I wonder is the following inequality is true/known:

Let $\Omega\subset \mathbb{R}^n$ be a (locally) Lipschitz domain which is the complement of a convex set, then $$ \int_{\partial\Omega} |u|^2 ds \le \int_{\Omega} |\nabla u|^2 dx + \int_{\Omega} |u|^2 dx $$ for any $u\in H^1(\Omega)$. It is easy to prove for example for the $C^2$ case: on introduces the tubular coordinates and uses the fact that the principal curvatures of the boundary are non-positive, then the problem reduces to $\Omega=(0,\infty)$, but I do not know how to approach the Lipschitz case. Versions for more general Sobolev cases would be of interest too.


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