Regularity results for mixed local and nonlocal double phase functionals

TL;DR

This paper establishes regularity results, including Hölder continuity and Harnack's inequality, for minimizers of mixed local and nonlocal functionals that combine fractional and classical derivatives.

Contribution

It extends De Giorgi-Nash-Moser theory to a new class of mixed local and nonlocal functionals with sharp conditions.

Findings

Proves Hölder regularity of minimizers.

Establishes Harnack's inequality for solutions.

Identifies sharp assumptions on parameters and coefficients.

Abstract

We investigate the De Giorgi-Nash-Moser theory for minimizers of mixed local and nonlocal functionals modeled after \[ v \mapsto \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\dfrac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dxdy+\int_{\Omega}a(x)|Dv|^{q}\,dx, \] where $0<s<1<p \le q$ and $a(\cdot) \ge 0$. In particular, we prove H\"older regularity and Harnack's inequality under possibly sharp assumptions on $s,p,q$ and $a(\cdot)$.