

77I) and A*(1 Di7 1) are in Ll(Q),

and it

and $2 jQ (w - +j) Di7 dx = 0.

Since snq$Di7 dx = -sn we conclude that

s R

7D&

dx and D,w = 0 almost everywhere

wDi7 dx = 0

for

in Q,

i = l,..., n.

Consequently, Sn WE dx = 0 f or every polynomial remark follows from the Weierstrass approximation

f, and the conclusion theorem.

of the

266

VICTOR

L.

SHAPIRO

The next remark we establish is the following: REMARK 2. Let u and v be in WI,*(S2), and suppose Qv < Qu weakly (A) in 9. Suppose also that $I is in C,l(s2). Then

Q(u, v; #+I < 0,

(3.3)

where Q(u, v; z,b+)is defined by (1.8).

Since #+ = max[#, 01, it follows that there exists a positive integer J > 0 such that #+(x) = 0 if dist(x, aQ> < J-i. Forj > J, define

$j(X) = I B(Qj-~)I-l j

B(O&1)

4+(x +

Y) dy,

where B(x, Y) is the open n-ball with center x and radius Y and 1B(x, r)l is its n-volume. Now for j > J, & is a nonnegative function in C,i(Q), and it follows from [3, p. 1451 that

B(O.j-')

for i = l,..., n. It also follows that so A(1 +i - #+ 1) dx -+ 0 Consequently, we obtain from sn G,(Du) Di(di - #+) dx -+ 0 JnF(x, u)($~ - $+) dx -+ 0 asj

from Jensen’s inequality and Fubini’s theorem and ssI A(1 D(& - #+)I) dx -+ 0 as j -+ CO. (1.6), (1.7), and (1.9) and (3.1) and (3.2) that as j -+ co. Likewise we see from (1.4) that + co. We conclude from (1.9) that that

$ Q(u, v; $j) = Q(u, v; ++).

(3.4)

But as stated earlier, & is a nonnegative function in C,l(fi) for j > J. Consequently, for j > J, Q(u, v; $J~),< 0, and (3.3) follows from (3.4). REMARK 3. Let u and v be in WJ(sZ) Qv < Qu weakly (A) in Sz. Then

Q[u, a; (u -

and (u - v)+ in W$*(G).

v)‘]

< 0.

Suppose that

(3.5)

It is clear from (1.4) through (1.7) that Q[u, v; (u - v)+] is well defined and finite. Set w = u - v. Since wf is in WiVA(sZ),it follows that there exists a sequence {&}F in Cam(G)such that ln A(1 4j - w+ I) dx and

r

-0

A[1 Dd(& - w+)l] dx + 0

267

WEAK COMPARISON PRINCIPLE

asj 3 00 for i = l,..., n. From (3.1) and (3.2), it follows that so 1$i - w+ / dx and ~n~Dt(~j-w+)~dx-+O asj-tco. Consequently, there exists a subsequence which, for simplicity, we also call {$j}F , such that & --f w+ and Di& -+ D,w+ as j -+ co almost everywhere in 52. But then it follows from [3, p. 1451 that +i+ -+ wf and D&+ -+ Diw+ almost everywhere in L?. From EgorofYs theorem and the uniform absolute continuity of

we then obtain that

s n

A(1 $j+ - wf I) dx

and

.cR

A(1 D&+

- w+)l) dx -+ 0

as j -+ co for i = l,..., n. From (3.2), we consequently conclude that lim [I +i+ - w+ [IA,0 = 0

j+m

and

ii& II WA+

- w+)lh

= 0

(3.6)

for i = l,..., n. Using (3.6) in conjunction with (3.1), we see from (1.8) that lim Q(u, V; $j+-) = Q(u, v; w+).

j-m

(3.7)

However, from Remark 2, we have that Q(u, v, &+) < 0. This fact in conjunction with (3.7) and the fact that w+ = (u - v)+ gives (3.5), and the proof of the Remark is complete.

4. PROOF OF THE THEOREM To prove the theorem, we set w=u-v

(4.1)

and observe by hypothesis that wf = max(w, 0) is in W$“(G). The conclusion of the theorem will follow once we show that w+ = 0 almost everywhere in Sz. By Remark 1, this fact in turn will follow once we show that Dwf = 0 almost everywhere in Q.

(4.2)

To establish this fact, we observe from Remark 3, (4.1), and the hypothesis of the theorem that Q(u, v; w’) < 0. (4.3) 505/36/2-7

268

VICTOR

L.

SHAPIRO

Set Sz, = (X : x in D and w(x) > 01.

(4.4)

Now, it follows that W+(X) = 0 almost everywhere in G - Q, . Also, from [3, p. 1451,we seethat Dwi- = 0 in Q - Q, and Dw+ = Dw in 9, . We conclude, consequently, from (1.8) and (4.3) that 0 >

a, {[G(Du) - G(Dv)] . Dw - [F(x, u) - F(x, v)]w} dx. s

(4.5)

Using (4. l), we see that the second term in the integrand in (4.5) can be rewritten as [F(x, 2, + w) - F(x, v)]w. From the hypothesis of the theorem and (4.4), we see that this expression is nonpositive. Since a minus sign precedes it in (49, we obtain from (4.5) that 02

s%

[G(Dv + Dw) - G(Du)] . Dw dx.

(4.6)

According then as A meets conditions (l.lO), (l.ll), or (1.12), we obtain respectively from (4.6) and the lemma that the following hold:

0 2 J W(I Dv I + I Dw I) - 41 Dv I)] I h I dx;

(4.7)

0 b 6, A’(( Dw ( 4-l) 1Dw 1dx;

(4.8)

0 2

(4.9)

fir

Ja,[A’(1 Dv I + 1Dw I) - A’(\ Dv 1+ 3 I Dw j 4-l)] I Dw 1dx. s

Next, we set 0, = {x : x in Q, and ) Dv(x)j < a>.

(4.10)

From the hypothesis of the theorem, it follows that I Dv j is in particular in Ll(sZ). Consequently, j Dv 1 is finite almost everywhere in Q, and we obtain therefore that I Q, - Q, I = 0,

(4.11)

where I E ) stands for the n-dimensional Lebesgue measure of E. Next, we recall that A’(0) = 0, A’(t) > 0 for t > 0, and A’(t) is strictly increasing for 0 < t < co.

(4.12)

WEAK COMPARISON PRINCIPLE

269

From (4.10), (4.11), and (4.12), we conclude respectively from (4.7), (4.8), and (4.9) [and therefore respectively from (1. lo), (1.1 l), and (1.12)] that

[A'(1 Dv j + 1Dw 1)- A'(\ Dv I)] 1Dw 1 = 0 a.e. in Q, ; (4.13)

A'(1 Dw j 4-l) 1Dw I = 0 a.e. in 52, ; (4.14)

[A'(] Dv ! + I Dw I) - A'(1 Dv / + 3 1Dw j 4-l)] I Dw / = 0 a.e. in Q, . (4.15) In all three cases above, it follows immediately from (4.12) that I Dw I = 0 almost everywhere in L?, , and therefore from (4.10) and (4.11) almost everywhere in L$ . From (4.4) and [3, p. 1451, we have that Dw+ = Dw in Q, . Therefore Dw+ = 0 almost everywhere in Qn, . Once again, from (4.4) and [3, p. 1451, we have that Dw+ = 0 in Q - Qn, . We conclude that Dwf = 0 almost everywhere in 52. Statement (4.2) is established and the proof of the theorem is complete.

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