dirichlet problem for demi-coercive functionals

Nonlinear Analysis, Theory, Methods & Applications, Vol. 10, No. 6, pp. 603-613, 1986. 0362-546X/86 $3.00 + ,00 Printed in Great Britain. Pergamon Journals Ltd.

D I R I C H L E T P R O B L E M F O R D E M I - C O E R C I V E F U N C T I O N A L S

G. ANZELLOTrI Dipartimento di Matematica, Universitg di Trento, 38050 Povo, Trento, Italy

G. BUTrAZZO Scuola Normale Superiore, Piazza dei Cavalieri 7, 56100 Pisa, Italy

and

G. DAL MASO Istituto di Matematica, Via Mantica 3, 33100 Udine, Italy and International School for Advanced Studies, Strada

Costiera 11, 34014 Trieste, Italy

(Received in revised form 15 October 1985; received for publication 20 January 1986)

INTRODUCTION

WE CONSIDER integral functionals of the type

F(u) = fo f(Du),

where f : ~N___~ [0, + ~] is a convex function with the following property: there exist a > 0, b >i 0, y ~ [R N such that

f (z ) >1 alzl - <r , z> - b ( 0 . 1 )

for all z ~ ~N. Integrands satisfying (0.1), and the corresponding functionals, will be called demi-coercive. Clearly, a demi-coercive integrand f may be zero on large subsets of ~ g and the functional F(u) will not control, in general, any integral norm of Du, though it is clear that the L 1 norm of Du is controlled by the functional with the addition of the integral fa<Y, Du>. The nice fact is that, in many problems, one obtains some control of this integral from the boundary conditions, and this gives the coerciveness of the Dirichlet problem in spaces of BV functions.

Examples of demi-coercive functionals arise naturally for instance in studying the equilibrium of elastic structures with unilateral constraints on the stress (see [1, 2, 4, 6, 9, 15]).

Many equivalent conditions to (0.1) are given in theorem 2.4 and a few examples ' a re discussed. The Dirichlet problem for demi-coercive functionals is then studied in Section 3. In Section 1 we collect several results about convex functions of measures, that we need in Sections 2 and 3.

1. CONVEX FUNCTIONS OF MEASURES

In this section we collect a few propert ies of the convex functions of measures. After the definition, given in (1.3), of the t ransformed measure f/z, we use lemmas 1.1 and 1.2 to extend the lower semicontinuity result by Goffman and Serrin (see [12]) to the general case of convex functions f with values in [0, + ~]. We conclude this section by considering the case/a = Du

603

604 G. ANZELLOTTI et al.

with u E BV(f~ ). For other work on functions of measures compare [3, 7] and their references. Let N be a positive integer. By a convex function on NN we mean a convex function defined

on NN with values on N tA {+ oo}. For every convex functional f we denote by dora f the effective domain of f defined by

d o m f = {z E ~N:f(z) < + o~}.

We say that a convex function f is proper if domf # : ~ . For every proper convex function f we denote by f= the recession function of f, defined for every z E ~N by

f=(z) = sup{f(z + w) - f(w): w ~ domf}. (1.1)

If f is lower semicontinuous, it is well known (see [14, theorem 8.5]) that f~ is lower semicontinuous and that

f(w + tz) - f ( w ) t

(1.2) f(w + tZ) - f ( w ) f(w + tz)

= lira = lim t ~ + ~ t t -~+~ t

for every z E ~n and every w E dotal . Let n be a positive integer and let G be a bounded open subset of R n. By an ~S-valued Borel measure on G we mean a countably additive ~S-valued function

defined on the o-field ~ ( G ) of all Borel subsets of G. For every ~S-valued Borel measure # let

#(E) = f~ #°(x) dx + #S(E) (E e ~ (G) )

be the Lebesgue decomposition of #, where the function #a : G ~ ~N is Borel measurable and Lebesgue integrable, and the ~N-valued Borel measure #* is singular with respect to the Lebesgue measure.

For every nonnegative proper convex function on ~N and for every ~N-valued Borel measure # on G, we define the nonnegative measure f# : ~(G)--* [0, + ~] by

(f#)(E) = f f ff \dl#sl{d#~ (x))d[#S[(x), (1.3)

where ]#*) is the total variation of/,~ and

d# * d>q

i s the Radon-Nikodym derivative of #* with respect to 1#*[. In order to study the properties of the measure f# we need the following lemmas.

LEMMA 1.1. Let f be a lower semicontinuous proper convex function on ~N and let ( f i ) i ~ l be a family of lower semicontinuous proper convex functions on ~N such that f = sup fi. Then fo¢= sup f?/.

iE l

Dirichlet problem for demi-coercive functionals 605

Proof. Since f,- ~<ffor every i E I, we obtain from (1.2) that f/~ ~<f® for every i ~ I, hence supfT ~<f~. Let us prove the opposite inequality. For every z E R N, w ~ d o m f we have i ~ l

f ( z + W) -- f ( w ) = sup f i ( z + W) -- supfi(a 0 < sup [ f i ( z + w) -- f i (w)] <~ supf? (z ) , i ~ l iE I i ~ l i E l

therefore by definition (1.1)

f ~ ( z ) = sup{f(z + w) - f ( w ) : w E domf} ~< supfT(z). • iE 1

LEMMA 1.2. For every nonnegative lower semicontinuous proper convex function f there exists an increasing sequence (fk) of finite-valued nonnegative convex functions such that f = sup fk.

kEl~

Proof. It is enough to take (see [5, proposition 2.11])

fk(z) =inf{f(w) + k l z - wl2:wEf f~N}. •

We are now in a position to prove the following lower semicontinuity theorem.

THEOREM 1.3. Let f be a nonnegative lower semicontinuous proper convex function o n ~ N ,

let G be a bounded open subset of Nn, let (/~h) be a (generalized) sequence of NN-valued Borel measures on G, and let ~ be an NN-valued Borel measure on G. If

lim f wd~,~ = f ~;d~, h (1.4)

for every continuous function q~ : G ~ N with compact support in G, then

(f/~) (G) ~< lim inf (f/~h) (G). h

Proof. By lemma 1.2 there exists an increasing sequence of finite-valued nonnegative convex functions (fk) such that f = sup fk. The results of Sections 2 and 3 of [12] imply that

kEN

(fk~) (G) ~< lim inf (f~#h) (G) ~< lim inf (fuh) (G) (1.5) h h

for every k E N. By lemma 1.1 and by the monotone convergence theorem we have

(f#) (G) = sup ( f , # ) (G). (1.6) kern

The thesis follows now from (1.5) and (1.6). •

R e m a r k 1.4. A careful inspection of the proof of theorem 3 of [12] shows that theorem 1.3 still holds if we assume that (1.4) is satisfied only for every q~ E C~(G) .

We denote by BV(G) the space of all functions u in L I ( G ) whose first order distribution d e r i v a t i v e s O l U . . . . . Dnu are bounded Radon measures on G (see [11, 13]). We denote by Du the ~n-valued Borel measure on G defined by D u = (DI u, . . . , Dnu). For every nonnegative

606 G. ANZELLO'Iq'I et al.

proper convex function on N~ and for every Borel set E C_ G we set

ef(Ou) = (fDu) (E).

We can prove now the following lower semicontinuity theorem for the functional fef(Du).

THEOREM 1.5. Let f be a nonnegative lower semicontinuous proper convex function on ~ , let G be a bounded open subset of ~ , let F b e a closed subset of En, and let (uh) be a (generalized) sequence in BV(G) which converges to a function u of BV(G) in the weak topology of distributions, in other words assume that

lim fG cpUh dX = fG (1.7)

for every cp E C'd(G). If Uh = U a.e. on G - F for every h, then

fc ae f(Du) <-liminf ~F f(DUh)" (1.8)

Proof. Let z ~ d o m f and let #h and # be the ~n-valued Borel measures on G defined by

~h(E) = (OUh)(E N F) + z m ( E - F) (E E ~(G))

u(E) = (Du) (E fq F) + zm(E - F) (E E ~(G))

where m denotes the Lebesgue measure on G. Let us prove that

limhJGf q0d~h = fG qod/x (1.9)

for every q) ~ C~(G). Since G - F is open and Uh = U a.e. on g - F, we have (DUh)(E) = (Du) (E) for every Borel set E C_ G - F, therefore

(1.1o)

for every q0 ~ CF(G). By (1.7) we have

limfaq~d(Duh)=-limf~uhDq~dx=-fauDqgdx=fcqgd(DU)h h

for every q~ ~ C~(G), therefore (1.9) follows f rom (1.10). By theorem 1.3 and remark 1.4 we have

(f/x) (G) <~ liminf (f#h) (G) (1.11) h


since

( f#h)(G) = f cnv f (DUh) + f ( z )m(G - F)

(f#) (G) = f c f (Du) + f ( z )m(G - F) f~F

a n d f ( z ) m ( G - F) < + 0% we obtain (1.8) from (1.11) and (1.12).

(1.12)

2. DEMI-COERCIVE CONVEX FUNCTIONS

In this section we study the properties and give a few examples of the convex functions which fulfil the following definition.

Definition 2.1. We say that a function f : ~N___~ ~ is demi-coercive if there exist two real numbers a > 0, b ~> 0 and a vector y E ~N such that

alz[ <~f(z) + (y, z) + b (2.1)

for every z ~ ~N, where (. , . ) denotes the usual scalar product in ~N.

Examples 2.2. The function f : ~ ~ ~ defined asf(z) = z + = max{z, 0} is probably the simplest nontrivial example of demi-coercive function. In fact (2.1) is satisfied taking for instance a = 1/2, b = 0, ~ = - 1/2. Other simple examples of demi-coercive functions on ~ are the functions f (z) = (z+) 2 and f (z) = (z - 1) +.

Example 2.3. The function f : ~2 ~ ~, defined by

f ( z ) = 2 + (z2)

for every z = (zl, z2) E ~2, is demi-coercive. In fact (2.1) is satisfied taking for instance a = 1, b = 1, ~ = ( - 1 , 0). Integrands of this type appear in [1].

For every lower semicontinuous proper convex function f on ~N we denote by f* the conjugate of f , defined for every z* E ~N by

f*(z* ) = sup{(z*, z) - f (z) : z E ~N}.

It is well known that f* is a lower semicontinuous proper convex function and that f** - - f (see [14, Section 12]).

In the following theorem, given a convex function f , we list some conditions on f , p , f* which are equivalent to the demi-coerciveness of f.

THEOREM 2.4. Let f be a nonnegative lower semicontinuous proper convex function on ~N and let z0 E domf . The following conditions are equivalent:

(a) f is demi-coercive; (b ) the re exist two real constants a and b, with a > 0 , such that f ( zo+ z ) +

f(zo - z) >I alz[ - b for every z E ~N; (c) for every z ~ •N _ {0} we have p ( z ) + f ~ ( - z ) > 0;

608 G. ANZELLOTTI etal.

(d) the cone {z E W’:f”(z) = 0} contains no straight lines; (e) the normal cone to domf* at 0 contains no straight lines; (f) the set {z E RN: 2f(z,) = f(z,, + z) + f(zo - 2)) is bounded; (g) for every z E R N - (0) there exists t > 0 such that

2f(z,) <f(zo + tz) +f(zcl - tz);

(h) there are no straight lines containing z. along which f is constant; (i) there are no straight lines along which f is finite and constant; (j) the set domf* has a nonempty interior; (k) there exists z* E RN such that J”(z)- (z*, z) > 0 for every z E RN - (0).

Proof. (a) j (b): it is obvious. (b) + (c): it is obvious. (c) G (d): it is obvious, since f” 2 0. (d) G (e): by theorem 14.2 of [14] the normal cone to domf* at 0 is equal to the cone

{z E W?f”(Z) = 0). (c)+(f):letg:lfP + [0, + w] be the convex function defined by g(z) = f(zo + z) + f(zo - z).

If we assume (c), then we have g”(z) > 0 for every z E RN - (0). By theorems 8.4 and 8.7 of [ 141 we obtain that the set {z E RN : g(z) s g(O)} 1s b ounded. Since f is convex, this set coincides with the set defined in (f).

(f) 3 (g): it is obvious. (g) 3 (h): it is obvious. (h) + (i): it follows from corollary 8.6.1 of [14]. (i) + (j): since f 2 0, it follows from corollary 13.4.2 of [14]. (j) ($ (k): see corollary 13.3.4 (c) of [14]. (j) + (a): since int (domf”) # 0 and f* is continuous on int(domf”) (see [14, theorem

lO.l]), there exist z” E RN, E > 0, b E R such thatf*(w*) G b whenever lw* - z*l G E. Let g: RN+ R be the convex function defined by g(w*) = b if (w* - z*I < E and g(w*) = + x if Iw* - z*/ > E. Since f s g, we have for every z E RN

f(z) = f**(z) 2 g*(z) = EIZ( + (z*, z) - b,

and (a) is proved. n

Example 2.5. Let K be a closed convex subset of RN and let f(z) = Idist(z, K)\” ((~3 1). Then f is demi-coercive if and only if K contains no straight lines (see theorem 2.4(h)). Functions of this type have been considered in [2, 91.

Example 2.6. Let K be a closed convex subset of RN with 0 E K and let A : RN+ RN be a linear symmetric positive definite operator. Consider the convex function TV : RN+ R defined

by

V(z) = { ‘,“2 z, If:,,:,:;,

Then the function f = I/J* is demi-coercive if and only if K has a nonempty interior (see theorem 2.4(j)). Integrands of this type arise in the study of the static equilibrium for elastic structures with unilateral constraints on the stress (see [4]).


The following theorem provides a coerciveness property of the convex integral functionals with a demi-coercive integrand.

THEOREM 2.7. Let f be a demi-coercive nonnegative lower semicontinuous proper convex function on N" and let g2 be a bounded open subset of N" with a Lipschitz boundary. Then there exist three numbers a > 0, c~ ~> 0, c;/> 0, depending only on f, such that for every u BV( )

a fca IDu]<- fca f (Du )+c , feca lul dH"- l+c2m( ~2 ), (2.2)

where/_p-1 denotes the (n - 1)-dimensional Hausdorff measure and m denotes the n-dimensional Lebesgue measure.

Proof. As f is demi-coercive, there exist a > 0, b ~> 0, y E ~n such that

alzl <~ f(z) + (y, z) + b (2.3)

for every z ~ R", hence

alz I ~ f~(z) + (y, z) (2.4)

for every z E ~". By (1.3) it follows that

a fca lDul fcaf(Du) + fca (r, Du) + bm(fa )

for every u ~ BV(f2). Integrating by parts the second term of the right-hand side (see [11, 13]) we get

afca]Dul<~fcaf(Du)+feca u ( Y ' v ) d H " - l + b m ( Q ) ' (2.5)

where v(x) is the outward unit normal to 092 at x, and (2.2) follows. •

3. DIRICHLET PROBLEM

In this section we shall consider a generalized Dirichlet problem for demi-coercive functionals in a formulation that is similar to the one used in the nonparametric Plateau's problem (see [10, 11, 13]). After giving an existence theorem, we study a set of boundary points where the datum is certainly attained.

We assume that f2 is a bounded open subset of ~" with a Lipschitz boundary; we denote by v(x) the outward unit normal to a ~ at x and by H "-~ the ( n - 1)-dimensional Hausdorff measure. Let f be a nonnegative lower semicontinuous proper convex function. For every q~ E L l ( a ~ ) we consider the functional

F (u) = fof(Du) + foa f~((qg - u)v)dHn-1

defined for all u ~ BV(g2). Let g be a function in L"(K2). According to the terminology used in the case of functionals

610 G. ANZELLOTYI et al.

with linear growth (see [10, 11, 13]) we shall refer to the minimum problem

min [F~(u) + ;agudx ] (3.1) ueBV(~)

as the (generalized) Dirichlet problem for the functional

faf(Du) + fagudx

with boundary condition q). We begin the study of problem (3.1) by proving the following lower semicontinuity theorem

for the functional F~.

THEOREM 3.1. Let (Uh) be a (generalized) sequence in BV(~2) which converges to a function u of BV(ff2) in the weak topology of L l ( f l ) , and let tp ~ LI(aQ) . Then

F~(u) <- liminf F~(Uh). h

Proof. Let ~ ' be a bounded open subset of ~" such that (2 C_ ff~' and let w be a function in Wl'l(~2 ') such that q0 is the trace of w on dQ (see [8]).

Define now

Vh={;h on if2,

on ff~' - ffl,

o n fl , U = I u ( w on g2' - f~.

It is easy to see that v belongs to BV(ffg) and that (Oh) is a (generalized) sequence in BV(f~') which converges to v in the weak topology of L l ( fg ) .

Since (see (11, 13])

we can write

Ol) la ~ = (q) - u ) l J n n-1 I ~ ,

DUhio~ = (q9 -- Uh)VH "-lIon,

F¢(u) = f f(Do),

F¢(Uh) = f f(DVh).

By applying theorem 1.5 with G = ~ ' and F = ~ we obtain

F~(u) = faf(Dv) <- liminf fa f(Dvh) = liminf

We are now in a position to prove the existence theorem for the Dirichlet problem.


THEOREM 3.2. Let f be a demi-coercive nonnegative lower semicontinuous proper convex function on N" and let f2 be a bounded open subset of N~ with a Lipschitz boundary. Then there exists a number Co, depending only on f and f2, such that the Dirichlet problem

man [ f a f ( D u ) + fa f~((cp-u)v)dHn-l + fogud ] u ~ BV(f2 ) f2

admits a solution for any boundary datum q~ ~ LI(0f2) and for any function g E L'(f2) with IlgllLo(a> < co.

Proof. We use the direct method of the calculus of variations. First we shall get a coerciveness estimate. We may assume that f satisfies (2.3) and hence (2.4) and (2.5).

For all u E BV(f2) we have

foa f~((q) - u)u)dHn-l + fo~ (Y'(qg - u)v)dHn-1

l>a l l u l d H n-' - a f IqoldH n-1 (3.2) . / ~0 f2

By adding (3.2) and (2.5) we obtain

faf(Du)+ fo F((ce-u)y)dnn-~ >~a{folDul+ f~ luldHn-~ }-c,(c;,f~,f) where

cl(~,f2,f) = fan cP(7' v) dH"-l + bm(~) + a ~oa IqgldHn-1

and it follows that

F~(u) + ~a gu dx >~ a {fa IDu]+ fort 'ul dHn-1 } - cl(q°' f2,f)

-- }lgll Ln(f2 ) II /~ll Ln/n- l(f] ) •

On the other hand, there exists a c o n s t a n t C2(~"2 ) 2> 0 such that (see [11, 13])

"u}lc./.-,(n> <<- c2(f2 ) {fQ IDul + for2 ,u] dH n-1 }

for all u ~ BV(f2). By (3.3) and (3.4) we obtain that, if

a

IIg{{L,(m < Co(f~, f) = c2(f2----)' then for all u E BV( f2 ) one has

F~(u) + fa gu dx >~ c3 {fn ,Du, + faf~ I u[ dHn-1} - C1 (li0 , ff2,f)

(3.3)

(3.4)

(3.5)

612 G. ANZELLOTTI et al.

where

N o w we set

c 3 = a -- cz(n)llgllL.( > > 0.

] If oc = + ~ the t heo re m is trivial. If oc < + ~, let (uh) be a minimizing sequence, i.e. a sequence such that

o~= lim [F~(Uh) + f gUh

By (3.4) and (3.5) the sequence (Uh) is bounded in L n/n- z(n ) and in BV(92 ), hence there exists a subsequence which converges weakly in L n/~- 1(~) to an e lement u 0 of B V ( n ). By the lower semicont inui ty t heo rem 3.1 we have

o~<~F~(uo)+ f guodx<~limh F~(Uh)+ f gUhdX]=Cr,

SO

F~(uo)+ f guodx=u~Bv(a)inf [F¢(u)+ f gudx]. I

If u is a solution of p rob lem (3.1), it is not t rue in general that u = go on 092. In fact, our functionals include in part icular the nonparamet r i c area functional , and in that case one may have u g: go everywhere on a n (see [11, 13]). On the other hand, our functionals may have super l inear growth in certain directions, and this translates into some rigidity of the boundary condit ion. In fact, if there exists w E B v ( n ) such that Fq:(w) < + ~, then the solution u of (3.1) satisfies in part icular

o F ((go -- u ) u ) d E n - l < + oc

and this implies that

(go(x) - u(x))v(x) ~ K r for H n-l - a.a. x E 092,

where Kf = {z E R" : f=(z) < + o~} is the convex cone where f ~ is finite, i.e. A n - Ksis the cone of all directions of superl inear growth of f .

I t follows that g o ( x ) - u ( x ) ~ < 0 / _ / , - l _ a . e . on the set { x ~ 0 9 2 : u ( x ) ~ K f } and go (x) - u(x) I> 0 /_/~-1 _ a.e. on the set {x E 0 n : - v(x) qi_ Kf}. Therefore , g0(x) = u(x) H n-1 - a.e. on the set {x E 092 : u(x) qF_ KfU ( -Kf )} .

For instance, if C is a closed convex cone in Nn and f(z) = (dist(z, C)) ~, then the da tum go is certainly at ta ined ( H n-1 a.e.) on the set

{x~ on: ~(x) ~ c u (-Q}.

Our results can be extended, with the same proofs , to the case of vector valued functions u ~ BV(Q) '~ , where m ~ > 1 is any integer. In this case, identifying ~m~ with the space o f all m x n matrices, the Jacobian matrix Du of a function u ~ B V ( n ) m is an Rm"-valued bounded R a d o n measure on 92. For every nonnegat ive lower semicont inuous proper convex funct ion


o f f on ~mn and every u C BV(Q) m we set fa f (Du)= (fDu)(f2). Given q0 ~ L l (3 f f2 ) m and g E Ln(f2)m we may consider the Dirichlet problem

min I r a f ( D u ) + fo f f ( ( q 2 - u ) ® ~,)dHn-1 + f gudx], (3.6) uEBV(g?) 'n f~

where, for every a E N '~ and b E ~n, we denote by a ® b the m x n matrix whose components are (a ® b)q = aib j. I f f is demi-coercive, the existence of a solution to problem (3.6) can be proved exactly as in the scalar case.

When m = n, we can also consider the space BD(f2) of all functions in Ll(g2)" whose strain tensor e(u) is an N"2-valued bounded Radon measure on ~ (see [16]). We recall that

(E(U))q = ½(Oju i + Diuj)

and that we identify ~.2 with'the space of all n × n matrices. For every nonnegative lower semicontinuous proper convex function f on R ~2 and every u ~ BD(f~) we set faf(e(u)) =

Given q) E L l ( 0 ~ ) n and g E L"(ff~)% we may consider the Dirichlet problem

rain [ f a f ( e ( u ) ) + foaf ((q)-u)Ov)dH"-l+ foguax] (3.7) uEBD(f~)

where, for every a, b ~ N~, we denote by a O b the n x n matrix whose components are (a @ b)ij = ½(aibj + ajbi). If f is demi-coercive, the existence of a solution to problem (3.7) can be proved exactly as in the scalar case.

R E F E R E N C E S

1. ANZELLO'I'I'I G., On the existence of the rates of stress and displacement for Prandtl-Reuss plasticity, Q. appl. Math. 41,181-208 (1983).

2. ANZELLOTTI O., A class of convex non coercive functionals and masonry-like materials, Ann. Inst. H. PoincarG 2,261-307 (1985).

3. ANZELLOTrI G., The Euler equation for functionals with linear growth, Trans. Am. math. Soc. 290, 483-501 (1985).

4. ANZELLOTTI G., Elasticity with unilateral constraints on the stress, preprint Dip. Mat. Univ. Trento (1984). 5. BREZIS H., Operateurs Maximaux Monotones, North Holland, Amsterdam (1973). 6. COMO M. ~ GRIMALDI A., A unilateral model for the limit analysis of masonry walls, Meeting on "Unilateral

Problems in Structural Analysis" Ravello, 22-24 September 1983). 7. DEMENGEL F. & TEMAM R., Convex functions of a measure and applications, Indiana Univ. Math. J. 33, 673

(1986). 8. GAGLIARDO E., Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni d i n variabili,

Rc. Semin. Mat. Univ. Padova 27, 284-305 (1957). 9. GIAQUINTA M. & OIUSTI E., Researches on the statics of masonry like structures Archs ration. Mech. Analysis

(1985). 10. GIAQUINTA M., MODICA G. & SOUCEK J., Functionals with linear growth in the calculus of variations, Comment.

Math. Univ. Carolinae 20, 143-172 (1979). 11. GIUSTI E., Minimal Surfaces and Functions of JSounded Variation, Birkh/iuser, Basel (1984). 12. GOFFMAN G. ~ SERRIN J., Sublinear functions of measures and variational integrals, Duke Math. J. 31,159-178

(1964). 13. MASSARI U. & MIRANDA M., Minimal Surfaces of Codimension One. Notas de Matematica, North Holland,

Amsterdam (1984). 14. ROCRArELLAR R. T., Convex Analysis. Princeton University Press, Princeton (1972). 15, ROMANO G. & ROMANO M., Elastostatics of structures with unilateral conditions on stress and displacement

fields, Meeting on "Unilateral Problems in Structural Analysis", Ravello (22-24 September 1983). 16. STRANG G. & TEMAM R., Functions of bounded deformation, Archs ration. Mech. Analysis 75, 7-21 (1980).

dirichlet problem for demi-coercive functionals

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