Regularity of absolute minimizers for continuous convex Hamiltonians

Abstract

For any $n\ge 2$, $Ω\subset\rn$, and any given convex and coercive Hamiltonian function $H\in C^{0}(\rn)$, we find an optimal sufficient condition on $H$, that is, for any $c\in \mathbb{R}$, the level set $H^{-1}(c)$ does not contains any line segment, such then any absolute minimizer $u\in A{M}_H(\Omega )$ enjoys the linear approximation property. As consequences, we show that when $n=2$, if $u\in A{M}_H(\Omega )$ then $u\in C^1$; and if $u\in AM_H(\rr^2)$ satisfies a linear growth at the infinity, then $u$ is a linear function on $\rr^2$. In particular, if $H$ is a strictly convex Banach norm $\Vert \cdot \Vert$ on ${\mathbb{R}}^2$, e.g. the $l_{\alpha }$-norm for $1<\alpha <1$, then any $u\in A{M}_H(\Omega )$ is $C^1$. The ideas of proof are, instead of PDE approaches, purely variational and geometric.

39 pages

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