Sharp numerical inclusion of the best constant for embedding $H_{0}^{1}(\Omega) \hookrightarrow L^{p}(\Omega)$ on bounded convex domain

Abstract

In this paper, we propose a verified numerical method for obtaining a sharp inclusion of the best constant for the embedding $H_{0}^{1}(\Omega) \hookrightarrow L^{p}(\Omega)$ on bounded convex domain in $\mathbb{R}^{2}$. We estimate the best constant by computing the corresponding extremal function using a verified numerical computation. Verified numerical inclusions of the best constant on a square domain are presented.