Dal Maso, Gianni; Garroni, Adriana New results on the asymptotic behavior of Dirichlet problems in perforated domains. (English) Zbl 0804.47050 Math. Models Methods Appl. Sci. 4, No. 3, 373-407 (1994). Summary: Let \(A\) be a linear elliptic operator of second order with bounded measurable coefficients on a bounded open set \(\Omega\) of \(\mathbb{R}^ n\), 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 \(f\in H^{-1}(\Omega)\), the solutions \(u_ h\in H^ 1_ 0(\Omega_ h)\) of the equations \(Au_ h= f\) in \(\Omega_ h\), extended to 0 on \(\Omega\backslash \Omega_ h\), converge weakly in \(H^ 1_ 0(\Omega)\) to the unique solution \(u\in H^ 1_ 0(\Omega)\cap L^ 2_ \mu(\Omega)\) of the problem \[ \langle Au,v\rangle+ \int_ \Omega uv d\mu= \langle f,v\rangle\qquad \forall v\in H^ 1_ 0(\Omega)\cap L^ 2_ \mu(\Omega). \] 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 \(w^*_ h\in H^ 1_ 0(\Omega_ h)\) 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. Cited in 46 Documents MSC: 47F05 General theory of partial differential operators 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35J25 Boundary value problems for second-order elliptic equations Keywords:asymptotic behavior of Dirichlet problems in perforated domains; linear elliptic operator of second order with bounded measurable coefficients on a bounded open set; compactness; \(\Gamma\)-convergence; method of oscillating test functions × Cite Format Result Cite Review PDF Full Text: DOI