$C^1$–boundary regularity of planar infinity harmonic functions

We prove that if $\Omega\subset \Rset ^2$ is a bounded domain with $C^2$-boundary and $g\in C^2(\Rset ^2)$, then any viscosity solution $u\in C(\overline\Omega)$ of the infinity Laplacian equation (\ref{eq}) is $C^1(\overline \Omega)$. The interior $C^1$ and $C^{1,\alpha}$-regularity of $u$ in dimension two has been proved by Savin \cite{S}, and Evans and Savin \cite{ES}, respectively. We also show that for any $n\ge 3$, if $\Omega\subset \Rset ^n$ is a bounded domain with $C^1$-boundary and $g\in C^1(\Rset ^n)$, then the solution $u$ of equation (\ref{eq}) is differentiable on $\partial\Omega$. This can be viewed as a supplementary result to the much deeper interior differentiability theorem by Evans and Smart.

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Published 27 December 2012