Illinois Journal of Mathematics

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

Julián Fernández Bonder, Julio D. Rossi, and Carola-Bibiane Schönlieb

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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 $A^*=\Omega\setminus B_d(x_0^*,r^*)$.

Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1111-1121.

Dates
First available in Project Euclid: 18 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258554352

Digital Object Identifier
doi:10.1215/ijm/1258554352

Mathematical Reviews number (MathSciNet)
MR2595757

Zentralblatt MATH identifier
1181.49006

Subjects
Primary: 49J40: Variational methods including variational inequalities [See also 47J20] 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 49K30: Optimal solutions belonging to restricted classes

Citation

Fernández Bonder, Julián; Rossi, Julio D.; Schönlieb, Carola-Bibiane. The best constant and extremals of the Sobolev embeddings in domains with holes: The $L^\infty$ case. Illinois J. Math. 52 (2008), no. 4, 1111--1121. doi:10.1215/ijm/1258554352. https://projecteuclid.org/euclid.ijm/1258554352


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