NEW RESULTS ON THE ASYMPTOTIC-BEHAVIOR OF DIRICHLET PROBLEMS IN PERFORATED DOMAINS

Let A be a linear elliptic operator of the second order with bounded m easurable coefficients on a bounded open set OMEGA of R(n), and let (O MEGA(h)) be an arbitrary sequence of open subsets of OMEGA. We prove t he following compactness result: there exist a subsequence, still deno ted by (OMEGA(h)), and a positive Borel measure mu on OMEGA, not charg ing polar sets, such that, for every f is-an-element-of H-1 (OMEGA), t he solutions (h) is-an-element-of H-0(1)(OMEGA(h)) of the equations Au (h) = f in OMEGA(h), extended to 0 on OMEGA/OMEGA(h), converge weakly in H-0(1) to the unique solution u is-an-element-of H-1(1)(OMEGA) AND L(mu)2(OMEGA) of the problem [Au, v] + integral-OMEGA uv dmu = [f, v] for-all v is-an-element-of H-0(1)(OMEGA AND L(mu)2(OMEGA). When A is s ymmetric, this compactness result is already known and was obtained by GAMMA-convergence techniques. Our new proof, based on the method of o scillating test functions, extends the result to the non-symmetric cas e. The new technique, which is completely independent of GAMMA-converg ence, relies on the study of the behavior of the solutions w(h), is-a n-element-of H-0(1)(OMEGA(h)) of the equations Aw(h)* = 1 in OMEGA(h) , where A is the adjoint operator. We prove also that the limit measu re mu does not change if A is replaced by A. Moreover, we prove that mu depends only on the symmetric part of the operator A, if the coeffi cients of the skew-symmetric part are continuous, while an explicit ex ample shows that mu may depend also on the skew-symmetric part of A, w hen the coefficients are discontinuous.