The best constant in a trace inequality
The author proves that the best constant \(c_ 0\) in the trace inequality \[ \| v\|_{L^ 1(\Gamma_ 1)}\leq c_ 0\|\nabla v\|_{L^ 2(\Omega)} \qquad \forall v\in H_{\Gamma_ 2}^ \perp (\Omega)=\{v\in H^ 1(\Omega),\;v=0\text{ on }\Gamma_ 2\} \] is equal to \(\|\nabla u\|_{L^ 2(\Omega)}\), where \(u\) is the solution of the problem: \[ -\Delta u=0\quad\text{in }\Omega, \qquad \partial u/\partial n=1\quad\text{on }\Gamma_ 1, \qquad u=0\quad\text{on }\Gamma_ 2, \qquad \partial u/\partial n=0\quad\text{on }\Gamma_ 2, \] where \(\Omega\) is an open connected bounded set in \(\mathbb{R}^ n\) with Lipschitz-continuous boundary, meas \((\Gamma_ 2)>0\), \(\Gamma_ 1\cap\Gamma_ 2=\emptyset\) and \(\Gamma_ 3=\partial\Omega\setminus(\overline {\Gamma}_ 1\cup\overline {\Gamma}_ 2)\) may be empty. Some examples and an application to the modified Steklov eigenvalue problem are also shown.
scientific article; zbMATH DE number 62784
scientific article