HARNACK'S INEQUALITY IN THE BORDERLINE CASE

Annales Academire Scientiarum Fennicre Series A.

I.

Mathematica

Volumen 5, ^1980, 159-1,63

HARNACK'S INEQUALITY IN THE BORDERLINE CASE

SEPPO GRANLTIND

1. Introduction

Harnack's inequality for

^general

quasi-linear elliptic

equations has been proved

by Serrin

_[13]

and Trudinger

_[1a].

In both

^papers

the proofis

based

on

the

iteration method introduced by Moser in _{[7] and [8].} In this

method the

lemma of John

and

Nirenberg _[4] is

essential.

We

consider

variational integrals of the form

(1.1)

I(u) - vu(x)) dm(x),

! '(*'

where GcR" is a bounded domain, u<W:(q, and the kernel F: GXR"-R

satisfies

the following conditions:

(1.2) The functions x*F(x,Vu(x)) are

measurable

for all u(W:(q.

(1.3) For a.e x€G the function z* F(x, z) is convex and alrl=F(x, r)=frlzl for all z€R", whete

a,

B>0 ^are

^constants.

Let _o€W)1q and write F*(G):{ueWl(G)lu-E<W:.r(G)}. A ^function

uo(%q(G) is an extremal for the integral ^(1.1) if I(u)>I(u) for all uQfr*(G). lt

can be proved that

uo

is locally Hölder continuous in G;

see

[6, ^Theorem

^4.3.1]

and [3, Remark 5.7]. Let uobe an extremal for the integral (1.1) and uo(x)>0 for x€G. Harnack's inequality

takes

the following form:

1.4. Theorem. Let B(*o,2r)cG. Then maxuo=cminus in B"(xr,r).

The constant

c

depends

only on alP

and n.

Our proof is based on an oscillation lemma for monotone functions in

^the

space

ILj ^(G)n C(G).

Lemmas of this type have been proved by

Gehring

_[1,

Lemma l]

and Mostow [9, Lemma4.l]. This ^makes it ^possible to avoid the lemma of John and Nirenberg. In the

case

n:2 ^{a similar method}

has been used

earlier for linear elliptic equations;

see

Gilbarg-Trudinger _[2, p.

200].

doi:10.5186/aasfm.1980.0507

SEppo Gn.a.NluNo

2. Preliminary

lemmas

We flrst give the definition of monotone functions.

2.1. Definition. Let GcÄ'

^be

a bounded domain. A function u€C(G)

is

monotone in

G

if

sup

rz(x): sup u(x) and inf u(x): ^inf a(x)

x€D x€AD x€D x€AD

for all domains D, DcG.

The following lemma gives an estimate for the oscillation of monotone func- tions in the space W:(q.

2.2. Lemma. Let u€C(G)aWi(G) and B"(xo,2r)cG. If u is monotone in

G, we haue

,ffi

^,,

^{'} =' (rr.!*,,rlY ^ul"

^dm)u"'

The constant

y

depends

only

on n.

Proof. The inequality can

be easily

derived from the oscillation lemma proved in _[9, Lemma4.ll;

see

also [1, Lemma l].

The next lemma gives a weak

maximum principle for

the extremals of

the iutegral (1.1). In what follows _E€.W:(G) will

be fixed.

2.3. Lemma. Let uo(%r(G) be an

extremal

for the integral (l.l).

^Then

usis

monotone

in

G.

Proof.

Suppose

that

there

is

a

domain D, DcG, and

a

point xo€D

such

that

uo(xo)=ryaxuo(x):a.

Define ä(x):16

_{uo@),

a} for x(D. Then ö<9""(D) and Vä(x):0

a.e.

on

^the

set {x(Dluo@)=-a}. It follows from the condition (1.3) that

I r@,Yå)dm(x)- [ r@,Yur)dm(x).

DD

This is a contradiction

since zo

is an extremal in the

class

$,o(D). Thus :tBUo(x) :

*r.rg

ro(r).

We prove the corresponding equation for minimum values exactly by the

same

argument. It follows that

uo

is monotone in

G.

Let usQF*(O be an extremal for the integral (1.1) and no(x)>0 for x(G.

Define u(x):fsg uo(x) fot x(G.

160

Harnack's inequality in the borderline case 161

2.4. Lemma. Let B(xo,4rl3)cG.

Then

*{,,nlYul" ^dm =

^co.

The constant co depends

only on alB

and n.

Proof. We may assume that uo(x)>(n-11't" for x(Bn(xo,4rl3). If ^this

^is

not the

case

we consider the function )ur, where 2>0 is large enough. The func- tion luo is an extremal for a variational integral, which is of the form (l.l)

and satisfies

the structure condition

(1.3)

with the

same constants

a

^ar,Ld §.

Let (€Cf"(B"(*o,arp)) _be a non-negative function such that ((x):l for

x(Bo(xs,r) and lY((x)l=crlr. ^We

^choose

h(x) : _q@)+-!Q-.

us(y)"-r'

Then hEfi*,(Bn(*o,4rl3)),

and

it

^has

the

generalized derivatives

h*,

: _{uo,,+ffi €*,-(n -D #ru*,} : (l -* - r#) ^uoxi+, #

<",.

Write 5: _{x6B'(x6 ,4r13)l((x):A}. ^Suppose that x(8"(xo, arl3)\^1.

^The

convexity and growth condition

(1.3) yields

F(x,Yh)= _(t -f, -, _#) ^rr.,y

^{u o) + @}

-, _# ^r(., # ?

^o

^r)

= ^(, -r, - rr#) ^F@,y

^{uo) -t} B

@7y- ^tv

^u,.

= f,., ^(1- @-'#)F(x'Yu') ^dm(x)

Bn(xg,4'

+PT#

Bn(xs,*f,,r,

lv€l'dm+ f F(x,Yuu)dru(x1.

It follows from this that

(n_1),,.,o,n{,.,#F(x,Yuo)dm(x)=B#u^6!n,,,,|vc|,,dm.

Since zo

is minimizing for the integral (1.1),

we get

I ^{F(x,yu) drn(x)} <- I F(x,vh) dm(x)

Bn(xs, arl}) Bn(xs,arl})

- I F(x,vh)dm(x)+{F(x,yh)dm(x)

Bn(xs, rlB)\S S

162 Srppo GnaNluNo

Notice that B"(*0, r)n,S:0. The condition (1.3)

implies --,

!,Y# ^{dru = -§-(å)}

^" u*,*!nrur

I

Y

^..ln

dm 5

^co'

Bn(xg, r)

The constant

c0 depends

only on al\

^and

^{n. Our lernma is}

^proved.

Letuobe an extremal for the integral (1.1) and uo(x)>0 for x(G. Let e>0 and define u(x):1sg (ro(r)+u) for x€G. The function

^u

is monotone in G

since zo

is monotone by

Lemma

2.3. We obtain a bound for the oscillation of u from

Lemma

2.2

and

Lemma2.4:

,,?,r,f,

^{u}

= v(*o!*,rlYul' ^dm)'t'

^{=- vctt".}

Then

we have

log (max

u6*e) -log ^(min

^u6

*e) =

^yrät"

on the set

-B'(xo,

r). Finally

we get

maxuo+s = st"t/" (minu6*e).

Harnack's inequality follows if we let e*0.

3.1. Remark. Let f:(fr,...,f): ^{G-R" be a} quasiregular mapping;

see [5], [10]. Re§etnjak [11],

[2]

^{has shown}

^that

^each

^of the coordinate functions of/mini-

mizes a variational integral of the form (1.1). Then it follows from Theorem

1.4

that Harnack's inequality is valid for the coordinate functions. This fact

has been

proved earlier by

Re§etnjak,

who

used

Serrin's paper

_[13].

3. Proof for

Theorem L.4

References

[1] GrHnrNc,

F. W.:

Rings and quasiconformal mappings

in

space.

-

Trans. Amer' Math. Soc.

103, 1962,353-393.

[2] Grr-ranc,

D.,

and

N.

S. TnuorNcrn: Elliptic partial differential equations of second order. - Springer-Verlag, Berlin-Heidelberg-New Y ork, 1977.

[3] Gn.rNruNo, S.: Strong maximum principle

for a

quasilinear equation

with

applications. ^- Ann. Acad. Sci. Fenn. Ser. A

I

Math. Dissertationes 21, 1978,1---25.

[4] JonN, F., and L. NRnNsrnc: On functions of bounded mean oscillation. - Comm. Pure Appl.

Math. 14, 1961, 415-426.

[5] Meni:ro,

O.,

S.

Rrcruln,

and

J,

^VÄrsÄ1,Ä: Definitions

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Ann' Acad. Sci. Fenn. Ser.

AI

^448,1969,

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[6] Monnrv, C.

B.:

^{Multiple integrals}

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Springer-Verlag,

Berlin-

Heidelberg-New York, 1966.

[7] Mosnn,

J.: A

new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations. - Comm. Pure Appl. Math. 13, 1960,457-468.

Harnack's ineqi;aiity in the borderline case 163

[8] Mosrn, ^J.: On Harnack's theorem for elliptic differential equations. - Comm. Pure Appl. Math.

14, 7961, 577-591.

[9] Mosrow,

G.:

Quasi-conformal ^mappings in z-space and the rigidity of hyperbolic space forms. - lnst. Hautes Etudes Sci. Publ. Math. 34, 1968,53-104.

UOl RESprNrl.r,Ju.

G.:

Space mappings

with

bounded distortion.

-

Siberian Math.

J. 8,

1967, 466-487.

[11] Rn§nrN.ur,Ju.

G.:

Mappings

with

bounded deformation

as

extremals

of Dirichlet

type integrals. - Siberian Math. J. 9, 1.968,487498.

[12]

Rr§rrNrn,Ju. G.:

Extremal properties

of

mappings

rvith

bounded distortion.

-

Siberian Math. J. 10, 1969,962-969.

[13] Srnnru,

J.:

Local behavior

of

solutions

of

quasiJinear equations.

-

Acta

Math.

111, 1964, 247-102.

[14] TnuorNcsn, N. S.: On Harnack type inequalities and their application

to

quasilinear elliptic equations. - Comm. Pure Appl. Math.20, 1967,721-747.

Helsinki University of

Technology

Institute of Mathematics

SF-02150

Espoo

15

Finland

Received 16 I|lday 1979

Revision

received 5

October

1979