COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, VOL.

XIII, 457-468 (1960)

A New Proof of de Giorgi's Theorem

Concerning the Regularity Problem
for Elliptic Differential Equations
J U R G E N MOSER

9 1 . The regularity problem for variational problems in several dimen-

sions in its simplest form requires to prove that the extremais of the integral

/ * f ( % >U Z B ' * * (9 %JdXl * . ax,,

under certain boundary conditions are necessarily analytic provided the
-
integrand f ( p l , p 2 , *, p , ) is a real analytic function of its arguments satis-
fying
n 12 n
(1 1 A-lIR 5c iWkP'(P)Ml 5 A Iff
k=l k,Z=l k=l

-
for all real PI, ' -,f i n , El, * * ., t,, ;Z being a positive constant. This problem
has been solved under various differentiability assumptions on u, mainly by
E. Hopf [l, 21. From the point of view of direct methods of calculus of
variations it is desirable to require only that 24 has Lebesque measurable first
derivatives for which

In a recent paper [4] de Giorgi succeeded for the first time to establish the
analytic character of the extremal under the assumption (2). Previously such
a result had been obtained for n = 2 by Morrey [3], but his proof could not
be generalized to higher dimensions. Simultaneously with [4] J. Nash [5]
solved the corresponding problem for parabolic equations by a completely
different approach.
The essential theorem in [4] concerns the Holder continuity of weak
solutions of linear elliptic differential equations which will be formulated
below. In this note we want to give a new derivation of this result. The
key lies in two theorems which are proven in Section 4. It was our aim to
derive these two theorems from a simple general principle which consists in
estimating the square integrals of convex functions f ( u ) of a solution %.
Theorem 1 will be obtained by such estimates applied to the powers f(u)
= Iujp, p 2 1, and for Theorem 2 it suffices to take log+ u-l. In Section 5
457
458 J. MOSER

we follow de Giorgi to derive his result from Theorems 1 and 2. In order to

bring out clearly the underlying ideas we refrain from any generalization
which would complicate the technical part.1 We want to mention that our
method also allows a simple proof of Nash's result [5].
This paper originated in a seminar in the course of which Dr. R. Peder-
son and Dr. E. Rodemich contributed several suggestions. Especially I want
to express my thanks to K. 0. Friedrichs who proposed essential simplifica-
tions and changes of the manuscript.
-
§ 2 . NOTATION.The real vector x = ( x l , . ., xn) varies over an open
set Q in which the real valued functions 'u(x),+(x) are defined; urndenotes the
vector (uzl, - * u.,) wherever it exists; a ( x ) denotes the matrix ak,(x)and
0 ,

TI

We assume that a is a positive definite symmetric matrix satisfying

(3) A-l(E, 6) 5 (E, a t ) IA(t,E )

for x in .f2 for all real t;3, > 1 is a constant (independent of x). I n the follow-
ing we will denote by c positive constants which depend on the dimension ?a
and A only. We will not distinguish these constants by subscripts but provide
that the constants c can be enlarged without invalidating the inequalities
considered.
Assume that u(x)is a weak solution of the partial differential equation

i.e.,
(4) s, (4zt auzvx =0

for any C" function $ ( x ) with compact support in 9.Throughout we assume

that a ( z ) , u ( z ) , zkrn2(x)are Lebesque measurable and that

jD(%, 'urn)dx <

so that (4)is meaningful. (Observe that (3) implies the boundedness of the
matrix a(.).)
Let 0'C SZ be a compact subdomain of SZ such that for every y E SZ' the
sphere Ix-yl < 6 is contained in Q.
'In [7] Stampaccliia rederives de Giorgi's result and studies second order differential
equations with lower order terms and the boundedness of square integrable solutions. Such
differential equations were also discussed by C. B. Morrey in [S] where the discreteness of the
spectrum for the corresponding eigenvalue problem is pyoven. These refercnces I owe to
L. Nirenberg.
 REGULARITY PROBLEM FOR ELLIPTIC DIFFERENTIAL EQUATIONS 459

THEOREM (de Giorgi). Let u(x) be a solution of (4) and

Then there exist two positive constants u, fi depending on n, I. and 6 only

suclz that
(6) lu(x)-u(y)l 5 Blx-yl“ for x , y E 0’.
Remark. In the formulation of condition (4) that u is a weak solution,
one can admit a wider class of functions (b(x),namely the functions for which

(7)
and which are of compact support.
9 3. It is well known that the weak solutions v = u of (4) satisfy an
inequality

whenever the sphere IX-X’~< p f o about x’ is entirely in a.The crucial

point of the following proof is the fact that also
= f (u)
satisfies (8) whenever f (u)is a non-negative convex function.
In fact it suffices that v is a non-negative “subsolution”2: (av,), 2 0.
To formulate this in a weak form we call v a subsolution if

(9) j+Q(+, av,)dx 5 0 for +(x) 2 0,

where (b(x) is of compact support and infinitely often differentiable.
LEMMA 1, The estimate ( 8 ) holds for any non-negative subsolutions of ( 3 ) ,
in particular for
8 = f(u),

where u ( x )is a solution of (4) and f ( u )a non-negative convex function, provided

tlie integrals in (8) exist. If v is a subsolution so is f ( v ) , @rovided f is non-
negative, convex, and monotone increasing.
Proof: Without loss of generality we take x’ = 0 assuming that
1x1 < p+o is contained in SZ. By ~ ( xwe
) denote a function of compact sup-
port in 1x1 < p+o with a piecewise continuous derivative. Then in (9) let
d ( 4 = “V2
which is non-negative and of compact support. Therefore, by (9)

analogy t o a subharmonic function

460 J . MOSER

Choosing for q ( x ) a function which is piecewise linear in 1x1 and is equal t o 0

for 1x1 > p f o and equal to 1 for 1x1 < p, we obtain jqzl S o-* and

which proves the estimate for subsolutions.

We show now that every non-negative convex function v = f (u)yields
such a subsolution. First we assume that f has a continuous second derivative
f”(u) which vanishes for 1uI > M ;the convexity implies f “ ( u ) 2 0. Let
y(x) 2 0 be of compact support and
d(x) = f ’ ( u ) y ( x ) .
Then
(d%> = (YZ9 a%)+f”w(%P
(14)
2 (ye,a%).
Integrating over x E Q yields
0 L J (y,, av,)dx;
this proves (9) for any y 2 0 of compact support which is infinitely often
differentiable. In fact, the assumption f” = 0 for lzll > M insures that
9, = f’YX+f’l@XY
is square integrable.
If here u is only a subsolution and f ’ ( u ) 0, then
d ( X ) = f ’ ( u ) y ( x )P 0
and the relation
0 1
1 (A?,au,)dx 2 j (y,, av,)dz,
implied by the above argument, shows that v = f ( u ) is a subsolution, which
proves the last statement of Lemma 1 in case f” is continuous and vanishes
for large arguments.
An arbitrary convex function f ( u ) can be approximated by a sequence
 REGULARITY PROBLEM FOR ELLIPTIC DIFFERENTIAL EQUATIONS 461

of twice continuously differentiable convex functions f,(zl) such that fi = 0

for large IuI, f , 4 f , f k ( u )--f f ' ( u ) , where f'(u) exists. Then for v, =fm(u)
one has

The existence of J v 2 d x implies that the right-hand side converges to

On the other hand, by Fatou's theorem

which proves the lemma.

LEMMA 2 . Let w be defined in 1x1 < p and w,w, be square integrable.
Then there exists u constant c, which depends on n and the choice of co such that

for every K in 1 5 K 5 n / ( n - l ) . Here N i s any measurable set in 1x1 < p

of measure m ( N ) 2 cclp". I n the following c i l will be fixed to be half the
volume of the unit sphere and c,, n 2 2, depends on n only.
Remarks. 1 ) For the case that the set N agrees with the sphere 1x1 < p,
Lemma 2 is contained in an inequality of S. Sobolev [S] (Section 7, formula
( 7 . 4 ) ) .For completeness we give a proof of Lemma 2 at the end of this paper.
2 ) The exponent K in (15) does not have to be 5 n./(n-l), however
the limitation K 5 n/(n-2) is essential.
3) The existence of the integral in (15) follows from the finiteness of
the integrals on the right.
Using Lemmas 1 and 2 it is possible to estimate the square integral of
wy in terms of the square integral of w for any non-negative subsolution w.
For this purpose let N be the sphere 1x1 < p in Lemma 2 and apply (8) to w:
462 J. MOSER

Assuming CT 5 p we have with a new G

which is valid for any non-negative subsolution w.

Finally we make the simple observation: If #o > 0 and
(17) 0 < 4 v 5 cv4;-1, v = l , . 2 , * * ~ ; f c1 >,
then

(17’)

where c1 = C K / ( K - 1 ) 2 . Defining the sequence y,, by

V+l-K-‘V
Y Y = c1 +”,
( 1 7 ) goes over into the inequality

0 < yv IyL1
which implies
KV
YY IYo *

This makes the statement (17’) evident.

tj 4. T H E O R E M 1 . Let v ( x ) 2 0 be a subsolwtion in the sense of ( 9 ) which
is defined in 1x1 < 2r. Then
(18) V y x ) 5 CY-”
SI, <25 v2dx
for almost all x irt 1x1 < Y .
Remark. This result will be applied to functions f (u) of a solution u,
where f is a non-negative convex function of u. For v = Iu/ one obtains n
bound
Ju15 Gs-nlz (jpx)”
for all x in a subset Q’ with the property that a sphere of radius 6 about
x E 12’ is contained in Q.
Proof: Since v ( x ) is a subsolution, so is
w = /v(x)/P= v(x)”
for fi 2 1 because f ( v ) = v p is a non-negative convex function with f ‘ ( v ) 2 0
for v 2 0. Let p = K” and
w,,= 8, v = 0, 1, 2, .) -
where K = n / ( n - 1 ) .
According to (16) we can estimate higher and hi,gher norms of v. Let
-
Pr 2 po > p1 > * be a sequence of positive numbers satisfying pVv1 5 Zp,,
 REGULA4RITYPROBLEM FOR ELLIPTIC DIFFERENTIAL EQUATIONS 463
then (16) applied to w = w , , - ~ , p = p v , u = ~ , , - ~ - 5
p ~pu yields

Choosing, for instance, pv = ~ ( l + 2 - ~which

) implies py-l 2 Zp, and

--= Pu 2v+1 5 3”)

Pu-1- P v
we find
4”5 c2 1 O K U q q - I s c’(b;-tl
with a new constant c. By a previous remark, (17), (17’), we conclude

v4w

Since the left-hand side tends to the essential maximum of zP, Theorem 1 is
established.
The following theorem represents a Harnack type inequality which
refers to non-negative solutions u. The assumption that .u is not identically 0
is expressed by the requirement that the set in 1x1 < r, where u > 1, has at
least the measiire c;’rn with an appropriate constant c,, > 0:
(19) m{u > I : 1x1 < Y ) > c,‘r’”.
THEOREM 2. Let u 2 0 be a solution of (4) in 1x1 < 2r satisfying ( 1 9 ) .
T h e n there is a constant c > 0 dePending on n and 1 only such that
r
~ ( z>)c-1 in 1x1 < --.
2
Proof:3 Using a procedure similar to that by which we derived inequality
(8) for all ZI = f ( u ) ,when f 2 0 is a convex function, we now derive

(20)

for functions v = f ( u ) for which also k = -e-f is a convex function. To

prove (20) we consider first functions f which are twice continuously differ-
entiable. Then the convexity of lit implies
(21) f”-f‘2 = efh” 2
- 0,
Let $(x) = f ’ y f z ) ,where y 2 0 is of compact support in 1x1 < Br. For f’ f 0
one has
(#z9 a%) = (Yz, a.x)+f”Y(% au,)
f”
= (Ym a%)+ -v(vx, 4.
Y2
*This proof was suggested by a similar argument for the two dimensional case in the
appendix of [6].
464 J. MOSER

We used this identity before in (14), estimating the second term by 2 0.

Now the following term represents the essential contribution: With (21)
we obtain
1
0 2 (yz a w x + J Y (vz2
I

or with y = q2, where r] again is a function of compact support in 1x1 < 2r,
we find
J-(9Vz)2d” 5 AJqZ(v,, av,)dx 5 %AJI(??,> a??vz)ldx
5 212 (jq:dx)” (1 (qvz)Zdx)%,
and with a new c
1 s
(r]vz)2dx5 c q:dx.
Choosing for q a function which is piecewise linear in 1x1 and equal to 1 in
1x1 < Y , one obtains (20).
We apply (20) t o
v = f ( u ) = Max {--log ( u + E ) , O), O<E<l.
Then
h = Max (-(~c+E), -1)
is obviously convex and v is defined because u > 0. Since by (19) v = 0
on a set of measure > c g l i n , Lemma 2 with K = 1 and ( 2 0 ) yields

Y-” jlZl
< r v2dx 5 ~ r 1,~
xl<r
- ~< c3.
v:dx

On the other hand f is convex and non-negative. Therefore Theorem 1gives,

for 1x1 < r/2,

whence, by the definition of v, -log (U+&) S c, or

Y
U+E 2 e-c in 1x1 < -
2
for allE in 0 < E < 1. For E .j0 one obtains Theorem 2 .
5
5 . Finally to prove the theorem of de Giorgi stated in Section 2 we
observe that, by the remark made after Theorem 1, a solution u satisfying
(5) is bounded in every compact subdomain Q’ by
lu(x)l < cs-n/2
provided a sphere of radius 6 about every x in Q’ is contained in Q.
I t is the aim to estimate the oscillation of u ( x )in 1x1 < p in dependence
 REGULARITY PROBLEM FOR ELLIPTIC DIFFERENTIAL EQUATIONS 465

on p *: Let
o ( p ) = Max u(z)-Min a(%),
1x1 <P 1x1< P

assuming that the sphere ($1 < p lies in SZ'. Obviously

o ( p ) < 2c6-*/2.

Fixing p = %r 5 6 and adding an appropriate constant to u (which does not

alter the oscillation) we can assume that
Max U ( Z ) = -Min U(X) = i o ( 2 r ) = hf.
I4<P 14<P
Then
M+ u U
M- U-
U
~- =1+-, - - 1--
M M M M
are also solutions of (4). They are both non-negative and at least one of
them satisfies condition (19) of Theorem 2 (the constant c;' being half the
volume of the sphere 1x1 < 1) depending on whether u 2 0 or zh 5 0 occurs
more frequently. Taking the first case we obtain by Theorem 2
Ui-M Y P
> c-l in 1x1 < -- = -- ,
M 2 4
or

In any case we arrive at

o ($) 5 M ( 2 - c - - l ) = o(p)(1-(2c)-l)

for p 5 8 . Applying the above estimate repeatedly we find for I= 4-mp

where cc = -*log (1- (2c)-I) = c.;' For every r 5 6 one can find an in-
teger fit 2 0 such that p = 4mr lies in

which gives

'In this argument we follow de Giorgi.

SMax, Min stand for the essential maximum, minimum.
6Added in proof: This inequality can be used to establish a Liouville tvpe theorem:
Any solution which is bounded in the whole space is constant. With w = limw(p) < UJ
one obtains 0 5 w (1-(2c)-l)w, hence w = 0. P-+W
466 J. MOSER

if 1x1 < 6 lies in 52'. Now let Q" consist of those points x in 0 for which the
sphere of radius 26 about x lies in Q. Then the sphere of radius 6 about x
lies in SZ' and we have for any two points x,y in a''
Izl(x)-zl(y)I 5 o(/z-y/) 5 C d - ( n / 2 ) - " I X - y / @
if lx--yl 5 6. On the other hand if Ix-yl > 6, one has trivially

which proves the statement (6) with

a = c;l, fj = C 6 6 - ( n / 2 ) - a .

5
6. PROOF OF LEMMA 2. Although the methods in the following proof
are not new, we add it for the sake of completeness. A recent proof of
Sobolev's inequalities can be found in lecture notes of L. Nirenberg (held at
Pisa 1958) which were not available to the author.
Proof: Let 1x1 < p and y range over any subset N of 1x1 < p with a
measure m ( N )> colp". Introducing polar coordinates for y with the center
x we define (y-x( = r ; y = x + r t , where 151 = 1. The proof is based on
the mean value theorem and the use of the Holder inequality:

or

Let 9 > 1 be a number as specified in the lemma and determine q such

that l/P+l/q = I., as needed for Holder's inequality. I n the following lines
we use the elementary inequality
 REGULARITY PROBLEM FOR ELLIPTIC DIFFERENTIAL EQU.4TIONS 467

u p f b p I( a + b ) ~5 2"-1(up+bP)
for u 1 0, b 2 0, which follows immediately by comparing l+xp and
( l + x ) p for x 2 0. Holder's inequality yields

From equation (22) we obtain, with p / q 2 p-1,

Noting that m (N)/p n> c;l we have, with a new c,

If fi 5 2n/(2n-l) < n / ( n - l ) , we have

so that

This inequality is stronger than (15). We derive (15) by applying

the last estimate to 7e11+~ in place of w. With 1 5 K 5 n/(n-1) and
p = ? K / ( K + ~5) 2n/(2n-l) one finds

using p ( l + ~ )= 2 K and l/$--+ = 1/2K the statement (15) fo~~ows

immedi-
ately.

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probleme, Math. Z., Vol, 30, 1929, pp. 404-413.
[?: Hopf, E., obey dett funktionalen, insbesondere den analytisrhen Chavakfer der Losi4nge)i
elliptischer Differentialgleichungen zweiter Ordnung, Math. Z . , \'d. 34, 1932, pp.
194-233.
468 J . MOSER

[3] Morrey. C. R . , On the solutions of quasilinear elliptic partial differential equations, Trans.
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[8] Sobolev, S., S z t v un thPovdme d'analyse fonctionnelle. Mat. Sbornik, Vol. 4, 1938, pp. 471-496.
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[I01 Stampacchia, G., Problemi al contorno ellittici con dati disrontinui dotati di solztzioni
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Received December, 1959.