On minima of radially symmetric functionals of the gradient

Nonlinear

Analysis,

Theory,

Methods

Pergamon

ON MINIMA

OF RADIALLY SYMMETRIC OF THE GRADIENT ARRIGo

CELLINA

S.I.S.S.A.,

and

& Applicotion.~, Vol. 23, No. 2, pp. 239-249, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/94 $7.00 + .@I

FUNCTIONALS

STEFANIA PERROTTA

Via Beirut 2-4, 34014 Trieste, Italy

(Received 31 March 1993; received for publication 14 July 1993) Key words und phrases: Existence, uniqueness and the qualitative properties, radially symmetric functionals. 1. INTRODUCTION IN THIS PAPER we consider the problems of the existence, the uniqueness and the qualitative properties (symmetry) of the minima to the problem

B MIxI, Iv&M + ww)l d-% s where B is the unit ball of IR” and the map u + g(r, V) is lower semicontinuous but not necessarily convex. Problems of this kind arise in domains as different as nonlinear elasticity, fluid dynamics and shape optimization, and, either for the problem of the existence of solutions or for the properties of the relaxed problem, are considered in [l-7]. In particular, the very same problem is considered in [7]. Our results present the following features: (a) no smoothness on g or h is required: g is either a normal integrand or a lower semicontinuous function; (b) the case h = 0 is allowed; in this case the assumption on g reduces, for the existence of solutions, to g being lower semicontinuous and growing at infinity, as is to be expected; for the uniqueness, in addition, to g** being strictly increasing, as is also to be expected; (c) the case h = au is allowed: for a # 0 our theorems yield at once existence and uniqueness of solutions with no further assumptions on g besides lower semicontinuity and growth at infinity. mjr&

2. BASIC

NOTATIONS

In what follows we shall assume that: IR”is endowed with the Euclidean norm 1. (, and B is the unit ball, whose measure is o,, . The (n - I)-dimensional Hausdorff measure of dB is nw, . The subgradient of a convex function g is denoted by ag. A map g: [0, l] x [0, a) + E is termed a normal integrand [8], if (i) for a.e. r E [0, 11, g(r, *) is 1.s.c. on [0, co); (ii) 3 a Bore1 function g: [0, I] x [0, 00) + @:g(r, *) = g(f, *) for a.e. r E [0, 11. Consider g: [0, 1) x [0, 00) + I!?. Let g : B x II?”-+ IT?be defined by g(x, r) = g(lxJ, [cl). Whenever the bipolar of g, g** is defined, by extension we call bipolar of g, g**, the map defined by g**(lxl, lrl) = g**(x, <). Remark that the map r -+ g**(r, r) is increasing. It is known [8], that g** is a normal integrand whenever g is and that g** satisfies the same growth assumptions as 5. 239

A. CELLINAand S. PERROTTA

240

3. MAIN

RESULTS

We shall consider the following problem (P) UEmin W,L”(B) B ]&?(I49IVu(x)l) + wx))l 5

(P)

k

where B = (x E R” : 1x1 < l], n L 2, g: [0, l] x [0, 00) + R is a normal integrand and h: IR -+ R is convex. We shall assume throughout the following growth assumption (

(GA)

there exist: a convex 1.s.c. increasing function v/: [0, 00) + [0, 00): sw(s) --* 00 as s + 00 and c~E L’([O, 11) satisfying

ig(r, s) 2 CX(T)+ w(s) for all s E [0, to), for a.e. r E [0, 11. The following result guarantees the existence of at least one radially symmetric solution to the minimum problem (P) associated to a convex function g. THEOREM 1. Let g be a normal integrand satisfying assumption (GA). Assume further that g ** = g. Let h: R --f R be convex. Then problem (P) admits at least one radially symmetric solution. Moreover, if h is monotonic and either g(r, *) or h is strictly monotonic, then every solution to (P) is radially symmetric.

Proof. Let u be a solution to problem (P). Consider the function fi defined by

(1)

u(oIx[) do.

It is our purpose to show that ti is a radially symmetric solution to (P). The symmetry comes from the very definition. (a) We claim that VU(x) = 1 x (Vu(olxl), n% 1x15 101= 1 i

ViqO)

=

w> do,

X#O (2)

0.

First remark that the above is true when u is of class C’. In this case, when 1x1# 0 one can differentiate with respect to the parameter x to obtain

while, for x = 0 (u(whi) - U(O))do (hi(w, VU(O)) + hi)ol&(hi)ol)

do

= 0.

Radially

symmetric

241

functionals

To show the validity of the above formula for any u in W’*‘(B), let us consider a sequence {u,), each uh of class C’ and u,, + u in W 1s1(B). Hence, from the previous result, Vi& satisfy (2). We are going to show first that the functions U,,defined by (1) converge to G strongly in L’. Set w to be -- 1 x XfO (Vu(&]), 01 do, w(x) = no, 1x1 i lo~=l x = 0.

i 0,

Hence, to prove the claim, it will be left to show that Viik converges to w strongly in L’(B). We have

I4wIxI) - u,Jwlxl)ldo dx l~(~lxl) - ~/J~lxl)~

do do

Through the same steps,

W(wIxl) - Wdwlxl), w>do dx I

i-s 1

lV4wlxl)

- Vu,(&])l

do do

1

joI=

B nwn

and, by applying again Fubini’s theorem,

Ilw- WJll 5 IIVU- wklll. The claim is proved. The above arguments defining ti out of u are similar to those employed in [9, lo] for a problem involving the Laplacian. (b) From the convexity of h, we obtain h@(x)) dx.

h(u(x)) dx 2 B

B

In fact h@(x)) dx I B

L B n%

F

h(u(wIxl)) dw dx

jloI=l

h(u(wr))r”-’

dr dw =

h@(x)) dx. B

A. CELLINAand S. PERROTTA

242

In particular the same computation

in the case h(u) = u yields S$x)dxa

~~W)dX.

(3)

To prove that U is a solution, it is left to show that

Since

IVti(x)( 5

-!non

IWI=

1

IWdxbl do

and s - g(r, s) is monotonic,

By the convexity of s r g(r, s) and Jensen’s inequality,

5

1 , ,= 1 w

B nwn SES

dlxl, IWdGl) do k >

1

=

5i loI=

g(r, jVu(wr)l)r”-’ drdo

=

IWx)l) d.x. Bg(lx19

0

Hence, ii is a radially symmetric solution to (P). (c) Assume now that h is monotonic and let us prove the result first in the case h is monotonic increasing. The map u --, g(r, u) is nondecreasing; by assumption, either g or h is strictly increasing. Let us show that every solution is radially symmetric. Let u be any solution and set ti to be ii(x) = &

u(o(x1) do. n i‘loI=

From the above, 6 is a solution. Consider the average w = +(u + a). By the convexity of the problem, w is a solution. In particular,

1

g(lxl, lVw(x)l)dx= ;

B

By the monotonicity

sB

&I,

IWx)l> dx + ;

!IB

&I9 Iwx)l> b.

(4)

and the convexity of g, we have

i!(lxl, IWx)l) 5 ~(lxl9tdwx)l + IWx)lN 5 MIxI, Iv4eo + MIxI9 IWx)l) sothat, by (4), the equality holds I449 IVw(x)l)= h493dwx)I

+ Iwx)lH = 3g
Set T(r) to be T(r) = suplu: g(r, u) - g(r, 0) = 0). The supremum is actually a maximum.

(5)

Radially symmetric functionals

243

We wish to show that for almost every X, IVw(x)I 2 T( [xl). In the case g is strictly increasing, and there is nothing to prove. Assume that h is strictly increasing. Remark that u --) g(r, u) is strictly increasing for u 1 T(r). Hence, notice the following property of w to be used later: for those x such that the gradient exists, whenever IVw(x)I 2 T(lxl), there exists n(x) 2 0 such that Vu(x) = A(x) Vii(x). In fact in this case the first equality in (5) implies that T = 0

and by the strict convexity of the Euclidean norm, this is true only if Vu(x) = n(x) Vti(x). We claim that the map r -+ T(r) is measurable. Fix E > 0. Since g(r, v) is a normal integrand, by [8, theorem 1.1, p. 2321, there exists a compact K, in [0, I], m([O, l] \K,) < E, such that the restriction of g to K, x IRis lower semicontinuous and the restrictions of r + g(r, 0) and of r --t ‘Y(T)to K, are continuous. In particular, there exists M,: g(r, 0) - a(r) I A4,. Hence, for every u satisfying g(r, u) - g(r, 0) = 0 for some r in K,, we have w(u) I g(r, 0) - a(r) I M, that implies Iv/ I V, for some V,. Consider a sequence (m) in K, converging to r* and set T* to be the limsup of T(r,). By taking a subsequence, we can assume that g(r, , T(r,)) converges toy satisfyingy - g(r*, 0) = 0; hence g(r*, T*) - g(r*, 0) I 0. g being nondecreasing in u, we have g(r*, T*) - g(r*, 0) = 0, hence T(r*) 1 T*, i.e. the restriction to K, of r -+ T(r) is upper semicontinuous. By Lusin’s theorem this proves the claim. Let us first show that IVES 1 T(lxl). In fact set A = (1x1: Ivu(x)/ < T(lxl)) and assume that m(A) > 0. Define u: [0, l] -+ IRby ~(1x1) = a(x). Remark that r + U(T)is locally absolutely continuous in (0, 11. In fact apply the change of variable formula to the transformation from Cartesian to polar coordinates, and obtain for the map U(T,19)= u(r#@, (9))that for almost every 8, the map u: r -+ U(r, 8) is locally absolutely continuous in (0, l] and u’(r) = ;

Since ti is radially (Vti(x), x/1x1), hence

symmetric,

(r,

8) =

($,

v4dQ,

a >*

I(vG(x), x/lxl>l = [VU(X)/; by the chain rule,

u’(x) =

Set a(r) to be 1; (u’(s)x ec&) + T(@xA(~))ds and fi : B --t II?to be U(x) = ~(1x1). Then, from the very definition, V(x) I U(x) and the strict inequality holds on a subset of B of positive measure, hence,

s

h(fY(x)) dx <

B

On the other hand,

s

h(fi(x)) dx.

B

&I9 Ivw)l) = !?(lXl~ IW~N)~ so that ti cannot be a minimum. Hence, (vz?(x)/ 2 T(lxl) a.e. To show that the same inequality holds for w, remark that by the definitions one also has

A. CELLINA and S. PERROTTA

244

so

that V&y)

s

1 x

=

nwrl

1x1lo\=1

and, by the definition and (3),

/Bti(x)dxr

(vw(wIxl), w>do

JBw(x)dx.

Assume that there exists a subset S of B, m(S) > 0, such that IVw(x)] < T( Ix]) for x in S. Since V w = * Vu + 3 Vii, on S we must have IVu(x)l < T(]x]); moreover, IVti(x)] must be equal to T( Ix]). In fact, if it is not so,

&I, IWx)l) < 3&4.4,IVWI) + 3&l, IWx)l) contradicting Set

(5). s, = (0: ]Vw(or)] < T(r)]

so that for r in a subset E

c

[0, l] of positive measure,

J

do > 0.

X&a)

10)= 1

Consider one such r in E. Since, for Ix] = r, T(r) =

IVti(x)l 5

& n

i

Iwl=l

IW~lxl)l da

on a subset of e(S,) of positive measure we must have lVw(wr)) > T(r). Again for r in E,

s

loI= 1

dr, lVw(wr)l)dw =

s

s(r, W-Ndw +

&

> loI= 1

SetXtobe{x=or:]wl

g(r,

h(wr)b

do

WS,)

g(r, T(r)) do =

)&II= 1

g(r,

Ivi@r)I)do.

= l,rEE).Then

Xg(lxl, IVw(x)l) dx

1V WW-) 1)do dr iVw(wr)l)dw)

dr

Radially symmetric functionals

245

For x not in X, by a previous remark, Vu(x) = A(x) Vn(x), so that VW(X)= i@(x) + 1) VzI(x); hence, on B\X, w itself is radially symmetric, i.e. w coincides with 2. By lemma 7.7 in [ll], VW = Vti a.e. in B\X. Hence,

a contradiction,

since w is a solution and h(w(x)) dx r B

s

h@(x)) dx.

B

Then m(S) = 0, i.e. for almost every x in B, Ivw(x)[ I ~(1x1). Hence, for almost every x, VU(X) = n(x) Vfi(x). This proves the theorem in the case h is increasing. (d) Finally, notice that the case of a decreasing function h can be reduced to the previous one by setting h(t) = M-0,

r E I?.

Now, h is increasing and each solution to the problem

$3 is radially symmetric. Since ri is a solution to (P) if and only if -5 is a solution to (P), we see that every solution to (P) is radially symmetric. To see how the assumptions of the previous theorem are sharp, consider the case h = 0 (h is monotonic, but not strictly monotonic) and let g = g(v) be the indicator of the interval [0, I] (g is convex and monotonic, but not strictly monotonic). Clearly there exist nonradially symmetric solutions to (P).

Remark.

The next results are concerned with existence and uniqueness of solutions for the minimum problem (P) when g is (possibly) nonconvex. We shall assume that g is independent on 1x1,

i.e.g(r, 0

= g(O.

Assume that: g(r, <) = g(c) is 1.s.c. and satisfies (GA); h: R --f R is convex and monotonic. Then the minimum problem (P) admits at least a solution u in W:*‘(B).

THEOREM 2.

Proof.

monotonic

(a) As in the proof of theorem decreasing.

1, it is enough to consider the case where h is

A. CELLINA and S. PERROTTA

246

Let tl be a radially symmetric solution to the convexified problem

Define u : [0, l] -+ IR by ~(1x1) = c(x). Remark that the map r -+ u(r) is locally absolutely continuous on (0, 11, and ~‘(1x1) = (x/lxl, Vll(x)). We are going to show that we can as well assume that Iu’I does not take its values (for r in a set of positive measure) on any interval (a, b), where g** is affine. This in particular will show that 1~’I takes its values where g and g** coincide, proving that u is a solution to the original problem. Let h:(u) be the right derivative of h at U; the map r --$ h:@(r)) is negative and bounded; consider H defined by I H(r) =

d-l/l

:(u(s)) ds.

s0 Assume g**’ = 01on (a, b). Consider first the case 01 = 0; in this case (a, b) = (0, T), and consider those r such that ju’(r)l I T. The same reasoning as in theorem l(c), implies that there exists another radial solution u such that Iv’1 2 T. Hence, we can also assume cr > 0. Set E, = (r : [u’(r)1 E (a + 6, b - 0)) and assume that for some 6, m(&) > 0. We are going to show that, under this assumption, it is possible to define a family (ti,) of variations of ti such that, for E small, the functional computed at fi, has a value strictly smaller than the minimum, a contradiction. Call E this E,,. Consider A(r) = (cY?’ - H(r))x&). Let rr and r,, rl < r,, be points of density for E and Lebesgue points for A, so that 1

r1+6 A(r) dr + A@,) = (CUT-~- H(r,)).

2 s r,

1

r,+6 A(r) dr -+ A(r2) = (CYT;-’- H(r2)).

2 s rz Since A is monotonic, A@,) - A@,) = 2rl. > 0. Set 6* = 6*(a) to be m[(r* - 6, r2) n E]. Since r, is of density for E, 6*/d + 1, i.e. 6* > 0. Let 8 be such that 6 < 8 implies A(r) dr - A@,) < a;

f I

r2_ A(r) dr - A(r2) s ?z 6

< a.

Set 7’ to be

and define q as q(r) = j’; q’(s) ds so that q(l) = 0 and, by the choice of 6*, q(r) L 0. Let us remark that u’(r) + &q’(r) coincides with u’(r) for r not in E, and, for r in E, is in (a, b) whenever E is sufficiently small. Hence, g**(lu’(r) + ar,~‘(r)I)is well defined and integrable.

247

Radially symmetric functionals

(b) Let us consider the family (U -t eq). By the mean value theorem there exists (r(r) E [0, e] and g:*‘(r) E ag**(lu’(r) +
g**WW + W@)l)- g**WWl) = ~&*‘Wr’(r). Moreover,

since the subgradient of h is monotonic,

there exists &(r) E [0, E] such that

Since N,(O) = 0 and q(l) = 0, we have

c 1

r”-%:@(r) + ~2{f)~(r))~~r) dr = -

1~~(~)~‘(~) dr.

i0

-0 Hence,

A, 5 E : [r”-‘&*‘(r) s 1 zz

e6

s

- H,(r)]q’(r) dr

r,+s*

1

[F’g,**‘(r)

TI

-

I&@)]~&)

‘2

dr - Er,_s V-k,**‘(r) a* i

- IIEW]x.&) dr.

For every fixed s, u(s) + &(s)q(s) converges to u(s) from the right, so that, being the subdifferential of a convex function monotonic, we have A!,&(s) + &(s)q(s)) --+k :(u(.s)) as E + 0. Moreover, since h is finite on R, there exists &I that bounds all the values of /h:(v)/ for u in a neighborhood of the image of the solution u. Hence, Ii,(r) --+iL;l(r) pointwise and is dominated by a constant. Moreover, for every r E E, g:*‘(r) = a, for every E sufficiently small. By integrating we obtain that, for every E sufficiently small,

Is 1

2

r,+&* [r”-‘g,**‘(r) - H&)]x&) I,

r2 ‘r_8 P”-*g?+“(~) - H&-)IxE(~) dr - $

1 Is

dr - $ S:‘+‘*/I(/.)dri

s

/i_64r)drl

< a

< a.

A. CELLINAand S. PERROTTA

248

Finally, for some positive E,

i.e. for some positive E, the function 6, defined by u’,(x) = z&j) + ~(1~1) a value for . . yields _ -the integral in (P**) less than the value-computed at 6, a contradiction. Remark.

In the case h =- 0, the assumptions of the above theorem reduce to lower semicontinuity and growth at infinity for g, as is to be expected.

3. Let g and h satisfy the same assumption as in theorem 2; in addition, assume that either g** or h is strictly monotonic. Then problem (P) admits one and only one solution.

THEOREM

Proof Assume that U, u: [0, l] -+ Ware such that the maps x -+ u(/x/) and x -+ u(\x/) are two distinct solutions to (P). There exists an interval (rt , r2) such that: u(r) > u(r), r E (r,, t-J; u(rz) = t.$r2) and, when rI > 0, z&r) = u(~r). Set w to be *(u + u). The map x -+ w((xj) is a further solution to (P**) so that C2 r”-‘[g**(iu’(r)() 5 1r, 1

r, r”-‘[g**(/u’(r)])

+ h(v(r))] dr + ;

+ h&(r))] dr

i r1

‘2 =

”Ir1

r”-‘[g**(lw’(r)j)

+

h(w(r))] dr.

The convexity of both g** and h implies, in particular, that ‘2

r”-‘h(w(r))

dr =

‘1 h being monotonic

” r”-‘h@(r)) rl

dr.

we infer

h(W)) = h(W)), The above is a contradiction

VI-62

(r,,rd.

to the existence of u and u in the case h is strictly monotonic.

(6)

Radially symmetric functionals

249

Assume now g** is strictly monotonic. Since h is convex, there exists at most an interval I on which h is constant and attains its minimum. Set t* to be sup(r i 1 : u(r) E I] and consider the map u* defined by U*(X) =

for r 5 r* for r > r*.

UP*) i u(r)

Then both i 01

I 01 s”-‘h@*(r))

dr 5

r”-‘h(u(r)) dr

and 1

1 F-'g**((u*'(r)j)

0

dr 5

r”-‘g**(lu’(r)l)

dr

0

hold, and the last inequality is strict in the case Iu*‘[ differs from Iu’l for r on a set of positive measure. Since u is a minimum, U’(T)must be 0 on (0, r*). From (6) one has that W(T)E I if and only if U(T) E I. Since w is a solution, the above reasoning implies that also w’(r) = 0 on (0, r*), i.e. u’(r) = w’(r) on (0, r*). The case r* > rl would violate the assumptions on rl , r,, u, u. Hence, on (rl , rJ the inequality w(r) > u(r) implies the inequality h(w(r)) > h&(r)), a contradiction to (6).

Remark. Whenever h is linear, h not zero, uniqueness (besides existence) is guaranteed simply by the lower semicontinuity and growth at infinity of g. REFERENCES 1. BAUMANP. & PHILIPSD., A nonconvex variational problem related to change of phase, Appl. Math. Optim. 21, 113-138 (1990). 2. GOODMAN J., KOHNR. V. & R~YNAL., Numerical study of a relaxed variational problem from optimal design, Comput. Meth. appl. Math. Eng. 51, 107-127 (1986). 3. KAWOHLB., Regularity, uniqueness and numerical experiments for a relaxed optimal design problem, in: International Series of Numerical Mathematics, Vol. 95. Birkhauser, Base1 (1990). 4. KOHNR. V. & STUNG G., Optimal design and relaxation of variational problems, I, II and III, Commun. pure appl. Math. 39, 113-137, 139-182, 353-377 (1986). 5. MARCELLINI P., Non convex integrals of the calculus of variations, in Methods of Nonconvex Analysis, Lecture Notes in Muthematics, Vol. 1446. Springer, Berlin (1990). 6. RAYMONDJ. P., Existence and uniqueness results for minimization problems with nonconvex functionals (to appear). 7. TAHRAOUIR., Sur une classe de fonctionelles non convexes et applications, SIAM J. math. Analysis 21, 37-52 (1990). 8. EKELANDI. & TEMAMR., Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976). 9. CELLINAA. & FLORESF., Radially symmetric solutions of a class of problems of the calculus of variations without convexity assumption, Ann. Inst. H. Poincare Analyse non Lineaire 9(4), 465-478 (1992). 10. FLORESF., On radial solutions to non-convex variational problems, J. Optim. Theor. Applic (to appear). 11. GILBARCD. & TRUDINCERN. S., Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977). 12. ZIEMERW. P., Weakly Differentiable Functions. Springer, Berlin (1989).