*Submitted Paper*

**Inserted:** 26 jun 2017

**Last Updated:** 26 jun 2017

**Year:** 2017

**Abstract:**

For $\Omega\subset R^N$ open bounded and with a Lipschitz boundary, and $1\le p<+\infty$, we consider the PoincarĂ© inequality with trace term $C_p(\Omega)\vert u\vert_{L^p(\Omega)} \le \vert\nabla u\vert_{L^p(\Omega;R^N)}+\vert u\vert_{L^p(\partial\Omega)}$ on the Sobolev space $W^{1,p}(\Omega)$. We show that among all domains $\Omega$ with prescribed volume, the constant is minimal on balls. The proof is based on the analysis of a free discontinuity problem.

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