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 \(q\in\{1,p,\infty\}\)

Mihai Mihăilescu, Denisa Stancu-Dumitru


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 \(\lambda(p,q;\Omega):=\inf_{u\in W_0^{1,p}(\Omega)\setminus\{0\}}{\|\;|\nabla u|_D\;\|_{L^p(\Omega)}}{\|u\|_{L^q(\Omega)}^{-1}}\) we recall some monotonicity results related with the following functions
(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\,,
when \(\Omega\subset \mathbb{R}^{D}\) is a given open, bounded and convex set with smooth boundary.



\(p\)-Laplacian; \(p\)-torsional rigidity; distance function to the boundary

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