Limits of relaxed Dirichlet problems involving a non symmetric Dirichlet form
The purpose of this paper is to extend some homogenization results to the case of nonsymmetric Dirichlet forms; the symmetric case has been previously studied by Biroli and Tchou. More precisely, let \(X\) be a locally compact separable connected Hausdorff space, \(m\) a given Radon measure supported on \(X\). Let \((a,D[a])\) be a nonsymmetric Dirichlet form on \(L^2(X,m)\) with strongly local symmetric and antisymmetric parts. The authors also assume that the symmetric part has all the properties required by Biroli and Tchou in the symmetric case. Then, under all these assumptions, the authors show that it is possible to define the domain of the form \(D[a]\) restricted to any open subset \(A\) of \(X\). After that they prove some compactness results in order to study the asymptotic behaviour of the solutions of \[ \begin{cases} Lu_h=f& \text{in} \Omega_h,\\ u_h=0& \text{on} \;\partial \Omega_h, \end{cases} \] where the domain varies as the parameter \(h\) goes to \(0\). The case of perforated domain and the relaxed Dirichlet problem are also considered.
scientific article; zbMATH DE number 1993719
scientific article