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

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