Regularity results for mixed local and nonlocal double phase functionals

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