Remarks on the homogenization method in optimal design problems

The method of Homogenization for problems of Optimal Design, developed by F. MURAT and the author, is recalled. It is shown how to avoid the characterization of effective properties of mixtures for a general functional that does not involve gradients. History of the subject In the early 70s, while Francois MURAT was working on some academic problems of optimization that had been proposed by Jacques-Louis LIONS [1], he found that a few of them had no solution [2]. His results were quite unexpected for me, and as we were sharing an office in Jussieu in those days, we had many occasions to discuss both his original proof and the various generalizations that he had then obtained [3], and the subject was so fascinating that it marked the beginning of a long and fruitful collaboration, although many of our results have been only partially published (in the sequel I will use "we" to mean F. MURAT and myself). Essentially the initial problem was to minimize J(o)= / \u(x)-z(x)\dx, (1) Jo where ~~7~(rf") + = f in (0,I),ti€#o(0>£)> (2) z 6 L(0,L) being given, and aeA={a£ L°°(0,L),a < a < f3 a.e. in (0,1)}. (3) F. MURAT was trying to apply the direct method of the Calculus of Variations, and he noticed that if a sequence an € A is such that an — f c a+ in L°°(0yL) weak • and — in L°°(0,L) weak *, then the an a . corresponding sequence of solutions un of (2) converges in HQ(0,L) weak to the solution tioo of d ( diioQ \ dx V dx / and J(an)-+J(a_,a+)= / |uoo(x) z(x)\ dx. (5) Jo He constructed then a particular sequence with a< a+, defined z = UQO, implying then inf J(a) = 0, and checked that it was impossible to have it = z in (2) for any a G A, by considering (2) as a differential equation for a, which had no solution staying between a and /? in the interval (0, L) for the choice that he had made. We were naturally led to characterize all the possible pairs (a-,a+) and more generally we proved that if a sequence U^ of measurable functions from an open set fi C R into R satisfies U^\x) £ K a.e. x € Q for a bounded set K> and {/() — {7> in L^ifyR?) weak *, then the characterization of all the possible limits C/ is U^°°\x) 6 conv(K), the closed convex hull of K, a.e. x £Q. This result might not have been stated in such a simple form before, and Ivar EKELAND told me that it had been implicitely used in some work of CASTAING and was related to a classical result of LYAPUNOV valid for a set endowed with a nonnegative measure without atoms, and indeed our proof extended easily to such a general case. Our 1 ';;••• :-'sitv Libraries characterization appeared quite useful when we tried to understand the more general situation where u is the solution of -div(agrad{u)} = / in fi, u £ H^Sl), (6) with aeA = {a£ L°°(fi),a < a < p a.e. in (12)}, (7) and one wants to minimize J ( ) x . (8) Of course, we had found that the main difficulty was to consider (6) for a sequence an converging only weakly, and to identify the weak limits of un and of an grad(un), but at that moment we were not aware that Sergio SPAGNOLO had already studied a similar question [4,5], and we were led to rediscover most of his results by a different approach which, after an improvement of our initial method that I based on our Div-Curl lemma, appeared more powerful. Although our names are rarely quoted nowadays, it is our method that almost everybody uses now, but many do not seem to understand that our notion of H-convergence is indeed much more general that the notion of G-convergence that S. SPAGNOLO had introduced, a reminder of its relation with the convergence of GREEN kernels. At a CIME session in Varenna in 1970, I had met S. SPAGNOLO who had asked me if some of my results about nonlinear interpolation had anything to do with his own results, which he quickly mentioned, but although I could say that there was no relation because the coefficients of his equations were not regular, I did not catch much about what his results really were. I think that after obtaining our initial results, we finally became aware of what S. SPAGNOLO had done through some work of Tullio ZOLEZZI, and one of his articles indeed puzzled us for a while, as we thought that one of his theorems contradicted some of ours [6]. F. MURAT had first identified what one calls now the effective conductivity of a layered material, and his formula had told us that, for N > 2, one could not characterize the limit of un even if one knew the limits in L°°(fl) weak • of h(an) for all continuous functions A, an information which I described later by using the notion of parametrized measures, until John M. BALL told me that the notion had actually been introduced by Laurence C. YOUNG, and should be called the YOUNG measure associated to the sequence an. The puzzling theorem in T. ZOLEZZl's article stated that if a sequence an converges weakly to a+ in L°°(n) then the corresponding sequence of solutions un converges weakly to the solution associated to a+. F. MURAT thought that some nuance in Italian might have tricked us in mistranslating what was meant, but as we were pondering if "debolmente" could mean anything else than weakly, it suddenly appeared that our mistake had been to read correctly weakly and to interpret it incorrectly as weakly *, as indeed it was the first time that we had seen any mention of the weak topology of L°°(fi) in a concrete situation; we understood then why there was a reference to an article of Alexandre GROTHENDIECK, who had shown that convergence in L°°(Q) weak implies strong convergence in Lloc(Q) for every finite p. In our initial proof we assumed that the sequence an € A was such that an —* a+ in L°°(0,L) weak • and > — in L°°(0, L) weak •, because these limits had played a role in the layered case, and that On « £?<"> = grad(un) — £ ( o o ) in L(fi; R) weak and DTM = an grad(un) — Z>(°°) in X (fi; R) weak. Using an integration by parts, we deduced that (£'.£>) — ( J E * 0 0 ) . ^ 0 0 ) ) in V'(Q) (or M(Sl) weak • ) , and we were led to identify the convex hull of the set (E,aE,a\E¥,a,-y parametrized by a € [a , /?] ,£€ R, and the explicit description of that convex hull gave us the missing link to prove the existence (for a subsequence) of a symmetric tensor a^ £ A, independent of/, such that D^) = a^JE^), and moreover that a_I < a^ < a+I a.e. in Q. Actually, our analysis provided the inequality (i?(°°) a+E^.D^ a^E^) < 0 a.e. in fi, which I will use later on. It is useful to mention that we had assumed no periodicity hypothesis on the coefficients of our equations, although we had been aware of that framework after reading notes of Enrique SANCHEZ-PALENCIA [7,8], but his work helped us understanding something more important. Up to that point, we had been dealing with abstract mathematical questions about Partial Differential Equations in variational form, using and improving results from Functional Analysis, and we had never used any physical interpretation of our equations for the quite simple reason that we were not so confident with the knowledge of Continuum Mechanics and Physics that we had been taught at Ecole Polytechnique. The new understanding that we obtained from reading the work of E. SANCHEZ-PALENCIA and comparing it to ours was that the weak convergence methods and the new H-convergence that we had been using (although the term was coined much later), were actually a new mathematical approach for modelling the relations between microscopic and macroscopic levels (I have learned since that the word microscopic should be replaced by mesoscopic when talking to people who think that microscopic only means the scale of atoms). In those days, relations between microscopic and macroscopic levels were only explained using a probabilistic interpretation and ensemble averages, and this is still the case in many circles. I was not writing much in those days, and the only articles that I wrote then were for the proceedings of a conference in Roma in April 1974 and another one at IRIA in June 1974 [9,10], and it was between these two conferences that we had discovered the Div-Cul lemma, in the process of reviewing all the situations that we knew where Ooo could be explicitly computed, and [10] is the earlier reference with a hint to that new philosophy about Continuum Mechanics and Physics which I have advocated for the last twenty years. In the same period some numerical computations about similar Optimal Design problems were performed in Nice, by Jean C 6 A and his team, and we were aware of the work of Denise CHENAIS, which meant that if one imposed some kind of regularity condition on an interface between two materials then a classical optimal solution could be found, while our work suggested that if one did not impose such a condition there might be no classical solution, in which case one would have to use generalized solutions corresponding to mixtures. In some simple cases we could propose a new relaxed problem that seemed to have much better numerical stability properties, and if I computed necessary conditions of optimality in [10], it was partly for telling J. CEA that there were stronger necessary conditions of optimality due to the fact that a classical optimal solution had to be better than all possible mixtures, but I could not convince him that if he refined his triangulations enough he might start seeing oscillations and that our analysis described what kind of oscillations were useful, so that it was not so important to resolve these oscillations numerically. He might have believed that the situation in his work with MALANOWSKI was general [11], while it was obviously the result of a small miracle due to the very special form of their function g, as they had g(x, w, a) = f(x)u. Had the computers been more powerful in those days, he might have discovered numerical oscillations in refining his triangulations, but the cost would have been prohibitive at the time and only coarse triangulations were used. The necessary conditions of optimality which I had computed considered a mixture of two isotropic materials, without constraints upon the proportions, and the necessary conditions of optimality that I had obtained were much stronger than the usual ones obtained by pushing the interface along its normal, an idea going back to HADAMARD. The classical idea, which is often only derived in a formal way (although F. MURAT <*K|, 1 -, (9) (a(*X-0 > d(*)£| for a f € R, a.e. x e Q >, P J and the basic result is that M(ot, /?; ft) is compact for the topology of H-convergence, i.e. from any sequence an G M(a,f3\Q), one can extract a subsequence am which H-converges to aejj G .M(a,/?;ft), i.e. if i) £ ( m ) — £ ( o o ) in L(Q\R) weak ii) £> = amE {m) — D ( o o ) in L(Q;R) weak iii) curlE stays in a compact of Hfo\ (ft; Ca(R ,R^j strong (10) iv) divD^^ stays in a compact of i/j~*(ft) strong then JD (00 ) = aefJE {oo) a.e. in ft. There exists a sequence of correctors P() G L(ft; C(R, R)) associated to the subsequence am, satisfying P<) — / in L(Q]C(R,R)) weak Q(m) _ a(m)p(m) _ fl^ in L 2 (&; C(R, R)) weak cur/(PA) stays in a compact of Hfo\ (ft;£a(/Z , R)"j strong for all A G R div(Q(>\) stays in a compact of #,~*(ft) strong for all A G R, and the role of P<) is to describe what oscillations must exist (at a microscopic level) in E^^ when one has only measured (at a macroscopic level) what the weak limit E°° is: if (10) holds then one has p(m)£.(oo) _ £»(m) _^ o in ^^(ft; R) strong, (12) and the convergence can be shown to hold in L^0C(Q)R ) strong for some p > 1 if one knows better integrability properties for f^°°) or if one uses MEYERS's regularity theorem. If all am are symmetric, then aefj is symmetric (but P^ ) is not symmetric in general), and in this case one has ifi) am-*a+ in L°° (ft;C,(R ,R)) weak • ii) (am)(a-)" in L°°(p;C8(R \R)) weak * then a . < ac/y < a+ a.e. in ft.