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
I(u) -
(1.1)
! '(*' 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.
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.
(1.3) For a.e
Let o€W)1q
and write F*(G):{ueWl(G)lu-E<W:.r(G)}. A function
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:
uo(%q(G) is an extremal
1.4. Theorem.
Let B(*o,2r)cG. Then maxuo=cminus in B"(xr,r).
on alP and n.
The
constant c depends only
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
160
2. Preliminary lemmas
We flrst give the definition of monotone functions.
2.1. Definition.
monotone in G if
Let GcÄ'
be a bounded domain.
u(x) and
sup rz(x): sup
x€AD
x€D
inf
x€D
A function u€C(G)
is
u(x): inf a(x)
x€AD
for all domains D, DcG.
The following lemma gives an estimate for the oscillation of monotone functions 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.
Harnack's inequality in the borderline case
2.4. Lemma. Let
B(xo,4rl3)cG.
Then
dm
*{,,nlYul"
The constant co depends only
161
= co.
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 function 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
Write 5:
{x6B'(x6
#ru*,
: (l -* - r#)
uoxi+,
#
<",.
,4r13)l((x):A}. Suppose that x(8"(xo, arl3)\^1.
convexity and growth condition (1.3) yields
F(x,Yh)= (t
-f,
-, #) rr.,y
= (, -r, - rr#)
Since zo is minimizing
u o)
F@,y
+@
uo)
-t
for the integral (1.1), we
B
-, # r(., # ?
@7y-
tv
o
r)
u,.
get
Iarl})F(x,yu) drn(x) <-Bn(xs,arl})
I F(x,vh) dm(x)
Bn(xs,
-
Bn(xs,
=
Bn(xg,4'
IrlB)\S F(x,vh)dm(x)+{F(x,yh)dm(x)
S
f,.,
+PT#
(1-
@-'#)F(x'Yu')
Bn(xs,*f,,r,
lv€l'dm+
f
dm(x)
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.
The
Srppo GnaNluNo
162
Notice that B"(*0,
r)n,S:0.
dru
!,Y#
--, r)
Bn(xg,
The constant
c0 depends
The condition (1.3) implies
only
= -§-(å)
on al\
"
I
Y ..ln dm 5
co'
u*,*!nrur
and n. Our lernma is proved.
3. Proof for Theorem
L.4
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}
dm)'t' =- vctt".
= v(*o!*,rlYul'
Then we have
log (max u6*e)
-log (min u6 *e) =
yrät"
on the set -B'(xo, r). Finally we get
= st"t/" (minu6*e).
inequality follows if we let e*0.
maxuo+s
Harnack's
3.1. Remark. Let
f:(fr,...,f):
G-R"
be a quasiregular mapping; see [5],
of/miniof 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
[10]. Re§etnjak [11],
[2]
has shown that each of the coordinate functions
mizes a variational integral
proved earlier by Re§etnjak, who used Serrin's paper [13].
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.
F.,
and L. NRnNsrnc: On functions of bounded mean oscillation. - Comm. Pure Appl.
JonN,
[4]
Math. 14, 1961, 415-426.
[5] Meni:ro, O., S. Rrcruln, and J, VÄrsÄ1,Ä: Definitions for quasiregular mappings. - Ann'
Acad. Sci. Fenn. Ser.
AI
448,1969,
l-40.
[6] Monnrv, C. B.: Multiple integrals in the calculus of variations. - Springer-Verlag, BerlinHeidelberg-New York, 1966.
A new proof of de Giorgi's theorem concerning the regularity problem for elliptic
Mosnn,
J.:
[7]
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