Regularity of the distance function to the boundary

Abstract

Let $\Omega$ be a domain in a smooth complete Finsler manifold, and let $G$ be the largest open subset of $\Omega$ such that for every $x$ in $G$ there is a unique closest point from $\partial \Omega$ to $x$ (measured in the Finsler metric). We prove that the distance function from $\partial \Omega$ is in $C_{loc}^{k,\alpha }(G\cup \partial \Omega )$, $k\ge 2$ and $0<\alpha \le 1$, if $\partial \Omega$ is in $C^{k,\alpha }$.

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