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

TL;DR

This paper introduces a verified numerical method to accurately estimate the best constant for the Sobolev embedding of $H_{0}^{1}( ext{domain})$ into $L^{p}( ext{domain})$ on convex domains, with verified results on a square.

Contribution

It develops a novel verified numerical approach to precisely include the best embedding constant for convex domains, improving accuracy over previous estimates.

Findings

Verified numerical inclusion of the best constant on a square domain.

Computed extremal functions for the embedding.

Enhanced precision in Sobolev embedding constants.

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.