COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, VOL. XIV, 577491 (1961)

K . 0. Friedrichs anniversary issue

On Harnacks Theorem for Elliptic Differential Equations*

J O R G E N MOSER

1. Introduction

The theorem of Harnack referred to in the title is the following: If u is a positive harmonic function in a domain D, then in any compact set D contained in D the inequality

rnax u 5 c min u, D D

holds where the constant c > 1 depends on D and D only. Equivalently, if 21 is normalized to take on the value 1 at some point of D one has

c-l 5 u 5 c in D with the same constant c.

This important theorem can be considered as a sharpened and quanti- tative version of the maximum principle. It is used in several existence theorems of potential theory since it establishes the compactness of a family of bounded harmonic functions,

In this paper we want to derive such a theorem for the solutions of uniformly elliptic differential equations of second order in selfadjoint form

where z = (xl, x2, - * , xn) and the matrix u = (avh(x ) ) is symmetric and positive definite. Moreover, the positive eigenvalues of u are assumed to lie between two positive constants, say between 1-1 and I , where the constant il > 1 is fixed for the following considerations. Otherwise the elements uvp(x) are only required to be Lebesque integrable functions.

The generality of the assumptions on the coefficients is called for if one studies nonlinear differential equations, where the coefficients uvp in (1.2) may depend on the solution u and its first derivatives for which no smooth- ness properties are available. In fact, one is lead to differential equations of this form if one studies the extremals of a variational problem

*This paper represents results obtained under Contract Nonr-285(46). Reproduction in whole or in part is permitted for any purpose of the United States Government.

51 7

578 J . MOSER

(1.3) 6 f F(w,)dx = 0, where F ( p ) is a twice continuously differentiable convex function of p = PI, - - - , pn). The first derivatives of w satisfy an equation of the form (1.2) with ayp = FP+. A recent famous result of de Giorgi [6] shows that the extremals w of (1.3) have Holder continuous first derivatives. We shall show in Section 5 that this result, i.e. the Holder continuity of the solutions u of ( l .2) , is a consequence of a generalized Harnack theorem.

Before stating the result we give formula (1.2) a precise meaning: For (weak) solutions of (1.2) in a domain D we require first of all that they belong to the closure of differentiable functions with the square norm

Such a function u will be called a solution of (1.2) if

$, (A> au,)dx = ? v , P'1 f, +&p%pd~ = 0 (1.4)

for any continuously differentiable function 4 whose support lies in a compact subdomain of D.

In analogy to potential theory we call u a subsolution if under the above assumptions

J D au,) dx 5 0 and 4 2 0.

Finally u is called a supersolution if -u is a subsolution. The main result of this paper is contained in

THEOREM 1. If is a fiositive solution of (1.4) in a domain D and if D' i s any compact set in D, then

Max M 5 c min u, D D'

wheere c depends on D, D' and 3, only. Remark. u > 0 in D means, of course, that the set where u I 0 in D is

of measure 0. Similarly, max, min in (1.5) stand for the essential maximum and minimum, respectively.

In Section 5 we shall show that the Holder continuity of the solutions is a simple consequence of Theorem 1. In Section 6 we follow Gilbarg and Serrin to study the behavior of solutions u near co using Theorem 1. In particular, it will be proven: If m is a bounded solution in 1x1 > p, then lim,++.m u exists. This result will be applied to minimal surfaces of n dimensions.

Like in all proofs concerning the regularity of a solution of elliptic differential equations, the argument is of an iterative nature, as for instance

ON HARNACKS THEOREM 579

in proofs where one shows that m-times differentiable solutions are (m+ 1)- times differentiable. However, the step in the iteration here is much finer and is measured in terms of the function

where m is the measure of D. This function @ ( p ) is monotone increasing and tends to the maximum or the minimum of zc in D as p tends to + co or - co. The iteration mentioned abovewillconsist in estimating @( (ni(n-2))p) in terms of a@). This method has been discussed in a previous paper [lo]. It is to be mentioned that the statement of Theorem 1 can be found at the end of the paper of J. Nash [7], however, no proof is given. For two dimen- sions such a result has been known before, see [3,4], but the methods cannot be generalized to n > 2.

I thank Professor K. 0. Friedrichs, whose encouragement and interest in the preliminaries of this work were essential for its completion.

2. Three Lemmata

The proof of the announced theorem will be based on general inequali- ties relating integrals of a function w = w(x) to integrals of the gradient of w. The inequalities are valid for general functions (for which the integrals considered exist) and not only for solutions of the differential equations. The first 2 Lemmata are due to PoincarC and Sobolev and are well-known. The third lemma was discovered by John and Nirenberg and is presented in a paper in this Journal. We shall state these inequalities and only justify the modifications of our formulation from the original statements.

We shall denote by Q = Qa@) a cube lxv-avl 5 h/2, Y = 1, 2, * * a , n, whose edges are parallel to the coordinate axis and are of side length 8. We use the notation

for the mean value of f over S. Of course, if the volume of S is normalized to be one, this mean value is simply equal to the integral. I - -

LEMMA 1 (PoincarC). If w, w, are square integrable in Q and

then

580 J . MOSER

(w-w,)"x 5 (A)2s, 37 w p c . s, For a proof we refer to [l]. The coefficient of the Dirichlet integral on

the right is 1-1, where A is the second eigenvalue of A u + h = 0 in Q, u having a vanishing normal derivative on the boundary of Q.

LEMMA 2 (Sobolev), If w, w, are square integrable, then for K = n/ (n- 2) one has

where defiends on n only. A particularly elegant proof for this result of Sobolev [2] can be found

in L. Nirenberg's paper [S]. There such an inequality is proven for functions of compact support. One can reduce the proof of this lemma to this case, as we want to show briefly: By a stretching of variables one can assume that h = 1, a = 0. We extend w to the cube Ix,[ 5 1 by reflection over the bound- ing hyperplanes of the cubes. Let +(x) be any continuously differentiable function with support in Ix,( < 1 which equals 1 in lxyl si. Then u = dze, is of compact support and by formula (2.4) in [S]

Since u: 5 2(+p+d2w:} ,

one obtains immediately the inequality of Lemma 2, where j3 still depends on the choice of 4.

LEMMA 3 (F. John, L. Nirenberg). If w i s square integrable in the unit cube Q(1): [q, 5 Q and if for every cube Q C Q ( l )

(W-WQf'dX 5 1, where W Q = $Qwdx, f Q then every power of w is integrable and even more: There exist positive constants a, j3, de$ending on n only, such that

The proof of a sharper version of this lemma. is presented in this issue [9]. The assumption there is that

f Q Iw-WQ[ dx 5 1

ON HARNACKS THEOREM 581

for some constant wQ which is a weaker requirement since by Schwarz inequality for any square integrable function f (s)

j Q P Z 2 (IQ I f l q . The statement of [9] is

jQ(l) ealw-woldx s /I for some constant wo. This implies that

$Q(ll e a ( W - W o ) ciz 5 p, jQ (1 ~ e-a(e-mo) ax < - 0. The product of these two inequalities gives Lemma 3.

3. Outline for the Proof of Theorem 1

By a covering argument the proof of Theorem 1 can be reduced to the case where D and D are two concentric cubes: D: 1x,/ < 2 and D: Iz,1 5 i. We shall indicate the plan of the proof for this case.

Let Q(h) denote the .cube Iz,I < h/2 and assume that u is a positive solution in Q(4). For the following we fix the choice of u and consider the function

where p ranges over all real values. We shall show that @ exists for all real values of f~ if 0 < h < 1. From Holders inequality it is clear that @@, 1) is monotone increasing in p. Moreover,

lim @($, 1) = maxu,

lim @($, 1) = minu, p++CQ Q (1)

p++ CQ Q (1)

so that the theorem takes the form

(3.2) @(+a), 1) I c,@(--, 1)

if zt is a positive solution in Q(4). Here like later on co, cl, * * * denote positive constants which depend on n and L only.

The proof will be given in two parts corresponding to the following two theorems :

If u is a positive subsolution in Q(4), then for ;b > 1 THEOREM 2.

582 J . hfOSEK

OY

THEOREM 3. If u > 0 is a supersolution in Q(4), then

OY

In particular, for $I = (1+K)/2, and for a positive solution in Q(4) of the differential equation one finds

max u = @(+a, 1) 5 c@(P, 2) 5 c@(#, 3) 5 c,@(-co, 1) = min u, Q(1) Q (1)

which is the desired inequality. The proof of Theorems 2 and 3 will be given in the following section.

Here we observe that Theorem 1 is an immediate consequence of (3.2). Namely, since under translating and stretching of the variables the in- equalities remain unchanged one has

max u 5 co min u

if u is a positive solution in Qa(412), where Q,(h) and Qu(4h) are concentric cubes of side length k and 412, respectively, with the center 2 = a. By Heine Borels theorem one can cover the closed set D (of Theorem 1) with finitely many cubes Qu(h) for which Q,(4h) C D. If this number is N , then one concludes from (3.3) that

&a@) Qa(h) (3.3)

maxu S c: min u if u > 0 in D, D D

which proves Theorem 1.

4. Proof of Theorems 2 and 3

In this section we want to show that Q(+co, h ) / Q ( - co, 12) remains bounded. This will be done by estimating Q($I, h)/Q(@, h) for $I > fi and deriving the desired estimate by iteration. For this purpose we present an

*Added in prool. As p approaches K the coefficient on the right-hand side tends to m. That the number K = FZ/ (n -2) has the correct value is exemplified by the truncated fundamen- tal solution rnax {Fa, M } which indeed is superharmonic (n > 2).

ON HARNACKS THEOREM 583

inequality for powers v = G of solutions % in which the square integral of the gradient of v is estimated in terms of the square integral of v.

L E m A 4. If u > 0 i s a subsolution in D and

v = G k , k >+, then foi any function q ( x ) with sufiflort in D one has

provided the integral on the right exists. The same assertion i s true for super- solutions if k < $.

Proof: The definition of subsolutions shows that

(4.1) I(+,, au,)dJ: 5 o for + 2 o in D, where 4 is a function with support in D. The proof rests on an appropriate choice of 4(z). In general if

er = I(.) = G k we shall set

4 = f/flq2 = lkltP-q2, where q is another function of compact support1. Because of

(sign f) (4,) a%) = (ff+f)r% au,)+2ff(a, a % h

one has

We distinguish the two cases k > from (4.1)

and k < #. In the first case one has

Lemma 4 is trivial for k = 0. Therefore we can assume k # 0.

584 J . MOSER

Dividing by the square root of the last integral one arrives at the desired inequality for k > $ with c3 = 3p.

For k < one can use exactly the same argument since then (2R--l)/[kl becomes negative and also the inequality (4.1) has to be reversed. This proves Lemma 4.

To get an inequality for @($, h) from Lemma 4 we set D = Q(h) and choose q so that it is = 1 in Q(h), where 0 < h < h 5 2h. To fix the ideas let [(x,) be a piecewise linear and continuous function so that [(z,) = 0 in (xll > h/2, c(zl) = 1 in lxll < h/2 and linear in the intervals h < 21x11 < h. Defining

q(x) = min [(x,), v=l,--,n

one finds that n L

17121 5 - h-h * Therefore Lemma 4 yields

2k 2 s Q (h) Combining this inequality with Sobolevs (Lemma 2), we find for h 5 2h

where y I c4 [(t -y (()+I]. 2k

2k- 1

Setting f i = 2k, which we require to be bounded away from zero by 191 2 4 = aA-2 3-12 8-1

(where cc is the constant of Lemma 3) , one has

if also 62 5 2h. Therefore 1 / K P

@(qb, h) = (1 v z x d x ) 5 c:/.(i -1)- (L) z / p @ ( f i , h) , Q (h 1 fi--1

-2/P 2 / p

@ ( K f i , 5 c6 (; -l) () @(f i , h) 9-1

(4.2)

ON HARNACKS THEOREM 585

provided 0 < h< h < 2h and p 2 q, fi # 1.

Similarly for 9 < 0

(4.3)

Now let P v = K V P ,

h = h, = 1+2-,

h = hv+l > -, h 2

then one finds

Iteration yields

P > 1, Y = 0, 1, 2, - - -,

v = 1,2;*- .

Since

lim sup @(#, h,) 2 @(+a, I), v+m

the proof of Theorem 2 is complete. For the proof of Theorem 3 we assume first that u > E > 0, where E

is a constant, thus ensuring the existence of @&, A) for p < 0. If the ine- quality has been proven under this additional condition, one can apply it to USE to get

For E -+ 0 one finds the desired inequality for positive supersolutions.

tains from (4.3) Applying the iteration, just described for f > 0, for 9 5 -q one ob-

(4.4) c,@(-m, 1) 2 @(-q, 2).

Similarly, since there are only finitely many K in the interval q d p < K, one has

586 J . MOSER

(4.5) @@, 3) 5 C l o q f q , 2) by finitely many applications of (4.2).

It remains to establish

(4.6) @((I, 2) I C11@(-% 2). For this purpose we set

1 v = log 21, 4 = - 7 2 ,

U

so that

(4% 9 a 4 = - T2 (vz > QV,) + 211 (7% 3 4. Since u is a supersolution, it follows that

V2(% QV%)dX d 2 (%, aqv,)dx I Q ( 3 ) IQ (3)

or, using Schwarz' inequality

q2v:dx 5 4A4 VEdx.

Let Q = Qa(h) be any cube in Q ( 2 ) and choose q appropriately, so that 77 = 1 in Q and has support in Q(3). One can achieve that lqzl 5 4/h so that

IQ (3) I Q W SQ v:dx 5 cla . 12-2 dx with c12 = 3n43A4, SQ

or by PoincarC's inequality

1, ( T J - - Z J ~ ) ~ ~ X 5 c12 dx for all Q C Q(2). s, This makes Lemma 3 applicable to the function w = C;;'~V = c:'~ log u

and yields

- where q = a/dc12 = C C / ~ P ~ ~ / ~ . Taking the q-th root, one arrives at

@(q* 2) 5 b"Q@(-q, 2), which proves (4.6).

Finally combining (4.5), (4.6) and (4.4), gives

@(A 3) 5 c10 - c11 * c,@(--, I ) , which proves Theorem 3 and thus also Theorem I..

ON HARNACKS THEOREM 587

5. Holder Continuity of the Solutions

We want to show now that the Holder continuity of the solutions at interior points is a simple consequence of Theorem 1. Let z = 0 be in the domain D of definition of a solution zb and assume 1x1 < 1 is contained in D. Let

M ( r ) = max u, p(r ) = min u, O < r < l , Izllsa 01 I T

which are understood as essential maximum and essential minimum. We apply Theorem 1 to the domains

Y D: Is1 < I and D: 1x1 5 Y = -

2

and set M = M ( r ) , M = M(Y), etc. Then

M-u, u-p

are positive solutions in D. Thus by Theorem 1

max (M-u) = (M-p) 5 c(M-M) = c min (M--21),a D D

(M-p) 5 c(p-p).

Adding the inequalities gives

c-1

c + l ( W - p ) 5- (M-p) .

Denoting the oscillation M ( r ) - p ( r ) of u in 1x1 < r by u)(r), we have

where 0 = (c-l)/(c+l) < 1. Iteration of this relation yields o ( y * 2-) I OYw(r) for Y = 1, 2, - * -, O < r < l ,

which leads to

This proves the Holder continuity of the solutions.

PBy homogeneity c is independent of r .

588 J. MOSER

6. Behavior of Solutions at 00

One can use Harnack's theorem to study the behavior of solutions near co, as was noticed by Gilbarg and Serrin [ 5 ] . But since Harnack's inequality was available only under some continuity assumption on the coefficients for n > 2 their results are not applicable to nonlinear equations. We give the discussion in a slightly different version.

Let M ( r ) = max u(x), p ( r ) = min u(x) ,

Ix[='P 1xI=r

which are continuous functions in 1 < I < co. From the maximum prin- ciple it is clear that M ( r ) cannot have two relative minima in 1 < Y < 00 since otherwise there would exist a relative maximum of u, contrary to the maximum principle (applied to an annulus), If such a minimum of M ( r ) occurs at Y = t , we consider Y > t where M ( r ) is increasing. Otherwise M is decreasing and we set t = co.

'If it occurs at r =z, the function p is decreasing in I > t. Otherwise p ( r ) is increasing and we set t = co. All together 4 cases arise from the 4 possible choices: t < 03 or = oo, t < 00 or = co.

In particular, if u is a solution defined in the whole space, M ( r ) and p ( r ) are defined for 0 5 r < co and it is clear (from the maximum principle) that t = z = 0. Hence M ( r ) is increasing and p ( r ) is decreasing.

If M ( r ) i s increasing, p ( r ) decreasing for r > ro, then M ( r ) - p ( r ) tends to 00 at least like a power of Y (.provided u i s not aconstant).

COROLLARY. T h e oscillation of a solution, defined in the whole space over 1x1 < I , grows at least like a fiositive power of r, firovided u is not a constant.

Proof: Consider u in the annulus

Similarly p ( r ) can have only one maximum.

THEOREM 4.

which plays the role of D in Theorem 1. For D' we take (XI = 1.

M(2r) -24, 24-p(2r)

M (2r) -p (r) I c( M (2r) --M (r) ) .) M ( r ) - d 2 r ) 5 C(P(+P(24)>

Since u assumes its maximum and its minimum at 27 in the solutions

are positive in D. Theorem 1 yields

ON HARNACKS THEOREM 589

where c is again independent of r , and depends on n, 1 only. Adding, one obtains for O ( Y ) = M ( r ) - p ( r )

w (27) +w (r) 5 c( 0 (2r) -w ( Y ) ) or

which proves the theorem.

and -,u(r) are not both increasing. We want to prove In particular, if u is a bounded solution in 1x1 > 1, it follows that M(r)

THEOREM 5. If u is a bounded solution (noncons!ant) in 1x1 > 1, then w ( r ) = M ( r ) - p ( r ) -+ 0 us r -+ GO. **

Hence limjsl-too u = urn exists.

COROLLARY. If u is a solution in 1x1 > 1, then the oscillation of 11 on 1x1 = r tends either to 0 or to GO.

Proof: Since M ( r ) , -,u(r) are not both increasing, there are these cases to be investigated: a ) M ( r ) , -,u(r) decreasing for r > ro , and b) &f(r) , p ( r ) monotone in the same sense. In case a) we consider u in the annulus D: I < 1x1 < 4r and on D: 2r = 1x1, where 7 > ro . Since u takes on the niaxi- mum and the minimum on the inner sphere, the previous argument gives

c-1

c f l w(2r) 5 -u)(r)

which implies w(p) 5 cp-am(r) -+ 0 for p -+ GO, GC > 0. Finally in case b) we can assume that M ( r ) , p ( r ) both are decreasing. Without loss of general- ity we can assume that

lim p ( r ) = 0, r - m

so that u > 0 in 1x1 > y o . Applying Theorem 1 to the function zl in the annulus

r 2

D: - < 1x1 < 2r, D: 1x1 = I , we obtain

M(r) 5 cp(r ) +. 0. **Added in proof. With an additional argument one can prove that ~ - ~ w ( r ) remains

bounded for r + co if n > 2.

590 J. MOSER

Hence

which proves Theorem 6. Finally we remark that a solution defined in the whole z-space cannot

be bounded from below unless it is a constant. Without loss of generality we can assume that N > 0.

O ( Y ) = M ( r ) - p ( r ) = ( c - l )p ( r ) -j. 0,

Theorem 1 shows that

0 5 u(.) 5 Czl(O), proving that zd is bounded from above as well. Thus by Theorem 4 N is a constant. ***

7. Minimal Surfaces in Several Dimensions

Consider an n dimensional surface z = f (z ) emhedded in the (n+l)- dimensional x, z = zl , * - - , x,, z-space, n > 2, In generalization of the two-dimensional minimal surfaces we want to study surfaces for which the n-dimensional volume is a minimum for a given boundary. Ignoring the boundary curves we set up the differential equations for such surfaces. The %-dimensional volume induced by the Euclidean metric \dzl2+ (d.z)% is given by

1 m7m.k and one obtains as Euler equations

where W = 2/l+(fJ2. The solutions of this differential equation are analytic and we assume that such a solution

z = f ( x ) in 1x1 > 1 is given.

In conclusion we want to prove here a result which for 2 dimensions can be given in a much sharper version and in fact is then contained in Bernsteins theorem.

THEOREM 6. I/ x = f (z) is a solution of (7.1) in 1x1 > 1 and i f (7.2) If,l < B in 1.1 > 1, then

~

***Added in pvoof . In a similar manner one can study the behavior of solutions near a finite singularity. In fact, Dr. Hans Weinberger wrote by informal communication of a proof that every positive solution in 0 < (21 < 1 which is unbounded in 0 < (21 < 1/2 can be estimated there by A I z I * - ~ < u(3c) < B ~ z [ * - ~ , where A , B are positive constants. A similar result was mentioned to the author by Professor Hasiey Royden

ON HARNACK'S THEOREM 591

exists.

COROLLARY. If f (z) i s defirted foy all 5 and (7.2) holds, then f(z) i s linear, i.e. represents a n n-dimensional hy perplane.

Proof: For some k = 1, - * * , n set u, = i&).

Differentiating the differential equation (7.1) with respect to xk, one obtains

where with f l y = f,,

The largest eigenvalue of this matrix is W-1 and the smallest is W". Hence under the assumption (7.2) one has uniform ellipticity with A = (l+B2)3/2. Theorem 6 is an immediate consequence of Theorem 5. The Corollary is a consequence of the corollary of Theorem 4.

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[3] Bers, L., and Nirenberg, L., O n lineav aad non-linear elliptic bouadary value pvoblems in the plane, Atti Convegno Intern. Equat. Deriv. Parziali, Trieste, 1954, pp. 141-167.

i4] Serrin, J . , On the Havnack inequality for linear elliptic differential equations, J . Analyse Math., Vol. 4, 1954-1956, pp. 292-308.

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Received February, 1961.