Stuart, C. A.

Toland, J. F.

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A VARIATIONAL METHOD FOR BOUNDARY VALUEPROBLEMS WITH DISCONTINUOUS NONLINEARITIESC. A. STUART AND J. F. TOLAND1. IntroductionLet Q be a bounded domain in U" which satisfies the cone property, and letp e L2(Q). We study the boundary value problem- Au(x) = f(u{x))+p(x), xeQ,(1.1)u{x) = o, x G anwhen / is locally a function of bounded variation on U. We do not suppose that / iscontinuous, but only that f(t) lies between f(t + ) and f(t — ) for all t e U.Experience teaches that, in general, solutions of (1.1) need not exist even under themost restrictive hypotheses (see [4], [6] and [7]) and that it is wise to consider insteadmulti-valued versions of (1.1), for which an adequate existence theory obtains (see [7]).The main results of this paper are contained in the existence theorems of Section 3for multi-valued versions of (1.1), and they are obtained using a variational approach inHilbert space. The situation under consideration is more or less the same as that of [7]but we find that our variational arguments give extra information which is unavailableby other means. We shall return to this later, but first we need to discuss what is meantby a variational formulation when / is discontinuous.The functional normally associated with (1.1) is= {^u(x)\2-F{u(x))-p(x)u(x)}dx,where F is a primitive of/. But in our case, when / may be discontinuous, # fails to beFrechet differentiate, and the usual notion of critical point is no longer available.To overcome this difficulty we decompose / as g — h, the difference of two non-decreasing functions on U (and this can be done because we suppose that / is locally ofbounded variation). Then we define a function J f — ^  on a suitable Banach space V, byf f34?(u)-g{u) = {i|Vw(x)|2 + H{u(x))}dx - {G(u{x)) + p{x)u(x)}dx .J JReceived 5 February, 1979.[J. LONDON MATH. SOC. (2), 21 (1980), 319-328]320 C. A. STUART AND J. F. TOLANDHere G and H are primitives of g and h respectively and so both # and tf are convexfunctionals on V. For such a functional 3^ — ^  there is a definition of critical point [9]which says that u e V is a critical point of 3tf — $ if and only ifn dJf (M) ^ 0 .Here 5Jf (u) and d^(u) denote respectively the set of sub-differentials of Jf and ^ at u(for further details and references, see [9]).It is clear that, in general, the fact that u is a critical point of Jf — <§ depends on «#"and on ^, rather than on the functional / whose value at each point of u e V is given byl{u) = 3V(U) — ${U). (For example, for a C°°-function / of one variable it is possible tofind 3/e and # , both convex functions of U, such that I(u) = 3^{u)-^{u) for all ueU,dtf{u) n d${u) ± 0 yet /'(u) f 0.)In Section 2 we will show that, provided the decomposition of/ is restricted to anallowable class, every critical point of tf -<& in L2(Q) is in fact in Wl0'2{Q.) n W2'2{Q)and satisfies (1.1) in the following multi-valued sense:- &u{x)-p(x) G/(U(X)) for almost all x e Q (1.2)wheref [/('-),/(' + )] if/(t-K/(*+),f(t) =Indeed because of the Sobolev embedding theorems we shall see that, in fewer than 4dimensions, there is a one-to-one correspondence between solutions of (1.2) and criticalpoints of #C — <§, whatever admissible decomposition of / we choose. In higherdimensions there exist solutions of (1.2) which are only critical points of Jf — ^  for somedecompositions of / and not for others. An example in Section 4 makes this claimexplicit.In Section 3 we consider the question of the existence of critical points of ^  — <$. Wewill suppose that / satisfies the one-sided growth condition used in [7], namelyl i m s u p — < A (1.3)where A is the first eigenvalue for the linear problem— Au(x) = Xu{x) on Q ,u{x) = 0 on dO..Under these hypotheses we show that, for an arbitrary admissible decomposition of/ there exists a minimiser u of Jf — & in L°°(Q). It follows at once that d^(u) # 0 andfrom Theorem 2.3 of [9] that u is a critical point of #£ — <&. Thus u satisfies (1.2).But Theorem 2.3 of [9] says more than that u is a critical point of Jf — ^ ; it says thatd&(u) ^ dJ^(u) (a much stronger claim than that djf (u) n d<&(u) # 0 ) . From this weshow that the set/(u(x)-)} (1.4)A VARIATIONAL METHOD FOR BOUNDARY VALUE PROBLEMS 3 2 1has zero measure. Thus u satisfies (1.2), but even more is true;u e W^-2(Q) n W2'2{Q) n L°°(fi) and— AM(JC) — p{x) e/(u(x)) almost everywhere on Q (1.5)where{ {fit)} i f /( t + ) ^ / ( t - ) ,7(0 ={ fit) if/(t-)>/(t+).In the last section we give two examples. In one we see how a solution of (1.2) may ormay not be a critical point of $P — <$ depending on the decomposition of/ chosen.In the other we show that (1.4) does not hold for all solutions of (1.5) (or indeed (1.1)).It is however always a property of a minimiser of #f — <§ if (1.3) holds. Thus the existenceof examples such as the one with which we finish often means that a non-uniquenessresult holds for solutions of (1.5). This observation is further elaborated in aforthcoming paper [8].Finally it is worth remarking that the classical variational approach to semi-linearelliptic problems may fail, not because / is discontinuous, but because we lack suitablegrowth behaviour of / at infinity. Our method of attack proves equally useful then.2. The variational formulationLet B denote the set of all functions / : U -> U which have bounded variation oncompact intervals. If fe B then for all teU, f(t±) = lim f(t±s) exists. Define the£-•0 +upward-jump-set (U(f)) and the downward-jump-set (D(/)) to be= {teU:f(t-)<f(t + )},= {teU:f(t + )<f(t-)}.Then for / e £, / is continuous except on the set D(f) u U(f). Put([/( '-) , /(* + )] UteM\D(f),fit) =(U(t + ), fit-)-]andI'{/(')} if teR\D(/),7(0 = •), fit-)-] ifteD(f).Thus / is a multi-valued mapping on U which fills in all the jumps of/, whereas / onlyfills in the downward jumps of / . LetBN = {fe B :/(0 ef(t) for all t e U}.The class BN was introduced in [7] and its usefulness for the problem (1.1) depends onthe following result which involves a slight extension of Lemma 2.2 of [7].322 C. A. STUART AND J. F. TOLANDLEMMA 2.1. Let f e BN. Then there exist two increasing functions g : IR -> IR andh : U -> IR which have the following properties:(i) f(t) = g(t)-h(t)foralltsU;(ii) g is continuous on U\U(f);(iii) h is continuous on U\D(f);(iv) /(0 = g(t)-fi(t) andj{t) = g(t)-fi(t)for all t e U.Remark. Clearly a decomposition with these properties is only unique up to theaddition of continuous increasing functions to g and h. We shall call a decomposition of/ , as the difference of two functions g — h, admissible if and only if g and h satisfy all theproperties of Lemma 2.1, and g(0) = 0.Hence we shall only consider / e B N , and those g,h, whose difference is anadmissible decomposition of / . LetG(t) = g(s)ds and H{t) = h{s)ds for all t e U.j Jo oThen both G and H are convex, continuous functions on IR. For each u e L2(Q) we shalldefine extended-real-valued functionals ^ and Jf as follows:{j;\Vu{x)\2 + H(u{x))}dx if u e W ^2(fi), H{u) e Ll{O),n+ 00 otherwise;{G(u{x)) + p{x)u{x)}dx if G{u) e. +oo otherwise.LEMMA 2.2. Both <& and 2tf: L2(Q) -> U u { + 00} are convex and lower semi-continuous.(i) An element v e L2(fi) belongs to the subdijferential of & at u if and only ifv{x) e g(u{x))+p{x)for almost all x e Q.(ii) An element v e L2(Q) belongs to the subdijferential of 3tf at u if and only ifue Wl2{0.) n W2'2(Q), and v{x)e -Au(x) + fi(u(x))for almost all xeQ.(iii) For each u e L2(Q), the set d^{u) n dJf{u) is either empty, or contains a singleelement.Proof. The convexity and lower semi-continuity results and parts (i) and (ii) arestandard (see, for example [1], or [2; Proposition 2.8 and Proposition 3.8]).A VARIATIONAL METHOD FOR BOUNDARY VALUE PROBLEMS 3 2 3(iii) Suppose that v,we d${u) n dJ^{u). Thenu e VFJ-2(Q) n W2>2{O),v(x),w{x)eg(u{x)) + p{x),v(x), w(x) e — Au(x) + fi(u(xj),for all x e E, where E ^ Cl and meas. (E) = meas. (ft).If x e £, then either u(x) G t/(/), in which case h is continuous at u(x), and thisimplies thatv(x) = w(x) = -or u{x) e R \ l / ( / ) and g is continuous at w(x), and this implies thatv{x) = w(x) = (^wThus v{x) = w(x) for all x e E and so almost everywhere in Q. Hence v = w in L2(Q).Now according to [9] a point u e L2(Q) is called a critical point for ^ f — ^  if and onlyifn dtf{u) ± 0.Because of our definition of admissible decomposition of/ we have the following result.THEOREM 2.3. Let f G BN and let g — h be an admissible decomposition of f. Anelement u e L2(Q) is a critical point of 3^ — ^  if and only if(i) ue^2(a)nr-2P;(ii) [ 1 - X ( « ) M H ) G L 2 ( Q ) ;(iii) x(u)h(u)eL2(Cl);and(iv) — A«(x)—p(x) e/(«(x)) for almost all xeQ.Here x{t) = \ifts U(f), and x(t) = 0 otherwise.Remark. One might feel that the critical points oiJtif — y should be independent ofthe decomposition / = g — h. That this is not the case is clear from an example inSection 4. However it follows from the above characterisation that, if u is a critical pointof J f — ^  and u e L°°(fi), then it is a critical point for every admissible decomposition of/ . This fact is exploited in the analysis in Section 3.Proof. If u e L2(Q) is a critical point of 3tf -<§, then u e Wfr2#l) n W2'2(Q) andthere exists v G L2(Q) such thatv(x)eg{u(x))+p(x),andU(X)G -Au{x)324 C. A. STUART AND J. F. TOLANDfor all x e E ^ Q with meas. (E) = meas. (Q). Thus[ z ( ( ) ) ] ( )andX(u(x))v(x) = X(u(xfor all xe E. Thus (i), (ii) and (iii) are established. Moreover we have also thatx{u(x))v(x)GX{u(x))lg(u(x)) + p(x)]and[ 1 ( M ) ] () e [1 - Z(uTherefore when xeE and /(w(x)) = 1,v{x) = - Au{x) + h(u{x)) e g{u{x)) + p{x).When xeE and x{u{x)) = 0,u(x) = g(u{x)) + p{x)e -Au{x) + fi(u{x)).Thus for all x e £-Au(x)-P(x)ef(u(x))and (iv) is established. Conversely suppose that (i), (ii), (iii), (iv) hold. Then-Au(x)-p(x) e ll-x{u(xmig{u(x))-h{u(x))-]+X{u(x))lg{u(x))-h(u(x))-]almost everywhere on Q. Letv(x) = [1 - z(u(x))] [g{u(x)) + p(x)-] + x{u(x))l - Au(x) + /i(u(x))] .Thenv e L2(Q),v{x)eg{u(x)) + p(x), andu(x) e — Au(x) 4- ^ (w(x)) for almost all xeQ.Thus v e d<&(u) n 5^(w), by Lemma 2.2, and the proof of the theorem is complete.COROLLARY 2.4. / / / e BN /ias an admissible decomposition which is such thatg{u) s I3{Q) for all ueWl0>2(Q) n W2'2(Q), then u is a critical point of &-<$ if andonly ifue W '^2(fi) n W2'2(Q), and -Au{x)-p{x) ef{u(x)) a.e. on Q.Proof Suppose that u e H^-2(fi) n W22(Q), and that - Au(x)-p(x) ef(u(x)). Itthen follows from the hypotheses that [1 — ^ («)]g(i/) e L2(Q). Thus, according toTheorem 2.3, u is a critical point of Jf — ^  if we can show that x(u)h(u) e l3(Q). It sufficesto show that if v(x) e g(u(xj) for almost all x e Q, then v e L2(Q).A VARIATIONAL METHOD FOR BOUNDARY VALUE PROBLEMS 325But g(0) = 0 and g is increasing. Therefore \v(x)\ < K + \g(2u(x))\ for almost allx e Q , where K = g{0 4-) — g(0 —). Hence ve L2(Q), and u is a critical point of 34? — $.Conversely if u is a critical point of Jf — ^ , then the conclusion follows at once fromTheorem 2.3.COROLLARY 2.5. For n = 1,2,3 and for every admissible decomposition off there isa one-to-one correspondence between critical points of M' —$ and solutions of (1.2) inW2>2{Q) n WJ'2(Q).f or n = A, if f admits a decomposition with g satisfying the growth condition\g(p)\<M + \p\aforM > 0, a e R ,or if n ^ 5 and / admits a decomposition with g satisfyingnM > 0 ,n - 4t/ten there is a one-to-one correspondence between the criticial points of 34? — $ (for thischoice of decomposition) and solutions of (1.2) in W2>2(fi) n W Q ' 2 ^ ) -/ This follows at once from the previous corollary, and the standard Sobolevembedding theorems of W2-2(Q) in L2(Q) which hold since Q satisfies the cone condition(see, e.g. [3]).Our next result gives emphasis to a certain type of critical point and derives itsinterest from the fact that all minimisers of Jf — ^  are of this type (see Theorem 3.1 ofSection 3).THEOREM 2.6. Let f have an admissible decomposition asg — h and let ubea criticalpoint oftf-y. If moreover, d^(w) £ dJf(u) then x(u{x)) = Ofor almost all xeQ, andconsequently,— Au(x) — p(x) ej(u(x))for almost all xeQ.Proof. It suffices to show that #(u(x)) = 0 for almost all x e O. So let v e d${u) and,for any t e U(f), setwhereThen w± belong to L2(Q) and w±(x) e g(u(x)) + p(x) for almost all xeQ. Sow±edy{u), and hence w±ed3^{u). It now follows from Lemma 2.2(iii) thatw+(x) = w_(x) = v(x) for almost all x e Q , and so #,(u(x)) = 0 for almost all x e Q .Since U(f) is countable, andZ(u(x))= I X,(u(x)),' 6 U{f)it follows that #(u(x)) = 0 almost everywhere.326 C. A. STUART AND J. F. TOLANDRemark. That #(u(x)) = 0 almost everywhere means that u takes values in theupward-jump-set of / on a set of zero measure.3. On minimisers of 34? — $Our considerations in this section concern the existence of solutions of theboundary value problem (1.5) under the same hypotheses as in [7]. The followingobservation is central to the analysis.THEOREM 3.1. Let 3tf and & be convex lower semi-continuous functional from aBanach space V into U (J { + oo}. Ifu is a minimiser of 3t' — <& in V, then{u) <= d3f{u).Proof. This is Theorem 2.3 of [9].THEOREM 3.2. Let f satisfy (1.3) and letg — h be any admissible decomposition for f.Then there exists a minimiser u o / / - ^ which is in Wl0'\Cl) o W2'\Ci) o L°°(Q) andwhich satisfies-Au(x)-p(x)e/(u(x))and x(u(x)) = 0 almost everywhere in Q. It follows that u is a minimiser ofjf — ^ for allpossible admissible decompositions of f.Proof. Let g — h be an admissible decomposition for / . Then by (1.3) there existconstants e, M > 0 such thatf(t) ^ M + (A-e)t for all t > 0andfit) ^ (A-e)r-M for all t > 0.HenceFit) ^ &A-e)t2 + M\t\ for all * e R .Now Fit) = Git)-Hit) and hence it follows that for all u e L2(Q),Note that equality need not hold since Giu), #(u) need not be integrable.We will prove that there exists an L00-minimiser u of / in L2(fl), so in consequence/(u) = *{u)-9(u).Therefore u is a minimiser for 3tf — <$ no matter which decomposition of / we choose.A VARIATIONAL METHOD FOR BOUNDARY VALUE PROBLEMS 3 2 7If e and M are chosen as above, then/(«) > I {hVu(x)\2-p(x)u(x)-±(A-e)u(x)2-M\u(x)\}dxn\Wu(x)\2dx ^ A |u(x)|2dx for all u e Wl'2.Hence / is coercive in Wl'2(Q), and bounded below, and it follows that any minimisingsequence {«„} has a subsequence (which we also denote by {«„}) which is weaklyconvergent to u, say, in WQ'2. But,F(t) ^ (A-e)t2 + M\t\,and it then follows by Fatou's lemma that / is lower semi-continuous in the sense thatwhen un —>• u in WQ'2Thus u is a minimiser for / .Finally to prove u 6 L°°(Q) we call upon a result of [5]. Putting G^t) = $At2-F{t)and G2(t) = -{At, it follows from Theorem 6.2 of [§] that u e L°°(Q). In [5] Gx andG2 are required to be C1, but it is easy to see that the result holds in our case as well. Therest of the proof follows by Theorems 2.6 and 3.1.4. Two examples(i) Let us consider Q. to be the unit ball in U5, and let us consider the boundary valueproblem-AM(X) = f(u(x)+l)7u{x) = 0, x e dQ .Then the radially symmetric functionis an exact solution which lies in W2'2(Q) n W^Q), but not in If{Q).Hence if we put g(t) = f(t +1)7 and h = 0 we find that u is a critical point of J f — 0 ,328 A VARIATIONAL METHOD FOR BOUNDARY VALUE PROBLEMSbut if we write g(i) = f(t +1)7 + exp t, and h(t) = exp t, then u is no longer a criticalpoint of 34? — <$ for this new decomposition.(ii) For each e e [ — 1,1] define the function[ - 1 if t < 0 ,fE(t)= £ if t = 0 ,( +1 if t > 0 .Then /£ is in BN and satisfies the hypotheses (1.3). Thus there always exists a minimiserof J f — # in L°°(Q), no matter what decomposition of fE we choose. For all e e [ - 1 , 1 ]the function u = 0 is a solution ofbut only when e = 0 is it a solution ofIn either case {x:f(u{x) + ) >f(u(x)-)} has full measure, and so 0 is not theminimiser of J f — $. Thus such solutions immediately guarantee a non-uniquenessresult. This idea is further developed in [8], where the question of what happens to thezero solution of — Au{x) = fE{u{x)) almost everywhere when e is changed from zero isalso examined more closely.References1. H. Brezis, "Monotonicity methods in Hilbert space and some applications to nonlinear partial differentialequations", Contributions to nonlinear functionals analysis (ed. E. H. Zarantonelfo, M.RJC.Wisconsin, Academic Press, New York, 1971).2. V. Barbu, Nonlinear semigroups and differential equations in Banach space (Noordhoff InternationalPublishing, Leyden, 1976).3. A. Friedman, Elliptic partial differential equations (Holt Reinhart and Winston, New York, 1969).4. I. Massabo and C. A. Stuart, "Elliptic eigenvalue problems with discontinuous non-linearities', J. MathAnal. Appl., 66 (1978), 261-281.5. G. Stampacchia, "On some regular multiple integral problems in the calculus of variations", Comm. PureAppl. Math., 16 (1963), 383-421.6. C. A. Stuart, "Differential equations with discontinuous non-linearities", Arch. Rational Mech. Anal., 63(1976), 59-75.7. C. A. Stuart, "Maximal and minimal solutions of differential equations with discontinuous nonlinearities",Math. Z., 163 (1978), 239-249.8. C. A. Stuart and J. F. Toland, "A property of solutions of elliptic differential equations with discontinuousnonlinearities", J. London Math. Soc. (2), 21 (1980), 329-335.9. J. F. Toland, "A duality principle for non-convex optimisation and the calculus of variations", Arch.Rational Mech. Anal., 71 (1979), 41-61.Departement de Mathematique, Department of Mathematics,Ecole Polytechnique Federale de Lausanne, University College,6, av. de Cour, Gower Street,Lausanne CH-1007, London WC1E 6BT.Switzerland.

A Variational Method for Boundary Value Problems with Discontinuous Nonlinearities

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