Lipschitz regularity for local minimizers of some widely degenerate problems

Abstract

We consider local minimizers of the functional \[ \sum_{i=1}^N \int (|u_{x_i}|-δ_i)^p_+\, dx+\int f\, u\, dx, \] where ${\delta }_1,\dots ,{\delta }_N\ge 0$ and $(\cdot {)}_{+}$ stands for the positive part. Under suitable assumptions on $f$, we prove that local minimizers are Lipschitz continuous functions if $N=2$ and $p\ge 2$, or if $N\ge 2$ and $p\ge 4$.

29 pages