Copyright
Guido Stampacchia.

On some regular multiple integral problems in the calculus of variations online

. (page 1 of 3)
Online LibraryGuido StampacchiaOn some regular multiple integral problems in the calculus of variations → online text (page 1 of 3)
Font size
QR-code for this ebook


NEW YORK UNIVERSITY
INSTITUTE OF MATHEWATICAL SCIENCES

i-raARY IMM-NYU no. 300

4 Washington Place, New York 3, N. Y. JULY 1962



NEW YORK UNIVERSITY
COURANT INSTITUTE OF
MATHEMATICAL SCIENCES



On Some Regular Multiple Integral Problems in
the Calculus of Variations



GUIDO STAMP ACCHIA



PREPARED UNDER GRANT NSF-G14520
NATIONAL SCIENCE FOUNDATION



On Some Regular Multiple Integral Problems in
the Calculus of Variations

Guldo Stampacchia



The research reported in this document has been
sponsored by the National Science Foundafcion under
Grant No. NSF-G14520. Reproduction in whole or in
part is permitted for any purpose of the United
States Government.



On Some Regular Multiple Integral Problems in
the Calculus of Variations*

Guido Stampacchia

Many interesting results on the differentiability of the weak solutions
of variational problems for the general multiple integral



/



F(x, u, gradu) dx

Q

recently have been obtained by E. R. Buley [l], C. B. Morrey [15] and
Ladyzkenskaya and Ural'tseva [10, 11].

The first results in this direction were found by E. De Giorgi [3] in
the case when F depends only on the grad u and F satisfies the condition

m|€|^ < F (p)?,€. < m|||^ (0 < m < M)

' ' - p.p. "^ i J -

The results quoted above require that F satisfies a condition of the following
type



m(i + |pr) l?r £ F (x,u.p)€.?. < M(i + |pr)^icr

i j



with r > - 1/2.



This paper was written while the author was a Temporary Member of
the Courant Institute of Mathematical Sciences, New York University, during
the academic year 1961-62. This Temporary Membership program is supported
by the National Science :^'oundation. Contract No. NSF—G 14520.



i • ':'■[



2

All these results depend on the fundamental theorems on the HBlder
continuity of the solutions of linear elliptic equations in divergence form where
the leading coefficients are assumed to be only measurable and bounded
functions. The first theorems of this type have been proved by De Giorgi [3],
Nash [18], Moser [16, 17], Extensions of these theorems have been given by
C. B. Morrey [13], Stampacchia [24], Ladyzhenskaya and Ural' tseva [10].

To begin with I remark that the classical variational problem of the



777^



minimal surface, where F = Vl + p is not included in De Giorgi's theorem
and not even in the more general theorem quoted above. I will prove in this
paper that there exists a Lipschitz function solution of the variational problem
for the integral



JrCu) = / f(gradu)



dx = min



provided that the integral is regular, i.e., F satisfies the condition
f (p)€.€. > for ? ^

and the boundary values satisfy the bounded slope condition (B. S. C. ) (that
implies Q. has to be convex; for a more precise statement see theorem (8. 1)),
This theorem is related for the double integrals to a well-known theorem by
Haar and the B.S. C. is related to the "three points condition" (see Rado [19]).
A similar theorem, using a different approach, has been proved, independently,
by D. Gilbarg [7].



Such a statement may fail when F depends on the function u, as we
shall show with a counter-example (§9).

In order to investigate what happens when the integral depends on the
function u too, I consider the integrals of the particular form



/ I f{grad u) + G(x, u) | dx .



I(u)

9.

If I(u) is uniformly regular, i. e. , f(p) satisfies the condition

f (p)?.?. > i/(l+P^)^U|^ (1/ > 0, - 1/2 < r < 0)

and Q. is strictly convex, then under suitable assumptions for the function
G(x, u) I shall establish in § 9 the existence of Lipschitz solutions and study
their differentiability properties.

The assumptions imposed on the function G(x, u) can be, for instance,
of the following type:

There exists a constant R such that, uniformly with respect to x,
we have, with a - 2(f + 1),

G (x, u) + R

min lim — z — > ~vA(a,n) ,

n— + 00 |u|

G (x, u)

max lim ^ < + oo ,

n— - 00 |uj

where 3< [(af-l)n+a]/(n-a) and A(Q;,a) is defined by (5. 3). In particular.



.n



if c" = 0, the constant A{2,Q.) is the first eigenvalue of the boundary value
problem: Au + Xu = in C2, u = on 9^. Applications of these theorems
to boundary value problems for some nonlinear- elliptic equations are given in
§10.

In order to obtain the results mentioned above, I need an extension of
the maximum principle for second-order elliptic equations in divergence form
(§ 2), some properties of the averages of convex functions (§3) and a generaliza-
tion of the method of the barrier functions (§6).

I want to thank Louis Nirenberg and Jlirgen Moser for several discus-
sions about the results of this paper.

§ 1. Notations and Terminology

If a is a domain of the Euclidean space e", dO. denotes its
boundary and Q its closure. If P^ ip.,P^, ' • ' .9 ) is a vector in E we put
p^ = p^ + p^ + • • • + p^. We shall say that a function u(x) belongs to C {^)
if u is continuous together with all the partial derivatives of order < m in
a. We shall say that a function u(x) satisfies a H8lder condition in ^ with
exponent \ (0 < X < 1) (or that u(x) is H8lder continuous in Q with
exponent X) if

|u(x') - u(x")|



sup r-

x'.x"ca |x' - x" I

x'^x"



< +00



We shall say that u(x) satisfies a Lipschitz condition in T or, that is, is a
Lipschitz function in Q if in the last definition it is a = 1.



. :r.H6{i



nr.itV



We shall denote by C . (C?) the class of the functions of C d)
whose m -order derivatives are HBlder continuous in Q with exponent X.

A domain '? belongs to C [C? e C ] if the boundary 8^ can be
covered by a finite number of open portions each having an equation of the form

(1.1) X. = w. (x. , X. , • • • , X. ) ,

n n 1 2 n-1

where the functions u. have continuous [Holder continuous (with exponent X)]

n
derivatives of order m.

A function iix) defined on d1 belongs to C idCl) [C J if k ^ 'k 1 u if u ^ •'



> k



If 1 < a < n we shall denote often by a* the number such that



i_ - i _ i

* ~ a ~ n



l.a
vVe recall that, for all functions u e H (f2), it is

o



_ o "x en

L m i L (Q)



n-2) Hull ,. < ^Jlu,

where 3^ is a constant depending only on a and n. Through all this paper
we denote by ^ the smallest constant for which the Sobolev's inequality (1. 2)
holds. vVe shall define the class i>Jc of functions G(x, u) defined for x e Q
and u c E' which are continuous with respect to u together with G (x, u)
for almost all values x of Q. and measurable with respect to x for almost all
values u of E' .

For instance, the function ^{x)ip(u) where i(x) c L (SZ) and



M^i) c C (E^) belongs to the class OJb.



{ K



§ 2. A Maximum Principle For Weak So lutions ol
Sell- Adjoint Elliptic liquations

Let a. (x) (i, i = 1, 2, ' • • , n) be measurable and bounded functions in



n such that



(2.1) a..(x)?.?. > i/UI^ iv > 0. |?|^=?.5.) .

ij 1 J - ' ' ' ' i 1



Consider the self-adjoint elliptic operator



(2.2) M(u) = (a. u )



A function u(x) cH (Q) is an M-subsolution [ IVI-supersolutionl if



(2.3) / a,,(x)u V dx < [> 0]

y ij x^ x^ -

Q

for all V c H (€1) such that v > in CI.
o —

vVe shall say that u is bounded from above on dP. by a constant
^ 3 max u if there exists a Cauchy sequence of functions u c C (r?) such

X c an '"

that



u < max u [k < rnin uj then the



s aj5i.>;o i^torl:-



u x£fn



ni ai u U tisi-t



•iiHi.



function u - | u | [u - | u | ] belongs to H^(r?) /^ if^iQ). ^
In the same way we define min u.

xcan

We shall prove the following

THEOREM (2.1) (Maximum Principle). K u(x) c H^(Sl) is an
M- subsolution [M- supersolution ], where M(u) is the operator (2. 2)
satisfying (2. 1) then, almost everywhere in ^

u(x) < max u(x) , [u(x) > min u(x)] .

x€d9. X € asi

PROOF: Let ^ be the max u; suppose there exists a set of
xeJQ
positive measure E where u(x) > ^. Then the function



In fact if u is a Lipschitz function we have < u ( c H (P.) and



and if u c H (2) there exists a sequence \ u 7 such that u — u in H (f2)

with u c C,(n) (and then Lipschitz functions) and u < max u on dQ;
moreover



Ktl



^,.,^, 2 »"»l'H\a) - M^\^,



Since almost everywhere < u i — < u [ and, on 9S2, \ u
equals u , we can conclude that \ ^ \ belongs to H (S^) and that
u - j u > belongs to H^(a).



[ u(x) - "? in E

v(x) = u(x) - ] u(x) I

in CI - E



belongs to H (12)^ is non-negative, therefore, from (2. 3)



/a..u V dx = / a..(x)u u
IJ X. X. J XJ X. X.



dx <



and consequently, from (2. 1), grad u = in E. It follows that the two
functions u(x) and < u(x) f have the same derivatives in 9., and, since



they coincide on 9f2, they have to be equal in Q. The conclusion is that
a. e. u(x) < ^ and the theorem is proved.

A similar argument proves the theorem for the M-supersolutions.

We want to prove a sharper maximum principle

THEOREM (2. 2) (Maximum Principle). If the function u(x) c H iCl)
is such that



(2.4) / (a..(x)u - f.)v dx < [> 0] (v c H^Q), v > 0)

/ ij x. J x^ - - o



where (2. 1) is satisfied and f. € L^(a) (i = 1, 2. • ' ' , n) (p > n), then there



exists a constant K depending on n,p, n such that



(2.5) u(x) < max u + f 5Z l^i i o

xcBQ '' L^D)



(2.6) [u(x) > min u - - ^^^ llf.ll ] .

' L%)



10



Consider the weak solution u € H (Q) of the equation



{a..u - f.)v = .

IJ X. J X.



vVe know that the following estimate holds [22, 23]



(2.7) "^ax |u| < ^ ^ hK^m ^P"^"^



Putting



it is easy to show that w is an M-subsolution and that max w = max u.

xcdQ x€da
Using (2. 7) and the theorem (2. 1) the proof of theorem 2. 2 follows.

VVe state here some results which we need later,

LEMMA (2. 1). Let lii(t) be a non-negative, not increasing function

for t > k , such that fo r h > k > k
- o o

(2.8) (J(h) < ^ | 1 and substitute ?^{ (|x-y| -p)/(R-p) )u in place of v.
Using Schwarz's and Cauchy's inequalities we get easily (2.14). In order to
obtain (2. 15) we write (2. 13) in the following way



/ia..(u-u) - (f.-a..u )>v dx =
r xj X.J ij X. ) X.



and we substitute S { ( |x-y | -p)/{R-p) )(u - u) in place of v.

We recall also the following

LEMMA (2.3). Under the same assumptions of the lemma (2. 2) if
f. c L^(a) with p > n, then u c C ^(Q') for all «' such that ^2' C «.

1 — ^— O, A

Moreover if Q C C and there exists a function u c H ' (Q) such that u-u

vanish on dCl. then u c C A€l), for some X such that (0 < X < 1).
Q^ X

The proof is a consequence of the extensions of De Giorgi's theorem
[10, 13. 24].



§ 3. Some Properties of the Averages of Convex Functions

vVe consider continuous convex functions in the whole space E . vVe
recall that a function f(x) is convex if



(3.1) «^^^^> < '^^'^2'^^'^



13



wh



enever x and x e E . The function f(x) is strictly convex if equality



1 2
holds only when x = x and f(x) is uniformly convex if there exists a positive

constant m such that



(3.2) f^'^i^) < '^^'^2'^'''^ ' mil^LZJLl)^



1^ 2

x + x



It is easy to see that (3. 2) holds if and only if the function f(x) - m|x| is
convex. Suppose f(x) c C (E ); it is well known that f(x) is convex if and
only if

(3.3) f (x)l.l. > for all x. ? c e'^

x.x. I J -

and f(x) is uniformly convex and (3. 2) holds if and only if there exists a
positive constant m such that

(3.4) f (x)i.?. > 2m|||^ for all x, ? c e" .

x.x. XI—''

We shall use the following property of the convex functions of one
variable

LEMMA (3.1). U f(x) is convex in e\ then for all |r| < ^

I'i ^\ « ^-^^ ^ f(x + X) + f(x - X) ^ f(x + X) - f(x - X) ^

(3.5) f(x + tr) < + — r

This means that the curve y = f(x + Z) for |r| < X remains below the chord
passing through the extreme points.

vVe consider again functions of n variables and we define the p -average
of the function f(x) in the point x



14



(3.6) f (x) = -^ / f(t)dt



I(x,p)

where w is the measure of the unit sphere and I(x, p) denotes the sphere with
center in x and radius p.

vVe show now the following

LEMMA (3. 2). If f(x) is a convex function, for each x € e", the
average f (x) is an increasing function of p.

PROOF: Let P,.Pp be two values of p with p < p . Since

A n n o

1 ^ S'^l 1 /
w(f (x) - f (x) ) = — / f(t) dt - — ~ / f(t) dt ,

Pn P, n / n n /

^ I(x,p^)

I(x,p )-I(x,p )

the proof of the lemma requires the following inequality



(3.7) ~ / f(t)dt < / f(t)dt .






I(x,p^) I(x,P2)-I(x.P^)



We consider, for each C c 1(0, 1), the function

f(x + rf )
which is a convex function of "Jr. Then, from the lemma (3. 1),



./ ^ -ex ^ f(x + Xg) + f(x - Xg) ^ f(x+Xg) - f(x- Xg) ^

f(x + .g) < ^ + ^ r



tin' ii-Vjr]0^i:, -C.d^ £0;i"''-;



.•r*qe ihw 'jrft }



"as



15



if |r| < X. Multiplying by t -X and integrating with respect to X

between p and p and with respect to ^ between and p ,



n -fi,



n 2 1 I - n I 2



^Ml ,n-l^^



n+1 p

1 /T f(x + Xg)-f(x-Xg) ,n-

,.1 y ^



If we integrate now with respect to ? over the surface Y^ =91(0,1) (|c| = l)i
we get

/ dw / f(x + t?)t-"'^df = / f{t)dt ,
y ^ /' ^(x.XC).f(x-Xg) ^n-l^ ^ r ^ r\,,,,,,,n-l



dX

f(t) dt



I(x,i2 2)-I(x.p^)



and



^,, / f(x+XC)- f(x-Xg) ,n-2^,

du / X dX =



Z ^^



■I-n



16

Thus the inequality (3.7) holds and the lemma is proved.

LEMMA (3.3). Let f(x) be a convex function in E . Then

2f-(x) < f (x^) + f (x^) ,

"~ 1 2 ■"

where x = (x +x )/2 and p =(p +p )/2.

PROOF: Denote by t = g.(f ) and by t = gJ'^) the linear

I 2

transformations which map I(x, p) onto I(x ,g)) and I(x ,p) respectively.



tj = gj(^) = x^ + 3- Ct- - x) ; tg = g^a) = X + ~ (o - x) .
P P



We have



J ^ (5/2)H, it is g(x+Xp) =\Ij{x + Xp) and

and then there exists a constant M such that

(g / J < m|€|^ .

^p(x) x.x. - '



.( A' :>



'jj-^ta srii jirsiv'



.iviu



I'i ^ '..



26

values ^(x) which satisfy the B. S. C. with the constant J^. If, moreover .
u(x) c C, (Q) n H^(n) then, a. e. in n



(4.4) max |u I 1 ^ (s = 1, 2, • • • , n) .



_ X



PROOF: Since u is a minimizing function, u(x) satisfies the identity



/



f (grad u)n dx =
p. ^ X.

for all n of H^(a).
o

We choose n = 9S/9x where S is in H (Q). Then, integrating
s o

2
by parts, we get (since u c H (S2) )



/



-^ f (grad u)g dx =
ax p. X.



and also

r ^P., at:

(4.5) / f (grad u) — ^ 1^ dx = (s = 1, 2. • • • , n) .

/ p.p. ax. ax.

Since p e H (^), using the lemma (4. 3) and the theorem (2. 1) we have (4. 4).

vVe want to show with an example that the previous theorem may fail
when \^ (u) depends on the function u too.

Consider the integral



27



(4.6) / (Yl+p^ - nu)dx .

where Q. is the sphere |x| < 1, in the class of all the functions vanishing
on 9f2. The function u = V 1 - x is a solution of such a problem, but (4.4)
does not hold.

§ 5. A Priori Bounds for Solutions of the Problem
J \ f(grad u) + G(x, u) | dx = min

We shall consider the variational problem for the integral



/ I f(p) + G(x. u)



(5.1) I(u) = / i f(p) +G(x.u) i dx

a

on the class of functions which assume the same boundary values (J> on dQ.

In this section we shall suppose that f(p) c C (E ) and G(x, u) is such that

the derivative G (x, u) is measurable with respect to x and continuous with
u

respect to u.

We shall subject the function f(p) or, f(p) + const., to the
assumption

I)

(5.2) f(p) > mIpT (- > 1, P > 0)

and we shall study the integral I(u) on the class of functions of H ' (f2)
assuming the boundary values i on dQ; denote such a class of functions by

1 Q" 1 0"

H,' (n). Of course, H,' (Q.) is defined only if ^ is the trace of a function



UL..t



aaal-^



: - ^3iS'ir"..n:



28



u of H '"(a). If ii"r„^-^., •



where L(c) is a suitable function of c depending on a and Q too. vVe used
the inequality: |a+b| ^ii^iuo'ii^BO; n'. "„l'r.)



'. ? iy.)^



30



a way that: iJt - (i +c) q/A(a, Q) > we deduce easily from (5. 7) and (5. 8)
the theorem.

The following theorem gives an a priori bound for the maximum
modulus of the solutions of the considered problem.

vVe shall assume that

B) The function G(x, u) may be written as a sum of two functions

G(x.u) = G^(x,u) + G2(x. u)
and that there exists a constant K such that for all |u| > K we have



(5.8) uG (x,u) >



and

(5.9) iGgJx.u)! < M(x)|u|^ .

where || M(x) || 7/q\ = M < + « with y > n/a and < 3 < Q'*(l - I/7) - 1.

THEOREM (5. 2). Let u(x) be a function minimizing the integral

1 a
(5. 1) on H ' (Q) where the assumptions I), B) hold. Suppose that H satisfies

the "cone propriety" and |(^| is bounded on 9f2 by ^, then u{x) is almost

everywhere bounded by a constant depending on m, K, M, ^, a, 0, fJ,



"lln'-^n)-



3

PROOF: vVe begin with the remark that if o; > n, the boundedness



o

The argument of this proof lies close to that used in [22, 23] for
proving inequality of type of (5. 7) for linear equations. The application to the
variational problems, first, is given in [10, § 10].



1 3

Online LibraryGuido StampacchiaOn some regular multiple integral problems in the calculus of variations → online text (page 1 of 3)