Mosco-convergence and Wiener measures for conductive thin boundaries
The main result reads as follows. Let \(R \leq \infty\) and \(F_{R}^{\epsilon}\) and \(F_{R}\) be the energy functionals defined in \(L^2(\Omega_R, d \mu^\epsilon)\) and \(L^2(\Omega_R, d \mu^\prime)\), respectively. It follows that \(F_{R}^{\epsilon}\) and \(F_{R}\) are local and regular Dirichlet forms. Assume \(R < \infty\). If \(\alpha\geq 0\) and \(0\leq \beta, \gamma, \nu \leq 1\), then \(F_{R}^{\epsilon}\) Mosco-converges to \(F_{R}\). Accordingly, the associated \(L^2\)-heat-semigroups \(T_{R}^{\epsilon}(t)\) and \(T_{R}(t)\) satisfy \[ T_{R}^{\epsilon}(t) \rightarrow T_{R}(t)\text{ as }\epsilon \rightarrow 0\text{ for every } t > 0, \] and the associated spectral measures \(E^\epsilon\) and \(E\) satisfy \[ E^{\epsilon}((\lambda,\eta]) \rightarrow E((\lambda,\eta])\text{ as }\epsilon \to 0 \] for every \(\lambda < \eta\) which is not in the point spectrum. In particular, \(F_{R}^{\epsilon}\) \(\Gamma\)-converges to \(F_{R}\), and as a consequence, every cluster point of the sequence \(u^\epsilon\) of the minimizers of \(F_{R}^{\epsilon}\) is a minimizer of \(F_{R}\). Namely, let \(a_m > 0\) and \(\epsilon(m) > 0\) be sequences tending to 0 as \(m \rightarrow \infty\) and let \(u_m \in L^2(\Omega_R, d \mu^{\epsilon(m)})\) satisfy: \[ F_R^{\epsilon(m)}[u_m] < \inf_{u\in L^2(\Omega_R, d \mu^{\epsilon(m)})} F_R^{\epsilon(m)}[u] + a_m. \] If \(u_{k(m)}\) is a subsequence of \(u_m\) converging to some \(\bar{u}\) in \(L^2(\Omega_R, d \mu^{\epsilon(m)})\), then \(\bar{u}\) is a minimizer of \(F_R\) and \[ F_R[\bar{u}] = \lim_{m\rightarrow 0} F_R^{\epsilon(k(m))}[u_{k(m)}]. \] Assume \(R \leq \infty\). If \[ \alpha \geq 2 \max \{\beta, \gamma\}\quad\text{and}\quad 0\leq \nu \leq 1, \] then the set of Wiener measures \(\{{\mathbb P}^\epsilon\}_{\epsilon > 0}\) associated to \(F^\epsilon\) is tight. Furthermore, if \(0\leq \beta, \gamma, \nu \leq 1\) and \(\alpha\geq 2 \max \{\beta, \gamma\}\), then \(\{{\mathbb P}^\epsilon\}\) weakly converges to the Wiener measure \({\mathbb P}\) associated to \(F\). In particular, \(F^\epsilon\) converges to \(F\) in both Mosco and \(\Gamma\) senses.
domain with highly conductive thin boundary
scientific article; zbMATH DE number 5956110
scientific article