## Abstract

We prove the local regularity of solutions to the problem of minimizing

${\int}_{\mathrm{\Omega}}\left[{L}_{\mathrm{\infty}}(\nabla v(x))+f(x)v(x)\right]\mathit{d}x\mathit{\hspace{1em}}\text{on}{u}^{0}+{W}_{0}^{1,\mathrm{\infty}}(\mathrm{\Omega}),$

where ${L}_{\mathrm{\infty}}$ is either $\frac{1}{2}{|\xi |}^{2}$ for $|\xi |\le 1$ and $+\mathrm{\infty}$ for $|\xi |>1$, or a more general convex, extended-valued function.

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