"Monotonicity with respect to p of the best constants associated with Sobolev immersions of type $W_0^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$ when $q\in\{1,p,\infty\}$"

Abstract

"The goal of this paper is to collect some known results on the monotonicity with respect to $p$ of the best constants associated with Sobolev immersions of type $W_0^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$ when $q\in\{1,p,\infty\}$. More precisely, letting $$\lambda(p,q;\Omega):=\inf\limits_{u\in W_0^{1,p}(\Omega) \setminus\{0\}}{\|\;|\nabla u|_D\;\|_{L^p(\Omega)}}{\|u\|_{L^q(\Omega)}^{-1}}\,,$$ we recall some monotonicity results related with the following functions \begin{eqnarray*} (1,\infty)\ni p&\mapsto &|\Omega|^{p-1}\lambda(p,1;\Omega)^p\,,\\ (1,\infty)\ni p&\mapsto &\lambda(p,p;\Omega)^p\,,\\ (D,\infty)\ni p&\mapsto &\lambda(p,\infty;\Omega)^p\,, \end{eqnarray*} when $\Omega\subset \mathbb{R}^{D}$ is a given open, bounded and convex set with smooth boundary."