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 $$v \mapsto \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\dfrac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dxdy+\int_{\Omega}a(x)|Dv|^{q}\,dx,$$ 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 $a(\cdot)$.