Gradient regularity for double-phase orthotropic functionals

Abstract

We prove higher integrability for local minimizers of the double-phase orthotropic functional \[ \sum_{i=1}^{n}\int_Ω\left(\left|u_{x_i}\right|^p+a(x)\left| u_{x_i}\right|^q\right)dx \] when the weight function $a\ge 0$ is assumed to be $\hat{I}\pm$-Hölder continuous, while the exponents $p,q$ are such that $2\le p\le q$ and $\frac{q}{p}<1+\frac{\hat{I}}{\pm }n$. Under natural Sobolev regularity of~$a$, we further obtain explicit Lipschitz regularity estimates for local minimizers.