Calculus of Variations and Geometric Measure Theory

P. Bousquet - L. Brasco - V. Julin

Lipschitz regularity for local minimizers of some widely degenerate problems

created by brasco on 04 Sep 2014
modified on 05 Aug 2015


Accepted Paper

Inserted: 4 sep 2014
Last Updated: 5 aug 2015

Journal: Ann. Sc. Norm. Super. Pisa Cl. Sci.
Pages: 29
Year: 2014


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

Keywords: Degenerate elliptic equations, Lipschitz regularity, Anisotropic problems