Gradient regularity for double-phase orthotropic functionals

TL;DR

This paper establishes higher integrability and Lipschitz regularity of local minimizers for a class of double-phase orthotropic functionals, under certain regularity conditions on the weight function and exponents.

Contribution

It proves higher integrability and explicit Lipschitz regularity estimates for local minimizers of double-phase orthotropic functionals with H"older continuous weights.

Findings

Higher integrability of local minimizers under specified conditions.

Explicit Lipschitz regularity estimates obtained.

Conditions on exponents and weight function regularity are critical.

Abstract

We prove higher integrability for local minimizers of the double-phase orthotropic functional \[ \sum_{i=1}^{n}\int_\Omega\left(\left|u_{x_i}\right|^p+a(x)\left| u_{x_i}\right|^q\right)dx \] when the weight function $a \geq0$ is assumed to be $\alpha$-H\"older continuous, while the exponents $p, q$ are such that $2 \leq p \leq q$ and $\frac{q}{p} < 1 + \frac{\alpha}{n}$. Under natural Sobolev regularity of~$a$, we further obtain explicit Lipschitz regularity estimates for local minimizers.