Poincaré inequality on subanalytic sets

Abstract

Let $\Omega$ be a subanalytic bounded open subset of ${\mathbb{R}}^n$, with possibly singular boundary. We show that given $p\in [1,\infty )$, there is a constant $C$ such that for any $u\in W^{1,p}(\Omega )$ we have $||u-u_{\Omega }|{|}_{L^p}\le C||\nabla u|{|}_{L^p},$ where we have set $u_{\Omega }:=\frac{1}{|\Omega |}{\int }_{\Omega }u.$

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