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 measurable coefficients on a bounded open set OMEGA of R(n), and let (OMEGA(h)) be an arbitrary sequence of open subsets of OMEGA. We prove the following compactness result: there exist a subsequence, still denoted by (OMEGA(h)), and a positive Borel measure mu on OMEGA, not charging polar sets, such that, for every f is-an-element-of H-1 (OMEGA), the 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 symmetric, this compactness result is already known and was obtained by GAMMA-convergence techniques. Our new proof, based on the method of oscillating test functions, extends the result to the non-symmetric case. The new technique, which is completely independent of GAMMA-convergence, relies on the study of the behavior of the solutions w(h)*, is-an-element-of H-0(1)(OMEGA(h)) of the equations A*w(h)* = 1 in OMEGA(h), where A* is the adjoint operator. We prove also that the limit measure 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 coefficients of the skew-symmetric part are continuous, while an explicit example shows that mu may depend also on the skew-symmetric part of A, when the coefficients are discontinuous.