Lipschitz regularity for local minimizers of some widely degenerate problems

We consider local minimizers of the functional $\sum_{i=1}^N \int (|u_{x_i}|-δ_i)^p_+\, dx+\int f\, u\, dx,$ where $δ_1,\dots,δ_N\ge 0$ 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 $p\ge 4$.

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