Lipschitz regularity for local minimizers of some widely degenerate problems

TL;DR

This paper proves Lipschitz regularity of local minimizers for a class of degenerate variational problems involving positive-part powers, under specific conditions on the dimension and exponent.

Contribution

It establishes Lipschitz continuity of local minimizers for a broad class of degenerate functionals with positive-part powers, extending regularity results.

Findings

Local minimizers are Lipschitz continuous for N=2 and p≥2.

Local minimizers are Lipschitz continuous for N≥2 and p≥4.

Regularity results depend on the dimension and the power p.

Abstract

We consider local minimizers of the functional \[ \sum_{i=1}^N \int (|u_{x_i}|-\delta_i)^p_+\, dx+\int f\, u\, dx, \] 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 $p\ge 4$.