Regularity results for mixed local and nonlocal double phase functionals

Abstract

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