Evolution equations with lack of convexity

,Vo,dinear Annlym. Theory. Merho& & Appkmm.r. Vol. 9. No. 12. pp. llol-lu3. 1985.

Rated m Great Britain.

0362-546X/85 $3 CO + .Ol 0 1985 Pcrgamon Press Ltd.

EVOLUTION

Scuola

EQUATIONS WITH LACK OF CONVEXITY

MARCO DEGIOVANNI

Normale Superiore, Piazza dei Cavalieri 7, 56100 Pisa, Italy

and

ANTONIO MARINO and MARIO TOSQUES Dipartimento di Matematica, Universita di Pisa, Via Buonarroti, 2, 56100 Pisa, Italy

(Received 12 July 198-t; received for publication 9 April 1985)

Key words andphrmer: Functions with lack of convexity, points which are critical from below, evolution equations, P-convergence.

INTRODUCTION

THE STUDY of the evolution equation

II’ = -(gradf) o U (1)

where f is a real function defined on a space X, has been widely investigated, even if the function f does not satisfy the classical regularity assumptions.

An important and well known theory on this subject is that of nonlinear semigroups [2, 5, 211 which has brilliantly developed and solved many problems of nonlinear differential equations.

For instance, a class of equations of this kind is represented by parabolic equations with convex constraints:

1

a,u(t, x) = Axu(t, x) if VI(X) < ~0, x) < v2 (4

a,+,~) = (&@J))+ if Vl (x) = u(f, x) < v2 (x)

d,u(t, x) = -(A&,x))- if 9% (x) < u(r, x) = tPz (x)

a,u(r, x) = 0 if VI(x) = ~(f, x), vS (x) = u(t, x)

Vl (x) s u(t, x) =s v2 (x)

u(r, x) = 0 ifxEaR

where VI, I/I~ : n+ R are two given functions with vy, 6 yr in a and qt s 0 =S vz on ~3S2. For this purpose one considers functionsf, satisfying the hypothesis of convexity, which are,

however, only lower semicontinuous and may assume the value +=. Among recent researches, the theory developed in [4] and [22] is one of the most systematical

studies of nonsmooth problems in a different setting.

This work was partially supported by a national research grant financed by the Minister0 della Pubblica Istruzione (40’%-1983).

1401

l-102 M. DEGIOVANNI, A. MARINO and M. TOSQUES

Some problems, which we have considered in our study of equation (l), can still be regarded as of parabolic type, but with a nonconvex constraint.

For our purposes it has seemed natural to consider a simple and direct extension of the notion of sub-gradient of a convex function (cf. (2)).

Here we want to emphasize two properties among the several ones enjoyed by this notion of subgradient.

First of all f is Frechet-differentiable in u if and only if f and -f have subgradient in U. Furthermore, in the concrete cases we have in mind, where X is a function space, the

existence of the subgradient off in u involves some regularity properties of U. We have also introduced a class of functions (cf. definition (1.4)) which contains convex

functions and which is fit to the lack of convexity of the problems we want to face. In particular, we emphasize the fact that, as in the convex case, no explicit hypothesis of

subdifferentiability is imposed. The class of functions introduced in (1.4) includes, for instance, the following ones (con-

sidered in [16, 181). Given Xi = P(O, 1; R”); h : R” + R of class C3 such that h-‘(O) is a compact hypersurface;

A, B:h(A)=S 0, h(B)< O,set

if u E H’.*(O, 1; UP), u(O) = A, u(l) = B,h(u(s)) s 0

+“o, elsewhere

and also given X2 = L*(Q) (52 regular bounded open subset of W”), ty,, q2 E M+Z(Q) : ql s y2 in R, vi S 0 S 7.&Z in aQ, set

:

41 ]Du~ZcLx,ifuEH~~2(R),

f*(u) = *

J” lUI*dx=r*, R

(r> o>, VI 6 u S q2 in S2

+x7 elsewhere.

In [16] the function fl is studied in order to establish the existence and multiplicity of geodesics .“with respect to an obstacle” (or in a manifold with boundary) joining two assigned points.

In [IS] the function f2 is studied in order to get the existence and multiplicity of the eigenvalues of the Laplace operator “with respect to an obstacle”.

Actually, in these examples, f is subdifferentiable in u if and only iff(u) < +m and u E @.*(O, 1; EP) (resp. u E HZ.*(Q)).

Therefore the subdifferentiability off in u involves some regularity properties of u. This is important, because the solution U(t) of (1)-for the functions f introduced in (1.16)-stays automatically in the set of the u’s where the subdifferential off is not empty, i.e. for every t, U(t) has a certain regularity with respect to the space variables.

Finally we remark that, in order to solve equation (l), a useful tool (which is based on the uniform convexity) asserts that, (cf. (l.l)), if X is a Hilbert space and f : X-, R U {+z} is a proper 1.s.c. function which is bounded from below, then for every E, A > 0, u in X, there

Evolution equations with lack of convexity 1403

exists u in X such that / u - u / < E and the function

w+ - WI2 +f(w)

has a minimum point in H (for a similar result, cf. [14]). In this paper we prove some of the results we announced in [7].

In this work, H will denote a real Hilbert space with 1.1 as norm and ( * 1 a) as inner product. If uEH, r>O, we set B(u,r)= {u/lo-ul<r} and if S2 is an open subset of H and

f: 524 IR U {-x, +=} is a map, we set

If u E D(f), we set D(f) = {Q(u) E R)

a -f(u) = I (Y E HI lim inf f(u) -f(u) - (4 fJ - u) 3 o

“--tY b-4 I (2)

and if u E H\D(f), we set a-f(u) = 0 (cf. [9,20]). We also put

D(d -f) = {u 1 a --f(u) + 0).

It is not difficult to see that a-f(u) is convex and closed for every u in H; grad-f(u) will denote the element of minimal norm of a-f(u), if d-f( u is not empty. We put also (grad-f(u) 1 = ) +X, if d-f(u) = 0.

If a-f(u) # 0, we say that f is “subdifferentiable” in u; 13-f(u) is the set of the “sub- differentials” and grad-f(u) is said the “subgradient” off in u.

Finally if (X, d) is a metric space and g : X + W U {+x} is a map, we put (cf. [9,19])

1 i max 0, limsupg(Uk(-$u)},ifg(u) < +m /Vgl(u) = “--LA u,

I +x, elsewhere.

SECTION 1

Letf: H-, R U {+=} be a lower semicontinuous (1.s.c.) function such that there exists Ae > 0 with

inf[& /u12 +f(u)]u E H} > --r.

Then, if 0 < A C A,,, u E H, the Yosida approximation off

fi(u) =inf[&lu-w12 +f(w)lwEH}

defines a real valued continuous map on IO, A,[ x H such that f(u) = prnf* (u), Vu E H.

Next we shall need the following theorem which is related, for instan;, to the result given in [14].

1404 M. DEGIOVANNI, A. ~MARINO and IV. TOSQUES

THEOREM 1.1. Let f: H* W U {f=} be a I.s.c. function such that D(f) f 0 and there exists a A,, > 0 for which

Then VA E IO, &[, VU E H, VE > 0, there exists a v E B(u, E) such that the function z ++

(l/24 Iv - z I* + f(z) has a unique minimum point on H.

Proof. Set m = inf{( l/(2&)) 1 z I* + f(z) / z E H} and let u. be a point of D(f). Let A, u, E be as in the hypothesis. Consider two strictly decreasing sequences (A&_ (a,),

and two real numbers A’, A” such that

0 < A’ < II < A, G A” < A,, lim Ak = A, lim dk = 0, bk 6 1. k k

By induction, we can find two sequences (vk)k, (Wk)k in H such that

c vi = U

&lVk - wk t* +f(Wk) sf,ik(vk) + dk, Vk 3 1

&bk - Wk12 +f(wk)+u* - wk_1 1’ +f(Wk_l), Vk a 2 k

h vk+l = wk + 2 (uk - w,), Vk 3 1.

Let M > 0 be such that /z 1 > M implies

Then 1 uk - u ( 6 E implies 1 wk 1 C M. In fact

~bk-wk~2+f(Wk)~~bk-UO!2+f(UO)+~k<~ t vk - zI*

+m-$-lz/*Sz;j 0 k

uk - zl* +ffz>, iflz12Mandif(vk --[SE.

Therefore I wkl C M. Now we can set

-1

Ak =An l- E

irk 2’fi(]u] + & + M) > ’ Vk 2 1.

Then (A,), is strictly decreasing to A and Ak < A’ if j E N is sufficiently large. Now we can prove, by induction, that

Ivk - UI 6 &and]vk - !&-i] s &2-k, Vka 1.

Evolution equations with lack of convexity

Clearly 1 u t - u 1 = 0. Therefore suppose

1 U; - U/ s E and / Vi - pi_ L( s cZei, if 1 G i s k.

Then 1 wkl s M and

s EZ-~-’ whichimpliesthatIu/,,, - ul s E.

Therefore (u~)~ is a Cauchy sequence which converges Now we remark that

to some 0 in B(u, e).

wk+l I2 + &lo,+, - Wk12 kc1

. I2 +fcWk) - e bkcl - Wk+l 1’ s 2~ "ok - Wk+l, k+l

+ 2Ak+, “+l 11 - wk I* + f&k) + bk - &luk - wk(2 -&uk+l - wk+1i2

k+l

=L~~k--wk~2+~(~k-_w*~~k-~k+~)+&‘k-wk+~~1

2Ak

+ 2Akcl L bk+l - wk I* +fAk@k) + dk - &ok - wki2

1405

-+k+l -wki2-f(uk+i -wkIwk -wk+l) 2A k+l

-+Iwk - Wk+liz* 2A

Therefore,

IW AkAk+l k+l - wkl* 6 2

lk - Ak+l dk

which implies that (W&)k is a Cauchy sequence for a suitable choice of (6k)k. Passing to the limit, as k goes to + a, in

& I"k - Wk12 +ftWk) SfAk(uk) + fik

we obtain the existence of a minimum point w of

z*$/u - zI* +f(r)onH.

-Finally for the uniqueness we remark that if we set B = w + (x/A)(u - w), for 0 < A c A < A, then w is the unique minimum point on H of P ++ (l/(2%)) I fi - z12 +f(z).

1406 M. DEGIOVAJWI, A. MARSO and ,M. TOSQ~~ES

Moreover

(6 - 01 = (1 - (K/A))/w - uj G (1 - (X/A))(M+ lul + E).

Therefore if 1 and A are sufficiently close, u is close to U.

Similar results concerning the function

instead of

are described in [15].

PROPOSITION 1.2. Let f: H -+ iw U {+x} be a 1.s.c. function. Then

vu E N.07 Zig in D(a-f) such that

lip wk = U, f(wk) s f(rc), limlfup (grad-f(w,) 1 G 2 1 Vfl (~1).

-- In particular we have that D(f) = D(c~ -f).

Proof. By the 1.s.c. off we can suppose 112 = inff> --r. We can also suppose that lVf[(u) > 0, otherwise we can take wk = K, Vk, First of all we consider the case u E D(f). Let (u& be a sequence COIWXgiIIg t0 u such that f(&) < f(U), for every k. Let (&k)k, (A& be two sequences of positive numbers such that

lip ck = 0, 1iF Ak = 0

lim lu- =O uk\ + Ek Et 7

k Ak Ak a 2(fh) -f@k))’

By theorem 1.1 we can find two sequences (Uk)k and (wk)k in H such that

\uk - ukl G &k, &IQ - wkj2 + f(Wk) = min 1

&i”k - z12 f f(z)jz E H).

Then

and

uk - wk -E d-fcWk)

Ak

Iuk - wki2 d t uk - d’ + 2Ak(f(u) -fbk))

s (Ek + juk - u[)* + 2jlk(f(U) - m)

which implies that IiF (u - wt 1 = 0.


Moreover we have

EZk fovk) +k -uki’ +f(Uk) s21,+f(“k) <f(U).

Finally, since 1 grad- f(wk) / s / (uk - wk)/&, It is sufficient to estimate 1 (uk - wk)/&l. Clearly we can suppose that j (uk - wk)/Akj > 2 / Vf I(u); therefore

I ~ ,f(u) -f(Wk) + I”k - d2 s,f(“) -ftWk) - luk-wkI Akluk - wkl b - wkl

x /u-ukl+&k+l

2Ak I vfl (u)

+(l”-ukI+Ek)2

2n: I Vf I (u> which implies lip sup I grad-f ( wk) I s 2 / ‘C’f I(u).

The extension to the case u E D(f) is obvious.

THEOREM 1.3. Let R be an open subset of a complete metric space (X, d). Let f: 6-, 54 U {+x} be a 1.s.c. function such

m = infCf( u) I u E S-2) > -x.

Suppose that there exist u in R, c > 0 such that

f(u) E Rand lVfl(u) 2 c, VU E R.

Then V&>O, 3 w in a 5’2 such that

(c - E)d(w, u) G f(u) - m, f(w) <f(u) - (c - E)d(W, u).

Proof. Choose E : 0 < E < c; we can define in Q rl D(f) an ordering, already considered in

[31, by def

01 s u* a f(uz) s f(Ul) - (c - E)d(U:, Ul).

Let & be the set of the chains containing u. Clearly {u} E d. Now we define in zA an ordering

by dcf

Al SA2wAA, CA*.

Let {Ai}i~r be a chain in (SQ, 5). It is easy to see that lJ Ai E d. iEI

Therefore, by Zorn’s lemma, there exists a maximal element A in s2. Since f is strictly decreasing on A, lim f exists and we have

A

Since

m 6 liyf G f(u).

d(v If(h) -fk*>I 17 u2 ) <

C--E ’

we can say that w = lim J E a exists, where J : A + iz is the map, because the set (A, J) is a A

1308 M. DEGIOVWNI, A. MARINO and M. TOSQUES

Cauchy net. Moreover by the lower semicontinuity off we have

f(w) <f(u) - (c - W(u, w>, Vu E A and (c - E)~(w, u) <f(u) - m.

It remains to prove that w E aS2. If w E 52, we have by the maximality of A and the previous estimate, that w E A and w is

the maximum of A. But we can find z E R, z # w, such that

f(z) Gf(W) - (c - 44Z, w)

since ( Vfl( w) 3 c. Therefore A U {z} is a chain which contains properly A, a contradiction. Therefore w E a Q.

We observe that the conclusion is in general false if we require E = 0.

Remark. Consider an infinite orthonormal system (eJ in a Hilbert space H. Then the function g defined by

I -J(lu/2_$)7 if/u/ai, u=Ae,,AElR

g(u) = 0, ifu=O

I +xc, otherwise,

is lower semicontinuous on H and IVgl(u) 5 1, tlu E H. Nevertheless for any r > 0, there is no w in H, such that 1 WI = r, g(w) s g(0) - r = -r.

Definition. 1.4. Let R be an open subset of H, f: Q + R U {+x} be a 1.s.c. function and $ : Q x IW*- LIP be a continuous function.

We say that f has a “$-monotone subdifferential” if

(a-pju- u)3 - (@64fWY 14) + d$AfW~ IPl>>lu - VI2 whenever u, u E D(d-f), CY E d-f(u), /3 E d-f(u).

If p L 1, we say that f has a “#-monotone subdifferential of order p” if

(a-plu- u)a --X(4 u,f(u)~f(u))(l+ IalP + I/v)lu - 4* whenever u, u E D(d-fj, LYE a-f(u), PE a-f(u), where x: S2* x R'-, R' is a continuous function.

It is not difficult to see that:

(a) f has a $-monotone subdifferential if and only if

such that

VUER, v(Uh)h, (Uh)h c Q, v(mh)ht @h>h

lip u,, = n, liy uh = u, nh # nh, (Yh E a-f(&),

PhEa-f(uh), tlhEN, s~p{If(“,)lVjf(u,)lVlru,IVIBhl}c+r,


we have that

inf (a/l - B,, I Uh - Uh)

/&I - d

> --“;

h

(b) f has a @-monotone subdifferential of order p if and only if Vu E R, V(uh)h, (u&, C Q, V((y,)h, (Ph)h such that liy uj, = u, lihm uh = U, uh f uh,

&II E a -fbdy Ph fi a -fm vh E N,

s”p{lf(Uh)IVlf(Uh)l}< +=c, h

we have that

cab - phbh - Oh)

?if (1 + 1 (Yh 1’ + Ip,, IP)lUh - oh)* ’ -=’

Therefore f has a $-monotone subdifferential if and only if

((u-/3]u-up - 4 (4 u*f(u),f(u)7 149 IBl>lu - VI2 whenever u, u E D(a-f), (YE a-f(u), j3 E a-f(u), where 6 : 52’ x R4 --+ 02’ is a continuous function.

Remark 1.5. (a) If g : Zf ---, R U {+=} is 1.s.c. convex function and c 3 0, then the function

f(u) = g(u) - cb12 has a @-monotone subdifferential, where @(z, x,, x2) = c.

(b) More generally, every (p, q)-convex function has a pmonotone subdifferential if $(z,xi, x2) = 16(q- + p]x21) (cf. (1.39) of [ll]).

(c) Let V be a closed submanifold of H with boundary of class C*. Let g : V+ $3 be a function of class c. Then if we define f : H+ R U {+=} by

g(u), if u E V

‘(lr)=I+= ifuEH\V 7

we have that fis a I.s.c. function and there exists a continuous function 8 : H--P R’ such that

f(u) af(u) + w - u> - %4(l+ IfN + IfWl + I4>lfJ - 4*

whenever u, u E H, cr E a-f(u). Therefore f is a function with a @monotone subdifferential with

@(z, x1 9 x2) = e(z>(l + 4 + Ix2 I)

where e: H-* lR+ is a suitable continuous function.

LEMMA 1.6. Letf: R ---, R U {+=} be a function with a #-monotone subdifferential. Let u be

1110 M. DEGIOVANNI, A. MARINO and M. TOSQCES

a point of R such that ) Vfj(u) C: + JZ and y, c, M, r, A be positive numbers such that

z E R and,f(z) kf(u) - y/z - Kj, if y(Z - K/ < 1

hi?> if(K)] + 1 + 6, M>Zy+c

yr< 1, su~{@(z,x~,x~)I/z - KI 6 r, (X1 1 c lid, 1x2) c hi} < +=

A i 2, 24y + c) < r, 2A sup{@(z, x1, x2) / /z - u/ 6 r, /x1 ) S M, /x2 I 6 M} < 1.

Then Vu in B(K, A c), 3 a unique w in B(K, r):

7 E a -f(w), U-W

If(w)IsM, T SM. I I (1.7)

Moreover w E B(u, r) and

(1.8)

if z # w and Iz - WI s r.

Proof. By theorem 1.1 applied to

we can find two sequences, (u& and (wJk in B(K, r) such that 1 uk - u ( G A/k and

Then

which implies that

1 C2y+c+-GM,

k

if k is sufficiently large. Therefore,

Iwk - UI s2A(y+c+f) <r

if k is sufficiently large. Then wk E B(K, r) and

(1.10)

uk - wk - E a-f(wk).

A


Moreover

and

f(Wk) >ff(u) - yr 2f(U) - 12 - M

if k is sufficiently large. Now by $-monotonicity of a-f we have

+ @ Wk,f(Wk), Iyi))jwh - wk/‘,

which implies

iwh -wk1(1-2Asup{~(z,x,,x~)(Iz-ul~r,Ix~l~M,IxlI~~))~~vh-vk~.

Therefore (w& converges to a point w of B(u, r) by (1.10). Moreover If(w) 1 s M, ) (v - w)/A I 6 M. Furthermore by (1.9) we have

&-W12+f(W)5~ Iv - z(z +f(z), Vz in B(u, r)

which implies (v - w)/A E a-f(w), since w E B(r.4, r). The proof of the uniqueness of the w’s satisfying (1.7) is the same as the proof that (iv& is

a Cauchy sequence.

Furthermore if z is a point of B(u, r) such that

by the same argument used in order to prove (1. lo), we find that

lv-‘l,M A-- 7 Iz-uI<r,+ z E a -f(z), If(z) I s fvl.

Therefore z = w.

THEOREM 1.11. Letf: 52 --, R U {+=I, be a function with a Q-monotone subdifferential. Then

VuinR:]Vf](u)<+~, t/c>0

34, r, A0 > 0 such that VA in IO, A,,], Vv in B(u, AC), 3 a unique w in B(u, r) such that

7 E d-f(w), If(w)/ sA4, l&q SM.

1412 M. DEGIOVANNI, A. MARINO and M. TOSQUES

Moreover w E B(u, r) and satisfies the property

f(,)+~~,-~~z<~(*)+~~~-zI’, ifzEB(u,r) and z # W.

Proof. It follows directly from lemma 1.6.

LEMMA 1.12. Letf: R --, R U {+=} be a function with a @-monotone subdifferential of order two. Let u. be a point of R, u a point of D(f) and M, E, r, A positive numbers such that

u E B(uo, r) C R, SUPOC(ZIrZ2,X*,X*)11Zi-UOICr, IXiISM,i=1,2}C+r

16Esup(X(ZI,ZZ,X1,XZ)IIZi--oI~r,]XiI~M, i=l,2}<1

f(z)kf(u)-s if /z-u,j<r; IfWl + ESM 9A.c<(r-(u-uo()2, 2A SUpol(z~~ Z2, Xi, -X2) II zi - uo I zG r,

lxil 4M9 i= 1,2)-C 1.

Then Vu in B(u, Y/(,4&)), 3 a unique w in B(u,, r):

Moreover w E B(uo, r) and

f(a+&lu- w/Z <f(z) + $ /!J - zy

if zEB(uo,r) and z#w.

Proof As in the proof of lemma 1.6 we consider two sequences (u~)~, (wJk such that

Iim ok = u, k

cWk)k c B("Oyr)

and

&bk - Wki2 +f(Wk) s$jvk-z[2 -t-f(z) if zEB(Uo,r).

Then

1 uk - wk 1’ 6 (uk - u12 + 2A(f(u) -f(wk)) s (juk - u[ + d(Ac))’ + 2j.E < GA&

if k is sufficiently large. Now

Iwk - &-J[ s Iwk - ukl + (uk - VI + Iu - u[ + ju -&I

~2~(~E)+Iuk_UI+~/(~&)+/U_UgI<r

if k is sufficiently large. Therefore


and

Moreover

IUk - Wk12 s4E

A ’

-M Sf(K) - E Sf(Wk) Sf(K) + & IUk - u1* sf(u)

+ (Iuk - 01 + d/(AE>)2 2A

GM

if k is sufficiently large. Applying the monotonicity of d-f as in the proof of lemma 1.6 we obtain

Iw/l - Wkl(l-(A+g&)SUpOI(Z1,Z2,X1,X2)IIZ; ---Ois~r

lx,1 s M, i = 1, 2)) d 1 u,, - uk 1.

Now we continue as in the proof of lemma 1.6.

LEMMA 1.13. Let f: 52 +- W U {+z} be a function with a @monotone subdifferential. Then there exists a continuous function 6 : Q X iw4 + lR+, such that Vu0 in R, with l Vf I(uo) < +x, we have

such that

where

f(u) af(u) + w - u> - ~(bfW~ r(uo),f(u>, 14) * 10 - 4*

Vv,uER, Va E a-f(u)

Iu - ~old(~o~f(~o)t v(uo),f(u)9 14) < 17

ID - UOI6(UO~f(~O)? v(uo>,f(u)7 IA> < 1

y(uo)=min{y~y~0,zES2,f(z)~f(u~)-ylz-2401, if ylz-uol<l}.

Proof. We want to apply lemma 1.6, around the point uo.

Let c, M, r and A be positive numbers such that

c=M+2+~/(lfWl), M = (If (uo) I + 2 + $1 + (MY + c>

?+= (1+ Y(~,))-‘~~~p~~‘~l~~P~~~z,~1~~*~l lz - uol S2P,

1x1 l c M, lx21 s M}< +=}

1 1 + Y(K”) + c z= r + SUP{$(Z* Xl 9 x2) II z - uo I

s r, 1x1 I 4 M, Ix2 1 s M} + 1.


It is easy to verify that

(&lJWI++ (suP~P~O/suP~~(~~~l~~~)l /z - W

s 2p, lx* / s Al. /x2/s iM}< +x})-’

+ supW.x,,x,)l Iz - 4 CT, lx, I s M, lx?/ S ill} is locally bounded when ~1~ runs in !A and M is fixed and it is non decreasing when A4 runs in p;pnd; isi;ed. Therefore l/(21.) is locally bounded as a function of (uo,f(rco). y(uo),f(u), ff in x .

Then there exists a continuous function 6 : R x WJ-, WC such that 4 3 l/(S). Now u + ALY E B(u,, r), because jr.4 - uoI + A( cyj < 2A + A j (~1 s AC. Moreover u E B(uo, r), because ju - uoI C 2A s t and

(u + J”4 - 24 E a -f(u)

A 7 If(u)1 s MT (u+Aa)-u <:M

/I -* Therefore by lemma 1.6 we have

&l(u + Aa) -u/z +f(u) + + La) - 012 +f(u>

if u E B(u, r) which is equivalent to

1 f(u)~f(u)+(culu-u)--_Iu-ul’, if uEB(u,r)

2A

and the conclusion follows.

Remark 1.14. Let f: Q * W U {+x} be a function with a @-monotone subdifferential. Then if (u&, is a sequence in D(d-f) converging to a point u in D(a-fl such that

siJPcf(% )I < +“, ydgrad-fWlI< +=

we have that hmf(U,,) = f(u).

Proof. It Suffices to apply lemma 1.13 with u. = u, u = uh, CY= grad-f(uJ, u = u, to get that limzup f(z+J S f (u).

THEOREM 1.15. Let f: R + R U {+=} be a function with a @monotone subdifferential. Then we have

/ Vf i(u) < += e a-f(u) f 0, tlu E R and

lvfb) = kwd-f(u)17 vu E D(d -f).

(For similar results cf. [7, 9, lo].)

Proof. It suffices to consider the case IVfl(z~) < +=, since

IVfl(4 =G Igrad-fWL vu f 51.


By theorem 1.11, we can find M, r > 0 and two sequences (A& (w& such that & > 0, lif Ak = 0 (w~)~ C B(u, r) and

We can also suppose that (u - Wk)/& is weakly Convergent to some (Y in H. Therefore lim wk = cl, and by lemma 1.13, there is a constant c, independent of k, such that,

we have

f(u) sf(Wk) + (F, u - wk) - c/u - wkj’? if cIu - UI < 1.

Now passing to the limit in k, we find

f(u) Sf(u) + (al0 - u) - clu - u/?, if c(u - u/ < 1

which implies that ICY E a-f(u), Moreover we have

fc”> >fcWk) + (cl - Wk(2

A - CIu - wkl* k

which implies

lu- wki $d -fcWk) + c,cl _ wk,

Ak I” - wki

Therefore, passing to the limit in k, we find

I grad- f(u) I G I4 s IV (4.

Now we introduce a particular class of functions satisfying the hypothesis of @-monotonicity for the subdifferential, which has further regulartity properties.

Definition 1.16. Let r$ : C-2’ x R3 + W be a continuous function. A lower semicontinuous function f : $2 + R U {+x} is said to be “@-convex” if

f(u) sf(u) + (4 IJ - u) - @(UY u,f(u)*f(u), I4 > I u - MI2

Vu in D(f), Vu in D(a -f), Vruin a-f(u).

It is easy to verify that every @-convex function has a &monotone subdifferential with a suitable 6.

Furthermore the following facts are equivalent: (a) f is @-convex; (b) VU E n, v(uh)h, (Uh)h c fi-2 with uh # &,, Vh E N, V(&)h with &j, E a-f(&), Vh E N and

liy uh = u, lim uh = u, h

SUhP{If(Uh)lVIf(uh)IVILYhl}c +=*

1416 M. DEGIOVANNI, A. MARINO and M. TOSQVES

we have that

inff(uh) -f(&) - (%/uh - uk) > -=.

k 1 uk - uk 12

A significant example of $-convex functions is constituted by the (p, q)-cc,nvex functions (cf. [lo, 11, 121.

In fact, if g is a (p, q)-convex function, it is easy to verify, by the results of [ll], that g is +- convex, where

@‘(zi, z2,xl,x2~x3> = %- + 64p’(lx1l + 1X21) + 4Pb31).

Moreover, if f : H --, R U {+s} is a 1.s.c. function and c > 0, then the function

u-f(u) + CJLq

is convex if and only if f is @-convex with

@(z,, zz, x1, x2, x3) = c.

In fact if f:H + IX! U {f~} is a 1.s.c. function which is +-convex with ~(zi, zz, x1, xz, x3) = c 2 0, it is easy to verify that g(u) = f(u) + c(u/* is 1.s.c. and @-convex with $(z,, zz, xi, x2, x3) = 0; therefore, by proposition (1.2)

inf(g(z)+jzl*zEH}>-*.

Set

gr(u) = inf ( &lU -z12 +g(z)lzEH , if O<J.<i,uEH.

I

By an argument similar to the one used in the proof of lemma 1.6, we can see that this “inf” is actually a “min”, that the minimum point J*(u) is unique and that JA is Lipschitz continuous of constant 1.

Moreover gA is a CL function and

grad g*(u) = ’ - i’(‘) ,

which implies that

(gradg&) - gradg&)lui - ~2) 3 0 and that gi is convex.

Since

g is also convex.

g(u) = ;iy* g*(u) Vu E H,

THEOREM 1.17, Let f : R -+ R U {+=} be a @-convex function. Then for every sequence (u& C R, (a& C H such that

1iF uk =uER, lir (Yk = @

in the weak topology of H

s;pcfbk)} < +=t ck E 8 -f(h), Qk,


we have

(Y E J-f(u).

Therefore for every MO, MI in R, the set

{u E S2 If(u) < AZ,,, 1 grad- f(u) 1 s AZ,} is closed in $2.

Proof. Since f is a I.s.c. function, sup{ If(uk) I} < + SC. Therefore there are two constants c,

6 > 0, independent of k such that

f(o) 3f(&) + (&In - %) - clu - QIZ, Vu in B(u, S) :f(u) Gf(u) + 1.

Then passing to the limit in k, we obtain CY E a-f(u).

THEOREM 1.18. Letf: Q + R U {+=} be a function with a @-monotone subdifferential of order two. Then the conclusion of theorem 1.17 holds.

Proof, We want to apply lemma 1.12. Put M = stp{ 1 f (uk) 1) and choose E, r, A > 0 as in the

assumptions of lemma 1.12. By decreasing A, we can also suppose that sup{ 1 cyI, 1’) c E/A. Then

we have, if k is sufficiently large, that uk + Acu, E B(u, I), uk E B(u, r) and

(uk + ‘y) - uk E d-f(&), If(&)/ 6 M, I(Uk+Aak)-uk~2 <4E

A -.

Therefore, by lemma 1.12, we obtain that

f(Uk) + & i(Uk + Ask) - Ukl* sf(z) + $ I(Uk + Ask) - d*~

Vz in B(u, r) which is equivalent to

fb) >f(uk) + (akb - Uk) - & Iz - Ukl*, Vz in B(u, r).

Then, passing to the limit in k, we obtain that CY E d-f(u)

Theorems 4.2, 4.3 contain some results more general than the previous ones (1.17) and (1.18).

THEOREM 1.19. Let f: Q + R U {+ 30) be a function with a Q-monotone subdifferential. Then

Vu0 in D(d-df), VM in W, 3r>O:

v(Uk)k in B(Ub r)y v((Yk)k in Hsuch that (Yk E a-f(Uk),

liy nk = n, lip ak = (Y in the weak topology of H

““kp cf(Uk)) s Mu s~p{bkI}sM

we have that CYE d-f(u).


Therefore, VMa, AI1 S M, the set

is closed; in particular, for every M, the function

restricted to the set (u E D(d-f)If(u) G M} is 1.s.c. on it.

Proof. By lemma 1.13, there exist two constants c, r > 0, depending on uo, M such that

u ++ I grad- f (4 I

f(v) Sf(Uk) + (4” - 4) - + - kx Vu in B(u,2r), VkinN.

Passing to the limit on k, we see that LYE d-f(u).

PROPOSITION 1.20. Let f : R --, A U {+Jc} be a function with a #-monotone subdifferential. Then

Vu0 in D(I~-f), vcao, 34, r, A0 > 0

such that

VA in IO, Lo],

there is a map

such that

Ji : RA + B(uo, 24

IJA(vJ - JA(VZ) I ,&Ml"l -“*I,

VA in IO, Ao], Vvl, v2 in S2*

v - JA(v) /I

E 8 -f (JA (“))

where

and Iv - 2.41 s AC}

Vu in Q,

N = {u E B(z.40, r)jf(u) 4 c, ) grad-f(u) ) G c}.

Moreover, if we define

fA(“) = &I” - Jd”)l’ +f(JA(“))

we have that fA : QA+ R is a C’ function and VA in IO, A,], Vu, it, v2 in QA

gradfA (“) = ” -:,(“), (gradfA(“l) - gradfA(“dl”l - “2)

2 -Ml”, - “2 1%

Evolution equations with lack of convexity

Furthermore, Vu in N,

1419

]gradfA(u)j s ‘gyd-LE)’ ~_-. lim J*(u) = u,

lim gradf*(u) = grad-f(u) I-0

and fA(u) converges increasing to f(u) (as k + 0). Finally N is closed and

f(o) Sf(u) + ((u\ u - u) - MI u - u1*

Vu in B(uo, r), Vu in N, Vcr in J-f(u) such that 1 aI s c.

Proof. First of all we can find, by lemma 1.13, M,, r > 0 such that

B(uo,2r) C Q, f(u) >f(u) + (alu -u) - M,~u - t.41’

Vu in B(uo, 2r), Vu in B(uo, r), Vcu in a-f(u) :f(u) 4 c, 1 &yI C c. Therefore the set

N = {u E B(uO, r)jf(u) s c, Jgrad-f(u)1 s c} is closed

and we have y(u) d (l/r) + (c + Mlr), Vu in N. By lemma 1.6, and decreasing r if necessary, we can find M?, Ms, )co > 0, such that

and VA in IO, A,,], Vu in N, Vu in B(u, J.c) there exists a unique JL(u) in B(u, r):

’ - ;‘(‘) E a-f&(u)), If&(u))1 s M,, 1’ - :,(‘) 1 s M2

moreover

fA(U> = &lu -J*(u)12 +f(.J&>) + - 4* +f(z), Vz in B(u, r).

As usual, by @-monotonicity, we have

(1 - AMd1Ji.W - JA(u~)I+~ - 021;

therefore, by a suitable choice of A,,, we get

IA - Jk(u2)I s 1 _ iM, 10~ - 021, VA in IO, LoI7 Vul, u2 in %,

because JL(u) is well defined if u E B(u,, AC) f~ B(u2, Itc), for some u1 and u2 in N. Now it is standard to show that fL is a C’ function on Q* and that

n - Jl(U) gradfA(u) = A .

1420

Moreover

M. DECIOVAWI, A. MARINO and M. TOSQUES

(gradfdud - gradfdu2)luI - u2>

2 lo1 ; u2’ (lfJl - !_I*/ - IJ*(u1) -J&J*)/)

2 - 1 _(;liMj Iul - u21* 2 -M41u1 - 42

for a suitable constant &I, and VA in 10, A,], Vu,, u2 in RA. Finally if u E N, we get, by @-monotonicity,

(1 - J.Mj)/gradf~(u)i = (1 - AMs) 1’ -pi/ 6 /grad-f(u)/.

Therefore for every u in N, we have pi? J*(u) = u and fA(u) converges increasing to f(u)

sincefis 1.s.c. andf(JA(u)) “fA(u) of. By theorem 1.19, we get, also, that liliO gradfA(u) = grad-f(u).

LEMMA 1.21. Let f: a + R U {+s} be a I.s.c. function such that

m = inf v(z)]z E R}) - =.

Suppose that there exists c > 0 such that:

(a) ]VJ(z) 2 c, t/z in R; (b) 3u, u in Q, with u f u and f(u) E 5X such that

f(u) sf(u) - CID - u].

Then there is a w in aS2 such that

Jw - u( sfy m, f(w) sf(u) - CIW - UI.

Proof. By theorem 1.3 we can find a sequence (wJk C aQ such that

IWk - u/ e f(u) - m c - (l/k) ’ f(Wk) sf(u) - (c - ;)I& - 4.

Since

lb - UI + Iu - UI s fb-4 - m c - (l/k)

+ f(u) -f(u) C

f(wd sf(u> - 4% - 4 + Iu - 4) + gy,r,“,

we can suppose that Ii,m(]wk - u( - (wk - u]) = (u - ul, otherwise we can choose a wk as w.

Therefore ii,m(] Wk - uj(u - u( - (Wk - UJU - u)) = 0.


After choosing a subsequence, we can also suppose that (w~)~ is weakly convergent to some w in H.

Then we have lw - uIju - u/ > (w - U(U - u) = lim(w, - D/U - u)

lip jwk - VI/U - ~1 31~ - ullu - UI

which implies (0 # cl) lim wk = w. k

Therefore the conclusion holds since f is lower semicontinuous

THEOREM 1.22. Let f: G + R U {+x} be a 1.s.c. function such that fin has a $-monotone subdifferential on CC!.

Suppose that there exists K in $1 with f(u) E R and suppose that:

(a) inf cf(u)lu E Q} > -=; (b) 3c > 0 : IVf](u) 3 c, Vu in R. Then there exists w in as;! such that

f(w) 6f(U) - CIW - UI.

Proof. By lemma 1.21, it is sufficient to find u in R such that u # U, f(u) S f(u) - clu - ~11.

If ]Vjj(~) = +r, it is obvious to find such a u. If ]Vfl(~) < +=, that is d-f(u) # C#J (cf. (1.15)), there exists by theorem 2.4 a Lipschitz curve

U: [0, T] -+ R such that U(0) = u, U’(t) = -grad-f( U(t))

a.e. on [0, T], f 0 U is Lipschitz continuous and

(f 0 v>‘(t) = -Igrad-f o U(t)]* a.e. on [0, 7’j.

Therefore, if we put u = U(r), we get

f(u) = f 0 U(T) s f(u) - rr /grad-f(U(O)l* dtsf(u) JO

- c I ’ IU’(t)l dr s f(u) - clu - ~1.

0

SECTION 2

In the following, R will denote an open subset of the real Hilbert space H.

Definirion 2.1. Let f : R + W U{+x} be a function.

If I c R is an interval with j f 0, a map U: I ---, L-2 is said to be a “mean maximal slope curve” (or briefly a “m.s. curve”) for f if:

(a) U is continuous on I and absolutely continuous on compact subsets of I”; (b) f 0 U is nonincreasing on I; (c) U(t) E -a-f(u(t)), a.e. on I.

1422 M. DEGIOVANNI, A. MARINO and XI. TOSQUES

PROPOSITION 2.2. Let f: R + fR U{+r} be a function with a $-monotone subdifferential. A curve U: I-+ R is a m.s. curve forfif and only if it is a m.s. curve in the sense of [9, 191,

that is:

(a) U is continuous; (b)foU(t)<+r ifrEZ,t>infZ;

(c) IU(t,) - U(ti)l c ifi iV~(U(r)) dr, if t, , t2 E I, t, < t2 ;

(d) fo U(t,) -fo U(I:; G - 1” jVfl(U(r))’ dt, if t,, t2 E I, t, < t2. r1

Proof. Suppose U is a m.s. curve for f in the sense of (2.1). Since f 0 II is nonincreasing and a-f( U(f)) # 0, a.e., (b) holds. It is easy to verify that

(fo U)‘(r) = -~U’(f)~~

if U and f o II are differentiable in t and U’(r) E d-f( U(r)). Moreover, in such a point, we have, for every sufficiently small h > 0,

fo U(t+h) -fOU(f) h

g - \vf/( q)) 1 vt + h, - W)l h

_ lu(t + h) - WI h E(lW + h) - W>l)

where E : R --, R is a function such that h~i~ E(S) = 0.

Therefore

-IU(W = (f O I/)‘(0 2 -IVfl(U(WWl

which implies [U’(t)/ 6 jVfj( U(t)). S’ mce the opposite inequality is obvious we have that

(f 0 v>‘(t) = -IU(f)12 = -jVJ’(U(t)), a.e. on I,

which implies (c) and (d), because f o U is nonincreasing. Now suppose that U is a m.s. curve for f in the sense of [9,19]. Clearly f 0 U is nonincreasing.

By (2.2) of [19], U is absolutely continuous on compact subsets of I”. Finally by (4.4) of [9] and by (1.15), we get that

U’(r) = -grad-f(U(f)) a.e. on 1.

THEOREM 2.3. Let f: C-2 + R u {+=} be a function with a @-monotone subdifferential. Let U1, U2 : [0, T[ + Q be two m.s. curve for f. Then, if 0 G t < T, we have

IW) - U2(t)l C IWO) - Li,(0)lexpiJr (@WI(~~

f(W)), IWN : 9(~2(hfo U2(sh IG(r)l>) d+

with the convention that 0 * 3~ = 3~.


Proof. It is sufficient to write the derivative of \U,(t) - Ur(r)J? and to apply the $- monotonicity.

THEOREM 2.4. Let f: Q -+ R u {+z} be a function with a @monotone subdifferential. Then

Vu0 E D(d-f), VK 3 0,3T, r > 0:

Vu in B(u,,, r) II D(d-f) with

f(u) 6 K, Igrad-f(u)1 s K

there exists a (K + 1)-Lipschitz continuous m.s. curve I/: [0, T[ -+ D(d-f) for f, such that U(0) = u, fo U is Lipschitz continuous.

Proof. Choose c > (Igrad-f(uO)l vf(ua) v K) + 1 and consider r, M, A,, N, S&, fk, J*, etc. as in the proposition (1.20).

We can also suppose, after decreasing AO, that

K < (1 - A,M)(K + l), &,c < ;.

Let u be in B(ua, r/2) n D(c~-f) such that

f(u) =Z K, Igrad-f(u)] S K.

For every A in IO, A,], let VA : [0, T,[ + RA be the unique solution of

Vi = -gradfA(UA) with CIA(O) = u

defined on its maximal interval of existence. Since we have (cf. (1.20))

(IUA(t + h) - UA(t)l’)’ = - 2(gradfA o U,(t + h)

- gradfn o U,(t)lU,(t + h) - U,I(~)) 4 2 M(IUA(t + h) - U,(t>12)

we get

/U:,(t)/ s IU;(O)l eMt s K e .nr

l-A&f

Moreover

IJ,@&)) - u,,I s lJ~(U,(tl) - U,(t)/ + /U,(t) - UI

+ IU - uoI s (A, + t) 1 _t 0

M eMf + ia Therefore, if T is such that

K

1 - A& erM<K+l,(Ao+T)c<~

1424 M. DEGIOV~I, A. M.WNO and M. TOSQUES

we have that, if f E [0, T,[ fl [0, T],

IJi(U&)) - uOI < r, Igrad-f(J@#))l s lgradf&W))/ < K+ 1.

Since f(J*( UA(r))) =z fA( Ui(r)) s f~(u) 4 f(u) C K, we have that

J,(U,(r)) E N, vr E [O, TA[ i-l [O, T].

Then

U,(t) E {u E Hj3u E N: 10 - UI i lL(K -t l)}, Vr E [O, T*[ n [O, T].

Since K + 1 < c, it follows that TA > T. Also we have:

(IV, - UP]*)’ = -2(gradf* o U, - gradf, 0 U,,,~UA - U,)

= -2(gradfA 0 UA - gradf, 0 U,IJn(Ud - J,u(U,))

- Z(grad fA 0 UA - grad f, 0 U,jA grad fl 0 U,, - ,LI gradf, o U,)

s 2M(lJ&U*) - JJu,)l)* + 4c’(A + ,u)

s 4M(lUl - u# + 43 (/I + p + 2M(A’ + $)).

Therefore, by the Gronwall lemma, we get

\U,(r) - U,(r)j2 s 4c2T(A + p + 2M(A* + $)) exp(4MT), Vr E [O, T]

which implies that (U,), is uniformly convergent on [O, T] to a continuous curve U. Clearly U is Lipschitz continuous of constant K + 1 and

lJ* (Ui (f)) - U(t)1 s IJL (U, 0)) - UA (01 + I UA (0 - WI c A(K + 1) + IU,(r) - U(r)l.

Therefore lip J*( U,) = U uniformly on [0, T] and U(t) E N, because N is closed (cf. (1.20)).

Clearly

Moreover

f( U(r)) s limAinf f(Ji (U,i (r))), Vr in [O, T].

f(W)) ~f(J*(W)>) - (U~WlU0) -Jn(UW>)

- MU(r) - J&J&))l* V(J&W>>)

- (c + MIW - JA (W))~)lW - J&‘dO)l

which implies that liAm f(Ji(Uk(t))) = f(U(f)), Tin [0, T].

Since

f&b(r)) -f(J#,(r))) = &iU,(r) - JLPAWI* +*

we have, finally, that liAmfi(Ui(r)) = f(U(r)), Vt in [0, 2-j.


But fA o CJi are equilipschitz nonincreasing functions; therefore fo CJ is a Lipschitz continuous nonincreasing function and

(fo U)‘(t) 5 -~Vfi(U(t))lU(t)~, a.e. on IO, T[.

Since for 0 d t, < t2 s T, we have

f~(W2)) -f&X(Q) = - I” IW>12 dt

r1

f-r2

s- J fl Igrad- f(-h(W)))12 dt

we get that

fo vt2> - fo U(t1) s

by the weak convergence in L’(0, T; H) of Vi to

- I ” IU’(t)l’ dr r1

U’ and

I-‘? fo U(t,) -fo U(t,) s - j IVN-‘(~>)2 df

r1

by Fatou’s lemma, theorems 1.19 and 1.15. Furthermore, by the previous estimates, we get also that

-IVZ-l(WW’(~)l s (fo v>‘(t) s -lwo12, a.e. on 10, T[ which implies that

]U(t,) - U(t,)] 6 i” /U’(t)1 dt d jr2 IVjj(U(r)) dt, (1 I1

therefore U is a m.s. curve for f by (2.2).

THEOREM 2.5. Let f: S2 - R U {+=} be a function with a @-monotone subdifferential. Let U: I+ 52 be a m.s. curve for f such that

Then

(a) U is Lipschitz continuous on compact subsets of Z\{inf I);

(b) for every t in !, there exists U; (t), U; is right continuous, j U; 1 is lower semicontinuous, I U; I is of bounded variation on compact subsets of Z\{inf I); U is differentiable and U; is continuous possibly outside of a countable subset of I and this happens where IV; I is continuous; moreover

I&O + h)l s lU;Wl exp(2Jfth @(Ws),f(W))7 Ill) b) I

if c E Z\{sup I), h 3 0, t + h E Z\{sup I) and U;(t) exists;

1426 M. DEGIOVANNI, A. M.WNO and M. TOSQUES

(c) if a-f(u(t)) f 0 in a point t of Z\{sup I), we have

U:(t) = -grad-f(U(t)), (fo U)l)(r) = -/grad-f(U(t))lZ;

(d) if for every c in R, the set {ulf(u) s c, [grad-f(o)/ G c} is closed in $2 (for instance, if f is @convex or d-fis @-monotone of order two), we have

a-f(u(f)) # 0, Vt in I\{infI},

f 0 U is Lipschitz continuous on compact subsets of I\{inf I) and

f oU(t,) -f 0 U(t,) = - 1” /grad- f(U(t))l* dt (1

Vt,, t2 in I (with t, f r,).

Proof. First of all let us consider t E Z\{sup fl, such that

l im inf I w + h) - WI < +cc

h+O+ h

By (2.3) applied to U(s + h) and U(S), we get

I W + h) - WI ~ I W + h) - WI exp ’ h h

b#W~ + h),

f(W+ h)), IWa+ h)l) + dW),f(W)), iIl’(o)l)do]

Vs b t, Vh > 0 with s + h E I. Therefore, for a suitable constant K, we have that IU’(s)l d K a.e. in I n [t, +r[ and (a) is

proved. Now suppose also that f( U(t)) < +=. Let r < T < sup I and let A : H+ 9(H) be the operator

defined by

Au =

I

b E a -fWl I4 6 W, if u E U([t, T])

0, elsewhere.

Clearly, by #-monotonicity, there exists a constant M 2 0 such that

(a - pIz.4 - u) 3 --A+4 - u/2

if Au # c$, Au # $ and (YE Au, /3 EAu. Then v’(s) E -AU(s) a.e. in [t, T]. Therefore by known results (cf. [2, 21]), U!+( ) s exists for every s in It, T[, Ul, is right

continuous, IU;(t)j is lower semicontinuous, U is differentiable and lJi is continuous where IU’+j is continuous.

Moreover, by the first part of the proof, we have

IWs>l s IW9l ex+f 9(W)7f(W4), IWN) do f

if s, f E I\{sup I}, s 2 t and U;(t) exists.


Therefore /I!& 1 is of bounded variation on compact subsets of Z\{inf Z) and the proof of (b) is complete.

Now suppose I E Z\{sup I), a-f(U(f)) f 0. Let V : [t, t + 6[ + D(l)-f) be the Lipschitz m.s. curve for f given by theorem 2.4 such that V(r) = U(t).

By theorem 2.3 we have that V = U in a right neighborhood of f. Therefore, by theorems 1.15 and 1.19, we have that /Vfl(U(s)) is lower semicontinuous in

a right neighborhood of t. Then by (b) of (2.4) of [19] and (b) of (1.3) of [19] we get

U;(t) = -grad-f(U(t)), (fo U):(t) = -/grad-f(U(t))]*.

Finally suppose that for every real c, the set {U E Q(f(u) d c, Igrad-f(o)] c c} is closed in $2. By (b) and (c) we have that a-f(U(t)) # 0, Vt E Z\{inf r).

Moreover, by lemma 1.13, for every t in Z\{inf Z} there are M, 6 > 0 such that

f(W)) W(Wt)) - Igrad_f(U(t,))lIU(t2) - WI)] - 1M]U(t2) - U(t,)]* if tt, t2 E ]t - 6, t + S[ fl I.

Therefore fo U is Lipschitz continuous on compact subsets of Z\{inf Z_I and

f(Wd) -fWd = - (” Igrad- f(Wl* dt r1

for every t, and t2 in I, with tt # t2.

THEOREM 2.6. Let f: S2 -j R U {+=} be a function with a @monotone that for every-c in W {u E Qp(u) < c, /grad-f(u)/ < c} is closed in R.

subdifferential such

Let Z : [0, T[ + Q be a m.s. curve for f which cannot be extended in a right neighborhood of T.

Then, if T < +“, we have that either there exists

!“: U(t) E as2

or

!ir (grad- f(W))1 v (-f(W))) = +=

and $WW7 fo W>, IWN is not integrable in any left neighborhood of T.

Proof. Suppose that T < + x. First of all it is clear, by theorem 2.5, that !iy Igrad- f(U(t))j

v c-f(W)) = + =, implies that @(U(t), f 0 U(r), IU’(t)l) is not integrable in any left neigh- bourhood of T.

Now suppose liz?f (grad- f(U(t))( V (-j(U(t))) < +x.

Since F-w,) - Wdl s (12 - h)“2 (1” Iv,(s)l2 dsy2

s (t* - t*)@ (f ‘:, U(t,) - f 0 U(t*)) 1/*

if 0 < fl < t2 < T, there exists !irnr U(t) = ti E ii.

1428 bl. DEGIOL’A.VNI, A. MARINO and IV. TOSQCES

Then, if r~. E Q, we have a-f(c) f 0, and the interval [0, T[ can be extended by theorem 2.4.

The following example shows that, in general, there does not exist a m.s. curve for a function fif the initial data does not belong to the domain of a-f andf is @-monotone of order greater than two (cf. (3.2)).

Example 2.7. Let (e,&, be an orthonormal system in H. Consider the functionf: H-, RU

{+x} defined by

c 0 ifu=O

1 _- log IUI

ifu=ie,,O<A=Zk

f(L0 =< 1

log IUI if u = Ae,, n 3 10, ;<A

1

n “lo

j+x elsewhere.

Then f is lower semicontinuous and YE > 0, ZlM > 0:

f(u) >f(u) + ((uju - U) - M(I + (++s)ju - U/2

whenever u E D(fl, u E D(Ff), ty E J-f(u). Moreover there is no ms. curve U for f&fined in a right neighborhood of zero such that U(0) = 0 and if we consider the m.s. curve V for f such that V(0) = e,/lO and [0, T[ is its maximal interval of existence, we have that T < +r and jiyf(V(t)) > --r.

THEOREM 2.8. Let f : S2 + R u { +=} be a function with a @-monotone subdifferential. Let (& c D(d-f) be a sequence converging to a point K of D(a-f) such that

s;pVW} < + =, S;P {Igrad- f(~h)l) < +=

and let U: [0, T] + D(d-f) be the Lipschitz continuous m.s. curve for f such that U(0) = K. Then we have that

(a) the Lipschitz continuous m.s. curve U, forf such that Uh(0) = r.+ (cf. (2.4)) are eventually defined on [0, 7’j and (uh),, converges to U in H’+(O, T; H) if 1 S p < +=;

(b) the sequences ((grad-f( Uh)j) h’ and (f OVA), are WcXItUEtlly bounded in L”(0, T) and liFf( Uh(f)) =f( v(t)), Vf in [O, T].

Proof. A similar result will be given in theorem 4.11 in the case of a sequence of functions (fh)h with some further assumptions.

Therefore we show, here, only the part of the proof which is not contained in that one. (I) First of all we prove that for every zi in D(a-f) for every M > 0, there exist r, t > 0 such

that if (oh),, is a sequence converging to a point u of B(ti, r), with

f(ud s M, Igrad- f(udl c M, (Vh E W,


then the Lipschitz m.s. curves V, V,, forfsuch that V(0) = u, V,(O) = uh are eventually defined on [0, t] with values in D(a-f), (v&, converges to V in H’,‘(O, t; H> and (Igrad-foV&h is bounded in L”(0, t). In fact by theorem 2.4 there are V, Vh : [0, t] + D(d-f) such that V, VA are (M + 1)-Lipschitz continuous m.s. curve for f with V(0) = u, V,(O) = uh and fo V, f 0 V, are Lipschitz continuous.

After decreasing the r, and in view of (2.4), we can suppose

f(z) af(ri) - 1 if z E B(u, 2r)

sup{#(z, xi, xZ)] /z - ci] S 2r,f(C) - 1 d xi 6 M, Ix:1 S M + 1) < -t-r.

Furthermore, after decreasing r, we can suppose that

Vh([O, r]), V([O, t]) C B(u, 2r).

Therefore by theorem 2.3 the sequence (v& converges to V uniformly on [0, r]. Moreover by remark 1.14 \ve have that lirf(u,J = f(u).

Then

lim;up

sf(u> -f(V(r>> = I’ lV’(412 b, 0

which implies that (Vh)h converges to V in H’.‘(O, t; H). (II) Now let (&)h and U : [O. T] * D(d-f) be as in the assumption and r/h : [O. Th[ + o(a-f)

be the locally Lipschitz continuous m.s. curve forfsuch that UJO) = r+,, defined on its maximal interval of existence (cf. (2.4)).

Then the same proof of (II) of theorem 4.11 shows that limhinf Th > T, (uh), converges to

U in H’**(O, T; H) and (Igrad-fo u&h is eventually bounded in L”(0, T). Therefore by remark 1.14 we have that

liyf(Uh(t)) =f(u(t)), tltin [O, T].

SECTION 3

In this section f : 52 * W U {+=} will denote a function with a $-monotone subdifferential of order two, that isfis a 1.s.c. function and

((u - piu - 0) z -x( u. U,f(~),f(~))(l + bl? + IP12>b - u12

whenever u, u E D(a-f), cuE a-f(u), p E a-f(u), where x: Rz X E?-, [w’ is a continuous function.

We shall denote, also, by @ : R X R ?+ R+ a continuous function such that

(a - Plu - 0) 3 -+?+A49 14) + @oAfW7 M>lIu - VI2

whenever u, u E D(a-f), CY E a-f(u), p E a-f(u), (cf. (1.4))


THEOREM 3.1. Let U,, Ut : [0, T[ -+ Q be two m.s. curves for f such that f( U,(O)) < +x and

W(0)) < +=. ThenifOGtCTwehave

where

lUt(t) - U~(t)l s lUr(0) - U/2(0)1 exp(K(t, UI, UZ)(~ +fo U,(O)

-fo U*(t) +fo U?(O) -fo u2 (4 >>

K(t, u,, u,> = sup(;l(U,(s), U,(S),fO U,(S),fO U,(s>)lO sx s t).

Therefore, if U,(O) = U,(O), we have that U, = U, on [0, T[.

Proof. By the same proof of theorem 2.3 we get

lUt(0 - u2(0 s lU1(0> - U2@>l exp (/‘x(UIb). U2(S),fo Ul(S)jfO~r2(~))

0

x (1 + lW~)12 + lWJl2N+ IU,(O> - U,(O>l

x exp(K(C Ul, U2)(t +fo U,(O) -fo U*(t) +f 0 U2(0) -f 0 U2(t))).

THEOREM 3.2. For every u in D(f), there is a m.s. curve U: [0, T[ - R for f such that U(0) = u.

Proof. By the lower semicontinuity off we can find r > 0, m E R such that B(u, 2r) C $2, f(z) 3 m for every z in B(u, 2r). After decreasing r we can also suppose

K = s”p{X(z* Y z27 x1 7 x2)l Izi - uI G 2r7 lxil s lf(u>l

+ In&i= 1,2}< +=.

By proposition 1.2 there iS a sequence (c& in B(U, I-) such that hm uk = u, f(Uk) S f(U),

d-f(Uk) # 0, Vh in N. Let Uk : [o, Tk[ --, B(u, 2r) be the m.s. curve forfsuch that u,(o) = Kk (cf. (2.4)) defined on

its maximal interval of existence. By theorem 3.1 applied to Uk(t f s) and U,,(r) and by theorem 2.5 we get

bd_f(UkW)l 4 lwd-f(“k(t))l exp(K0 + 2(f(U) - 4

ifOat<r<Th.

(3.3)

Then by theorem 2.6, we have ,“rnk /Uk(t) - u[ = 2r or Tk = +m.

Since

[U,(t) - u/ s IUk(t) - ukl + r s d(f(f(U) - m)) + r

if T > 0 is such that d(T(f(u) - m)) < r, we have that Th > T. By theorem 3.1 it is

Iuk(f) - udf)i =S bk - uk( exp(K(T + 2(f(u) - m))) if t E [0, T].


Therefore (U,), converges uniformly on [0, T] to some continuous curve U: [0, T] --, B(u, 2r).

Moreover U E H’s’(O, T; II) because (cf. (d) of (2.2))

ir Igrad- f(U,(r))l’ dr = /r /UA(t)l’ dt <f(u) - m. 0 0

By Fatou’s lemma we have

,I&= inflgrad-f(Uh(t))l c +a, a.e. on]O, T[.

Then by theorem 1.18, (3.3) and remark 1.14 we have that for every t with lim inf Igrad-f( U,,(t))] C +m there is a subsequence (Uhk)k such that h

Therefore d-f( U(t)) # 0, Vr in IO, T] and f 0 U is nonincreasing in [O, T]. Finally, Vt in IO, T[, let V: [0, S[ --, B(u, 2r) be the m.s. curve for f such that V(0) = U(r)

(cf. (2.4)). By theorem 3.1 we get

Iv(s) - Uh(r + s)l G IU(r) - uh@>l exp(K(T + 2(f(u) - d)

ifossC(T- t) A 5. Going to the limit as h + += we obtain that V(s) = U(t + s) in a right neighborhod of zero; in particular we get that U;(t) = -grad-f((U(r)) by (2.5).

Since, Vc in R the set {u E @f(u) s c, Igrad-f(u)1 SC} is closed in Q (cf. theorem 1.18), from theorem 2.5, we deduce the following theorem.

THEOREM 3.4. Let U: I-, Q be a m.s. curve for f. Then

(a) Vr E Z\{inf 4, d-f(U(r)) f 0

and

vt,, t2 E I, (tl f t,)f(u(t2>) -f(~(t,)) = - Li2 bd-f(uO))12 dc

(b) if a-f(U(t)) # 0 f or a certain t in A{sup I}, we have

U:(t) = -grad-f(U(t)); (fo U);(t) = --lgrad-f(U(0)12, Igrad-f(U(t + h))l

6 lgrad-f(u(9)l exp(K(h + 2f(u(O) - 2f(uO+ h)>>> (3.5)

whenever h B 0, t + h E Z and where

K=SUp(X(U(sl)v u(s2),f(U(s,)),f(u(s,>)Itssi st+h,i= 1,2),

moreover +( U(s), f( U(s)), 1 u’(s)~) is integrable on [t, t + h],


(c) if I is the maximal interval of existence and sup I < f=, we have

Proof. Let f, T E I be such that t 6 T, a-f( U(r)) # 0. By theorem 3.1 and theorem 2.4, U is Lipschitz continuous and +( U(s), f OU(s), /U’(s)/) is

essentially bounded in a right neighborhood of t. Therefore, by theorem 2.5, it is

U;(t) = -grad-f(U(t)), (f 0 U):(t) = -/grad-f(U(r))j’.

Moreover, by theorem 3.1 applied to U(s + h) and U(s) we get

IWs)l s IWt)l exp(C - 0 + 2f(W - 2fW)))

for almost every s in It, T[ where

K = supk(W,), U(s*),f(U(s,)),f(U(sz>)lf s 4 <s, i = 1,a

therefore $( U(s), f(U(s)), jU’(s)l) is essentially bounded in [t, T]. Now the proof of (a) and (b) can be completed by means of the results of theorem 2.5. Finally, suppose that I is the maximal interval of existence and that sup I C fx. By (b) we have that I lisEl f(U(t)) > --3c implies that l\rn,;,“p /grad-f( U(r))/ < +z.

Therefore by theorem 2.6, we must have lJn,d(U(r), JQ) = 0.

Remark 3.6. Let U: [0, T]-, S2 be a curve such that u E H’qo, T; H), supcfo U(t)] 0 < f < ZJ < + =, U’(r) E - a-f(U(t)) a.e.

Then f o U is nonincreasing and therefore U is a m.s. curve for f.

Proof. Let V: [0, S[ + R be the m.s. curve for f such that V(0) = U(0) (cf. (3.2)) defined on its maximal interval.

Put

Then

I ’ x(W), v(@,f 0 u(S>,f 0 v(S))(l + Iv’W2 + IV’Wi’> dd

0

Since

S K(s) S + ( jrlv’(i)“di+foU(0)-foV(s)). 0

ww - V(W) s 2x(W), V(4Tf 0 u(4,f 0 V(6)) x (1 + Iu’(S>12 + Iv’(S>i*)~U(S> - V(S)i2

we have, by the Gronwall lemma, that U(s) = V(s) on [0, T] II [0, S[. Then, by theorem 3.4, we have that S > T and f 0 U is nonincreasing.


THEOREM 3.7. If (u,Jh is a sequence converging to a point u E D(fl such that

s;p(f(u, )1< +=

and if U: [0, T] + R is the m.s. curve for f such that U(0) = U, we have

(a) the m.s. curves CJ,, for f (cf. (3.2)) such that Uh(0) = &, are eventually defined on [0, T] and (uh)h converges to U uniformly on [0, T];

(b) liy f( Uh(t)) = f( U(r)), Vr in [0, q, the sequence (/grad-f( Uh(t))l) is eventually bounded

in L”(t, T), Vt in IO, T] and the sequence (f0 u,,),, is bounded in L”(0, T); (c) (Vi), converges to II’ in LZeE(O, T, H), if 0 < E

f(u).

S 1 and in L*(O, T, H) if lihmf(uh) =

Proof. A similar result will be given in theorem 4.9 in the case of a sequence of functions (f&, with further assumptions. Therefore we show here only the part of the proof which is not contained in that one.

(I) First of all we prove that for every ti in D(fi and for every M in R, there exist t, r > 0 such that if (Uh)h iS a sequence converging to a point u in B(fi, r) with s;p f(uh) S M, then the

m.s. curves v,, v for f such that v,(O) = uh, V(0) = u are eventually defined on [0, t] and (VJ converges to V uniformly on [0, t]. In fact we can find r > 0 such that B(ti, 2r) C Q and m = infCf(z)l Iz - tij 6 2r) > -x. After decreasing r we can also suppose that

E = SUp(x(Z1,Z2, X1, .X2)1 IZi - Lil S 2r, [Xi/ S IMI + Iml, i = 1,2} < +z.

Let (Uh)h be a sequence (which we can suppose contained in B(ri, r)) COnVerging to a point u

in B(ti, r). Let vh : [0, th[ + B(ri, 2r) be the m.s. curve for f such that V,(O) = uh (cf. (3.2)) defined on its maximal interval of existence. By theorem 3.4 we have that th = += or

lim lti - v,(t)) = 2r. t* Th

Since 1 v,,(f) - ti 1 < d(t(kf - m)) + r, if t > 0 is such that d(t(M - m)) < r, we have that th > t, Vh E N. The same result hold for V. Moreover, by theorem 3.1, we get

Iv,(f) - v(f>I s bh - 0 I exp(R( t + 2(M - m)))

which implies that (vh)h converges to V uniformly on [0, t] and step (I) is proved. (II) Now let U : i0, T] + Q be the m.s. curve for f such that U(0) = u and CJ,, : [0, Th[ --, R,

the m.s. curve for f such that u,(O) = &, defined on its maximal interval of eXiSti%Ce. We Set

s = sup{f E [o, T] 1 f < limhinf Th, ( uh)h converges to u uniformly on [0, f]}

and we continue as in the proof of theorem 4.9.

1434 M. DEGIOVANNI, A. MARINO and IV. TOSQCES

SECTION4

In this section we shall deal with the I-convergence of the functions considered in the previous sections.

For sake of completeness we recall the definition of I-convergence; the general results are contained, for instance, in [6].

Definition 4.1. Let X be a metric space; if f, fh : X --;, R U (-2, +X} (h E f%) are functions, we write that f = lY(X-) lihm fh if for every u in X

(a) f(u) S lim inf fh(uh), for every (uJh converging to U;

(b) there exis:s a sequence (u&, converging to u such that f(u) = lirfh(uh).

THEOREM 4.2.Let f:Q + R U {+x} be a sequence of Q-convex functions (with $ independent on h), such that

Then

f= lY(Q-) lip fh.

with f(u) > --p and

s;pifh@h)} < +“Y

v(ah), weakly convergent to ix with a,, E d-fh(uh), we have

auE d-f(u) andf(u) = liFfh(uh).

Proof. Clearly there is a constant c such that

syp(\fh(“h)/) c c~ s~p{/~h~~sc.

For every u in R, let (Uh)h be a sequence in 52 such that

lip uh = u and lip fh(uh) = f(u).

Then, by $-convexity

fh(uh) 3fh(uh) + (mhbh - uh) - d+h, Uh,f(Uh),f(Uh), bhI)I”h - uh12.

Therefore there exists a continuous function 8 : Q X R + R, such that

fh(uh) afh(uh) + bhbh - uh) - B(Uh,f(Uh)>bh - uh?.

Going to the limit as h* +“, we get

f(u) a”(u) + (40 - u) - qbf(4)Iu - u12, VUER,

which implies that cz E a-f(u).


Now consider a sequence (LiJ,, such that

Then, as before,

lim Lib = u, h

li~fh(G) = f(u).

going to the limit, we get lim;upf,(uh) <f(u) which concludes the proof, by r-convergence.

the definition of

THEOREM 4.3. Let fh : Q 3 W u (+=} be a sequence of functions with a @-monotone subdifferential of order two (with x independent on h, cf. (1.4)).

Then, iff= r(Q-) lipfh, the same conclusion as in theorem 4.2 holds.

Proof. Let (&J, be a sequence such that li&m &, = u and lihmtnfh(fih) =f(~).

Let M be a constant such that Ifh(u,,)] s M, ) (~1 =S M, f(J,) + 1 s M. We want to apply lemma 1.12 to the functionsfh, with U, &, instead of no and u respectively. Let r, E > 0 be such that

By the definition of r-convergence and after decreasing r, we can suppose that.

and

f(u) 3fh(tifJ - ;, (l.ih - u ( s 5.

Therefore

fh(z) 2fh(fih) - .z if Iz-u/S-.

Finally we choose a positive number A such that

%a<; ~2~supcX(Z~,zZtx~,xZ)J Izi --uI

s T, /Xi 1 s M* i=l,2)<1

and Ai@ < E. Now we can apply lemma 1.12. First of all (ult + ,Icy,J E B(&, t/(k)). moreover

uh E B(u, r) and


Therefore, by lemma 1.12, we have

Uh + Aah) - Uh (? +fh(Uh) s & j(Uh + I.&) - ,‘/2

+fh(z), if 12 - UJ Cr which implies that

fh(z)3fh(uh)+(cyI,/z_uh) -&/z-,hj’, if /z-LL(/<r.

Now we proceed as in the proof of (4.2).

Definition 4.4. Let X be a metric space and fh : X-t w U { - x, fx}, be a sequence of functions. We say that the sequence Cfh)h is “asymptotically locally equicoercive” (or briefly “a.1.

equicoercive”) if for every bounded sequence (u& in X such that sypcfh4(uk)} < +=, for

some subsequence(f,,,), there exists a converging subsequence (uk,)jS

Remark 4.5. If v&, is an a.1. equicoercive sequence such that f = l-(X-) lip fh , then f is locally

coercive, that is for every bounded sequence (L+)~ such that s~p~(rr,)} < + 2, there is a

converging subsequence (Uki)j even if the functions fh are not, in general, locally coercive.

THEOREM 4.6. Let (X, d) be a complete metric space, fh : X-, W U {+TJ} be an a.1. equicoercive sequence of 1.s.c. functions and f: X + R U {+=} be a function such that

f = T(X-) IiF fh.

Then for every u in X, there exists a sequence (&,)h such that

lim uh = n, h lipfh(“h) =f(“)t limpdvfh I(uh) s Icfl(u).

Proof. By (4.1) we can suppose that 1 Vf I(u) < + =. If the conclusion fails, we can find E > 0 and a subsequence (fhk)k such that

IVf/&)[ 2 Ivfl(u) + &, ifkE N, /U - Uj d &,fhk(U) “f(u) + E.

Let (~2~)~ be a sequence such that IiF tiik = u and hFfht(&) = f(u). By definition of r-

convergence, there is r > 0, k,, E N, such that, Vk s k.

inf{fht(U)(IU-~ikI~r}>-“, B(iik, r) C B(u, s)

and

fhk(&) “f(u) + &*

Now for every p in 10, r], k a k0 we can apply theorem 1.3 to fhkiYL where the complete

metric space Yk is equal to


Therefore we can find a sequence (w& such that

/W+kI =p, fhk(Wk) Sfhk(fik) - (WI(u) + ;) I Wk - 41.

Then, by the a.1. equicoercivity, there is a subsequence (Wk,)j converging to w, which implies

that

lw-ul=o, f(w)sf(u) - (lw(4+~)lw-~l.

Therefore IVfl(u) 2 I Vfl(u) + (c/2), which is absurd.

THEOREM 4.7. Let R be an open subset of a Hilbert space H. Letfh : 52 -+ R u {+x} be an a.1. equicoercive sequence of lower semicontinuous functions and f : S2 -B R U {+x} be a function such that

f= T(S2-) lipfh.

Then Vu E D(a-f), VCY E a-f(u), there exist two sequences (uJh in D(a-fh), (Qh such that ffh E d-fh(ujJ and

lim uh = u, h

lipfh(uh) =f(u), l$n cyh = (Y.

Proof. First of all we can suppose, by the properties of T-convergence, that (Y = 0; otherwise we can consider the functions uwfh(u) - (culu - u), u -f(u) - (crlv - u).

Then by theorem 4.6 there is a sequence (uh)h such that

lihm uh = u, lim fh(vh) = f(U), h

lim lvfh 1 (uh) = 0. h

Now, by proposition 1.2, we can find two sequences (uh)h, (Qh such that &h E d-fh(&) and

1 IUh-uhls-,

h Ifh(“h) -fh(uh)l s ;T

Therefore lip uh = U, li? fh(Uh) = f(U), Iif ah = 0.

THEOREM 4.8. Let fh : R ---, R U {fx} be an a.1. equicoercive sequence of functions and f : Q + R U {+x} be a function such that

f = r(n-) hm fh. h

Then: (a) if, for every h, fh has a @-monotone subdifferential (with $J independent on h). f has a

@-monotone subdifferential; (b) if, for every h, fh is Q-convex (with $I independent on h), f is @-convex.

1438 M. DEGIOVAW, A. MARINO and M. TOSQUES

Proof. We prove (b); in an analogous way, we will get (a). By theorem 4.7 and definition 4.1, Vu in D(d-f), VaE a-f(~), Vu in R there are three sequences (u&, (a&, (v&. such that cu, E a-f(&), likm u,, = uj li? VA = 0, liFfk(uh) = f(u), liyfk(uk) = f(V) and

lim aI, = a. Since k

fk@h) 2fk(uk) + (+h - Kh) - @k, Vk9fk(Uk)rfk(Uk)j bhbh - d2

going to the limit, we get that f is +-convex.

We observe that theorem 4.8 is ingeneral false, if the sequence (fh)k iS not a.1. eqUiCOerCiVe as remark 2.5 of [12] shows.

THEOREM 4.9. Let f, fh : Q ---, R U {+=} be functions with a Q-monotone subdifferential of order two (with x independent on h, cf. (1.4)).

Suppose that f = lY(Q-) likmfh and that (fh)h is a.1. equicoercive.

Then, if (&)h is a sequence converging to a point u in D(f), such that

s@fh(“k)} < +=

and if U: [O, T] + R is a m.s. curve forfsuch that U(0) = u, we have: (a) the m.s. curves u,, forfh such that u,(O) = uh (Cf. (3.2)) are eventually defined on [0, T]

and (uh)k COnVergeS to u Uniformly On [0, T]; (b) for every c in IO, T] we have that likmfk(u,,(f) = f(u(f)), the sequence (Igrad-fk 0 uk])h

is eventually bounded in L”(t, T> and the sequence (fk 0 u,&, is eventually bounded in L”(0,

T); (c) the sequence (&)h converges to u’ in L2-E(0, T, H) for every & : 0 < & 6 1 and in

L’(O, c H) if liFfh(uh) = f(u).

Proof. (I) First of all we prove that for every ti in D(f) and for every M in Z! there exist r. r > 0 such that if (ok),, is a sequence COnVerging to u in B(fi, r) with s;p fh(uh) G M, then the m.s. curves vh, V for fh and f such that V,(O) = uh. V(0) = v are

eventually defined on [0, t] and vk conver- uniformly on [0, t]. In fact let r > 0, ho E I\, be such that if h 2 ho, 1.z - ti/<2rthen B(u,2r)cR,fk(z)~f(ii)-l

K = sup~(z,, z2,x1,x2)] ]Zi - ti] G 2r,

lx;] < If(fi)l + IMI + 1) < +x.

Therefore f (z) L f (22) - 1 if 1 z - ti 1 < 2r. If vh : [0, t h[ -+ B(ti, 2r) is the m.s. curve for fh such that v,(O) = uk, defined on its maximal interval of existence, we have, by theorem 3.4, that t/, = += or lim / V,(t) - til = 2r. On the other hand

r* rh

jVh(f) - ri] c Iv,(t) - uk( + /uh - ri] < d(t(M-f(fi) + 1)) + r’.


Therefore there exists t > 0, such that r,, > t, Vh 3 ho. Now we have, Vt,, tZ in [0, t], that

lV,(rz)- V,(QP~/(I~z -UM-fW+ 1))

which implies that (V,), is uniformly equicontinuous. Moreover by the a.1. equicoercivity we have that {Vh(r) 1 h 2 ho} has a compact closure in H, for every t in [0, t].

Therefore, by Ascoli-Arzela theorem, there exists a subsequence, which we continue to denote by V,, converging to a continuous curve V: [O, t] --, B(uo, 2r) uniformly on [0, t]. Moreover V E HL,*(O, t; H) because J; 1 V;(s) I2 ds s A4 - f(ti) + 1. Now by (3.5), we have that

Igrad-fh(Uh(t))I s Igrad-fh(I/h(s))1exp(K((t-s)

+ Wh(VhW -fh(vh(t))))) if 0 < s 6 t G t. Therefore squaring and integrating in s, on [0, t], we get

t]grad-fh(V,,(t))]2 G (M -f(C) + 1) exp(K(2r + 4(M -f(C) + 1))).

Then by theorem 4.3 we have that Vt in IO, t],

limfh(Vh(f)) =f(V(t)), hS limhinfjgrad-fh(Vh(t))/

I grad-f(V(t)) I (4.10)

which implies, also, that f0 V is nonincreasing in 10, t] and (fo V)‘(t) 2 -1 Vf] (V(f)) I V’(f) I a.e. on IO, t[. Since (Vi)h is weakly convergent to V’ in L’(0, t; H), because

we get that

f(VO*>) - fW>) 6 - I 12 I V’(s) I2 ds, if 0 C t, G I2 d t 11

which implies that

-]Vf](V(f))]V(t)] < (fo V)‘(r) G -]V(t)12, a.e. injO, t[

and

IV(t,)-V(t,)l ~~“~v’(r)lds~(121V~l(V(~)dc, if 0Gtt<r2Gt. 11 I1

Furthermore, applying Fatou’s lemma to

fh(Vh(t2)) -fh(Vh(tl) = -1” kPd-fh(Vd@)12 dS

I1 we get, by (1.15),

fOV(‘2)-fOV(f,)S - I ‘* Ivf12vw) h7 if 0 d fI 6 t2 s t. II


Therefore, by theorem 2.2 V is a m.s. curve for f in the interval IO, t]. In particular

vi(t) E - ~3_f(V(t)) a.e. in IO, t[.

Moreover V E kF2(0, t; H) and sup(fo V(f) IO s t c t} < +=. Then, by remark 3.6 fo V is nonincreasing and V is the m.s. curve for f such that V(0) = o; the sequence converges to V on [0, t] and step (I) is proved.

(II) Now let U: [0, T] + R be the m.s. curve for f such that U(0) = K and U,, : [0, Th[ 3 R be the m.s. curve for fh such that U,(O) = &,, defined on its maximal interval of existence.

Set S = sup{t E [0, T] 1 t C limhinf Th, u,, converges to u uniformly on [0, f]} we must prove

that S = T and that the supremum is a maximum. If S < T. consider the r and r associated with U(S) (E D(f)) and M = s~pcfh(uh)} according to the previous step. Lets be in [O. S[ such

that S - s < r/2 and 1 U(s) - U(S) 1 C r. Then fh( uh(S)) d M and lihm U,(s) = U(s). Therefore

applying the previous Step to oh = Uh(s), u = U(s), we contradict the definition of S. Then S = T. In a similar way we see that the supremum is a maximum and (a) is proved.

Now, since U([O, Tj) is compact we can find m in W, p > 0, h,, in IW such that

fh(z) 2 m, if d(z, U([O, T]) s o. h > h0

and

K, = supCX(~~,~~,~~t~~)/d(zi, U([O, 7.1)) GP,

l&l s WI + Id, i= 1,2}< +x.

Therefore, as in the step (I) we get

dWd-fh(%(t))12 S (M - m) exp(Ki(2T + 4(Jf - m))), vt E IO, z-1.

Then ( 1 grad-fh o Uh I) is eventually bounded in L”(t, 7): by (4.3), hF fh(uh(f)) = f(U(r)) Vt

in IO, T] and (fh 0 uh), is bounded in L”(0, 7’); therefore (b) is proved. Now, since

,:I u;,(t>l* dt=fh(h) -f#d-))

and since UA converges to U’ weakly in L*(O, T; H) we have by point (b), that Ul, converges to U in L’(O, T; H) if Ii? fh(uh) = f(u).

NowletsbesuchthatO<&51;thenifOC6<T,

jT/U;~~-~dr=~~lU~~~-idr+~T,U;I~-Fdr 0

+-o’jU;,ldl)

d

1 - (J2) T . (3"2 +

I :Cl,I*-' dt.

b

Now by the previous argument and the point (b) we get

lihmlrlU;12-rdr~~*lU’,?-‘dt b b


which implies that

lim;up ji/li;~‘-‘dr4ji~[/l12-‘dr 0 0

which concludes the proof.

THEOREM 4.11. Let f, fh : Q ---, R U {f=} be @-convex functions (with @ independent on h) such that

f= r(S2-) IiFf,.

Suppose that the sequence (fh),, is a.1. equicoercive. Then if (I.& is a sequence converging to u E D(a-f) such that

“YP cfh(rGl)I < +=t s\pIlgrad-fh(uh)II < +=

and if U: [0, T] ---, S2 is a Lipschitz continuous m.s. curve for f such that U(0) = u, we have that

(a) the Lipschitz continuous m.s. curve Uh for f,, (cf. (2.4)) such that U,(O) = u,,, are eventually defined on [0, T] and V, converges to U in H’*p(O, T; H), vp : 1 d p < fm;

(b) for every t in [0, T], li~fh(~h(f)) = f(U(f)); the sequences (Igrad-fho V,]), and

(fh 0 U,), are bounded in L”(0, T).

Proof. (I) First of al1 we prove that for every ii in D(a-f) for every M > 0, there exist r, t > 0 such that if (v&, is a sequence converging to u in B(t.i, r) and fh(uh) s M, I grad-fh(uh) I s M, then the Lipschitz m.s. curves V,, V for fh and f such that V,(O) = oh, V(0) = u are defined on [0, r], (Vh) converges to V in H1v2(0, t; Z-I) and (I V’ I) h h is bounded in L”(0, t). In fact we can

find a r > 0, ho E N such that B(Li, 2r) C R and

m=inf(fh(z)Ilz_liIa2r,h~h,}>-~.

Since (fJh are +-convex, they have also a &monotone subdifferential for a suitable 6 (independent on h). After decreasing r, we can also suppose, that

K= sup{~(z,xl,x2)~ Iz - UI =~2r, Ix1 1 Q M

+ Iml, lx2 l s M + 1) < +=.

Let (Uh)h be a sequence converging to a point u in B(ti, r), With fh(Uh) 6 M, 1 grad-fh(Uh) I c M. We have Uh E B(cZ, T). Let V,, : [0, Th[ --, B(ti, 2r) be the Lipschitz COntinUOUS m.s. curve for fh such that V,(O) =

nh, I grad-fh(vh(f)) I < M + 1 for every I in [0, Th[, defined on its maximal interval of existence

‘c;$;$m,“,;3~&

(M + 1)t < r and M eZKr < M + 1.

Then by theorem 2.5, it must be

Igrad-fi,(Vh(r))I G MeZKf.

1442 M. DEGIOV.UNI, A. MIARIU.UO and M. TOSQUES

Therefore, by (2.6), we have that t < Th and V, : [0, t]- B(ci, 2r) is (M + 1)-Lipschitz continuous.

By theorem 4.2 we get that Vr E [O, t] liFt;l 0 V,(t) = f 0 V(t) which implies that f o V is

nonincreasing on (0, t], In order to prove that V’(r) E -a-f(V(t)) a.e. in IO, t[ we go on as in the proof of theorem

(4.9). (II) Now let (uJh, U: [0, 7’j + R be as in the hypothesis and let U,, : [0, T,[ -* R be the

locally Lipschitz continuous m.s. curve for fh such that Uh(0) = ub, defined on its maximal interval of existence (cf. (2.4), (2.5)). Put S = sup{rE [0, T] j t < limhinf T,, (I/,), converges

to U in H’,2(0, t; H), 1 grad-f,, 0 U,,/hl is bounded L’(0, t)}. We state that .S = T and the supremum is a maximum. Suppose that S < T. Put ci = U(S) and let M be such thatj$,(uJ s M, / grad-f( U(t) 1 + 1 s M,

if tE[O, T]. Let r and t be associated to U as in step (I). Clearly we can suppose that S + (t/2) =S T. Let s be such that

S-i<s<andIU(r) - U(S)1 <r, Vr in [s, S]

to get an absurd, it suffices to prove that

limhinf T,, > S + (t/2),

[ y$converges to U in H’y2(0, S + (7’2); H) and (/grad-f, 0 Uhj)h is bounded in L”(0, S + t

If limhinf Th C S + (t/2), we can find a subsequence (Uhk)k such that Thk < s + t and

li?Igrad- fhk 0 Uhk(s’)l = Igrad-fo U(S’)/ f or a suitable s’ in Is, S[. Therefore by step (I),

we deduce that Tht 3s’ + t>s + t> Tht. In a similar way we prove that ( Uh),, converges to U in Ht+*(O, S + (t/2)); H) and

((grad-f,, o u,,I),, is bounded in L’(0, S + (t/2)). Therefore S = T and in a similar way we can prove that S is a maximum. Then (a) and the second part of (b) hold. The remaining part of (b) follows by theorem 4.2.

REFERENCES

1. AUBIN J. P., lyufhemarical Merhods of Game and Economic Theory, North-Holland, Amsterdam (1979). 2. BREZIS H., @jtrateurs Maximaux Monotones, Notes de Mathematics, Vol. 50, North-Holland, Amsterdam

(1973). 3. BRCNDSTED A. & ROCKAFELLAR R. T., On the subdifferentiability of convex functions, Proc. Am. math. Sec.

16, 605-611 (1965). 4. CLARKE F. H., Optimization and Non-Smooth Analysis, John Wiley (1983). 5. CRANDALL M. G. & LIGGE~ T. M., Generation of semi-groups of nonlinear transformations on general Banach

spaces, Am. J. Math. 93, 265298 (1971). 6. DE GIORCX E., Generalized limits in calculus of variations, Topics in Functional Analysir, Quademo della Scuola

Normale Superiore di Pisa (1981t81). 7. DE GIORGI E., DEGIOVANNI M., MARINO A. & TOSQUES M., Evolution equations for a class of non-linear

operators, Atti Accad. Naz. Lincei 75, l-8 (1983). 8. DE GIORGI E., DEGIOVANNI M. & TOSQUE~ M., Recenti sviluppi della r-convergenza in problemi ellittici,

parabolici ed iperbolici, MFhods of functional analysis and theory of elliptic equations Proc. Inr. Meeting Dedicated ro the Memory of Professor Carlo Miranda, Greco Ed., Liguori, Napoli (1982).


9. DE GIORGI E., MARINO A. & TOSQUES M., Problemi di evoluzione in spazi metrici e curve di massima pendenza, Arri Accad. Naz. Lincei 68, 18G-187 (1980).

10. DE GIORGI E., MARINO A. & TOSQ~ES M., Funzioni (p, q)-convesse, Arri Accad. Naz. Lincei 73, 61-t (1982). 11. DEGIOV.UWI M. MARXNO A. & TOSQ~_!ES M., General properties of (p, q)-convex functions and (p, q)-monotone

operators Ric. iMat. (Naples) 32. 285-319 (1983). 12. DEGIOVANNI M., MARINO A. & TOSQUES M., Evolution equations associated with (p. cl)-convex functions and

(p, q)-monotone operators, Ric. Mar. (Naples) 33, 81-112 (1984). _ .

13. DEGIOVANNI M.. MARINO A. & TOSQUES M.. Critical ooints and evolution eauations. in Mulfifuncrions and Integrals, Lecture Notes in Marhemarics 1091, 18-t-192, Springer, Berlin (1984). ’

14. EDELSTEIN M., On nearest points of sets in uniformly convex Banach spaces. 1. Lond. Marh. Sot. 43. 375-377 (1968).

15. EKELAND I., Nonconvex minimization problems. Bull. Am. marh. Sot. 1, W-474 (1979). 16. MARINO A. & SCOLOZZI D., Geodetiche con ostacolo, Boil. Un. Mar. Iral. 2-B. 1-31 (1983). 17. MARINO A. & SCOLOZZI D., Punti inferiormente stazionari ed eouazioni di evoluzione con vincoli unilaterali non

. convessi, Rc. Sem. Mar. Fis. Milan0 52 393-414 (1982). 18. MARINO A. & SCOLOZZI D., Autovalori dell’operatore di Laolace ed eouazioni di evoluzione in presence di

ostacolo, Problemi Differemiali e Teoria dei Punti Critici. Pitagora. Bologna (1984). 19. MARINO A. & TOSQLJES M., Curves of maximal slope for a certain class of non regular functions. Boll. Un. Mat.

Zral. 1-B. 143-170 (1982). 20. MARINO A. & TOSQUES M., Existence and properties of the curves of maximal slope (to appear). 21. PAZY A., Semigroups of non-linear contraction in Hilbert space, Problems in Non-linear Anal,vsis. Prodi ed..

C.I.M.E. Varenna, Cremonese (1971). 22. ROCKAFELLAR R. T., Generalized directional derivatives and subgradients of non convex functions, Can. 1. Mach.

*‘, 257-280 (1980).

Evolution equations with lack of convexity

Documents

Transcript of Evolution equations with lack of convexity