Computable majorants of constants in the Poincaré and Friedrichs inequalities (Q1947432)

Let \(\Omega\) be a bounded open Lipschitz domain of \({\mathbb{R}}^{n}\). Let \(\Gamma\) be the boundary of \(\Omega\). Let \(H^{1}(\Omega)\) denote the Sobolev space of functions of \(L^{2}(\Omega)\) with first order distributional derivatives in \(L^{2}(\Omega)\). Let \(C_{P}(\Omega)>0\) be the (Poincaré) best constant such that \[ \|v\|_{ L^{2}(\Omega) }\leq C_{P}(\Omega) \|\nabla v\|_{ L^{2}(\Omega) } \] for all \(v\in H^{1}(\Omega)\) such that \(\int_{\Omega}v\,dx=0\). Let \(C_{F}(\Omega, \Gamma_{0})>0\) be the (Friedrich) best constant such that \[ \|v\|_{ L^{2}(\Omega) }\leq C_{F}(\Omega, \Gamma_{0}) \|\nabla v\|_{ L^{2}(\Omega) } \] for all \(v\in H^{1}(\Omega)\) which have zero trace on the subset \(\Gamma_{0}\) of positive measure of \(\Gamma\). The author proposes a way to compute upper bound for \(C_{P}(\Omega)\) and \(C_{F}(\Omega, \Gamma_{0})\) in case \(\Omega\) can be decomposed into a finite family of simpler domains for which such constants can be estimated easily.