CONVERGENCE IN ENERGY FOR ELLIPTIC OPERATORS**University of Pisa

A problem in the optimal design of networks under transverse loading

“…[15]) G-convergence is in the peculiar circumstance of being equivalent to weak convergence, and the topological properties that are required for a solution to exist can be assessed in a more familiar context. The situation is close to that encountered in dealing with the one-dimensional problems where G-convergence is equivalent to weak convergence of the inverses of the operator coefficients, [16,17], so that one could in fact obtain existence results without mentioning G-convergence. Velte and Villaggio [18], for instance, follow this approach in a paper which raised our interest in the present problem.…”

“…On the other hand, F is continuous with respect to L (Q)-convergence when it is regarded as a functional of wa . By recalling that G-convergence of a sequence {ok} to some a implies that the corresponding sequence of solutions w k of problem (1.1) converges to wa in L (Q) (see for instance [16,17]) it follows that F is also (/-continuous. Therefore, solutions of (1.5) exist provided that we can prove that G-convergence is equivalent to weak*-L°°(Q) convergence on the set S".…”

“…Then, denoting by H[g(Q!) = {u e h\Q') I u -g g //p(Q')} with Q' c n and g G H\Q'), one obtains from [16,17] min / okw,]dx^ min / a;w ,, dx,…”

Relaxation Through Homogenization for Optimal Design Problems with Gradient Constraints

“…The notion of homogenization is given in the theories of G convergence and H convergence, see (Ref. 9) and (Ref. 10).…”

On general two-scale convergence and its application to the characterization of G-limits

“…G-convergence was first introduced by Spagnolo ([22], [23] and [24]). Later, Murat and Tartar generalized the concept under the name of Hconvergence, see [15], [16], [17], [25], [26].…”