The best constant and extremals of the Sobolev embeddings in domains with holes: The $L^\infty$ case

Abstract

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain. We study the best constant of the Sobolev trace embedding $W^{1,\infty}(\Omega) \hookrightarrow L^\infty (\partial\Omega)$ for functions that vanish in a subset $A\subset \Omega$, which we call the hole. That is we deal with the minimization problem $S_A^T =\inf\|u\|_{W^{1,\infty}(\Omega)}/\|u\|_{L^\infty(\partial\Omega)}$ for functions that verify $u|_A = 0$. We find that there exists an optimal hole that minimizes the best constant $S_A^T$ among subsets of $\Omega$ of prescribed volume and we give a geometrical characterization of this optimal hole. In fact, minimizers associated to these holes are cones centered at some points $x_0^*$ on $\partial\Omega$ with respect to the arc-length metric in $\Omega$ and the best holes are of the form $A^*=\Omega\setminus B_d(x_0^*,r^*)$ where the ball is taken again with respect of the arc-length metric.

A similar analysis can be performed for the best constant of the embedding $W^{1,\infty}(\Omega) \hookrightarrow L^\infty( \Omega)$ with holes. In this case, we also find that minimizers associated to optimal holes are cones centered at some points $x_0^*$ on $\partial\Omega$ and the best holes are of the form $A^*=\Omega\setminus B_d(x_0^*,r^*)$.