Sharp Sobolev inequalities with interior norms

Abstract

In this paper, we establish some new sharp Sobolev inequalities on any smooth bounded domain \(\Omega \subset{\Bbb R}^n\). Let \(S_1\) and S be the sharp constants corresponding to the Sobolev embedding and trace inequalities respectively. If \(n\ge 4\), there exist constants \(A(\Omega)\), \(A_1(\Omega)>0\) such that \(\forall u \in H^1(\Omega)\)\(\) and \(\) If \(n=3\), for any \(k_3 >3\), there exist constants \(A(\Omega, k_3), A_1(\Omega, k_3)>0 \) such that \(\forall u \in H^1(\Omega)\)\(\) and \(\)