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 $$v\mapsto {\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{\mathrm{\mid }v(x)-v(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+sp}}}dxdy+{\int }_{\mathrm{\Omega }}a(x)\mathrm{\mid }Dv{\mathrm{\mid }}^{q}dx,$$ 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 $a(\cdot )$.