Some remarks on Sobolev type inequalities
Two Sobolev type inequalities for balls are extended to arbitrary smooth bounded domains. In the case of a ball, the first inequality is due to Brezis and Lieb, the other one to Escobar. Theorem 1. Let \(\Omega \subset \mathbb{R}^ n\), \(n\geq 5\) be a bounded domain with smooth boundary. Then there exists a constant \(\lambda (\Omega)\) such that for all \(u\in H^ 1 (\Omega)\) \[ | \nabla u|^ 2_{2,\Omega}+ \lambda (\Omega) | u|^ 2_{2, \partial\Omega} \geq 2^{-2/n} S| u|^ 2_{2n/ (n-2), \Omega}. \] (Here \(\lambda (\Omega)\geq {{n-2} \over 2} H(\Omega)\), \(H (\Omega)\) is the maximal value over \(\partial \Omega\) of the mean curvature \(H(x)\) and \(S= \pi n(n-2) (\Gamma (n/2)/ \Gamma(n))^{2/n}\).) Theorem 2. Let \(n\), \(\Omega\) be as in Theorem 1. Then there exists a constant \(\lambda (\Omega)\geq {{n-2} \over 2} H(\Omega)\) such that for all \(u\in H^ 1 (\Omega)\) \[ | \nabla u|^ 2_{2, \Omega}+ \lambda (\Omega) | u|^ 2_{2, \partial \Omega}\geq S^ 1| u|^ 2_{{{2(n-1)} \over {n-2}}, \partial\Omega}. \] Here \(S^ 1= \inf\{ \int_{\mathbb{R}^ n_ +} | \nabla u|^ 2 dx\); \(\int_{\mathbb{R}^{n-1}} | u(x',0) |^{(2n-2)/ (n-2)} dx'= 1\}\) is the best constant in the trace imbedding \(H^ 1 (\mathbb{R}^ n_ +)\to L^{(2n-2)/ (n-2)} (\mathbb{R}^{(n-1)})\).
scientific article; zbMATH DE number 720498