Notes on best constants in some Sobolev inequalities (Q1321504)

The purpose of the paper is to evaluate the best constant \(c_ 0\) in inequalities of the type \(\| v \|^ 2_{L^ 1(\Omega)} \leq c_ 0 \{\| \nabla v \|^ 2_{L^ 2(\Omega)} + \| v \|^ 2_{L^ p(\Gamma_ 1)}\}\) where \(\Omega_ 0 \subset \Omega \subset \mathbb{R}^ n\) are ``regular'' open sets, \(\partial \Omega = \Gamma_ 1 \cup \Gamma_ 2\), meas \((\Gamma_ 1)>0\), \(p=1,2,v \in H^ 1(\Omega)\). One can compute the best constant by solving convenient linear elliptic boundary value problems. Finally, the author shows by an example that for inequalities involving \(L^ 1\) norms and \(L^ 2\) norms it is not always possible to compute the associated best constants by solving linear elliptic boundary value problems.