Interior $C^{1,\alpha }$ regularity of mixed local-nonlocal $(p,q)$-energy minimizers for $p\le sq$

We establish the local $C^{1,\alpha }$ regularity of minimizers for functionals of the form $$w\to {\int }_{\mathrm{\Omega }}(\mathrm{\mid }\mathrm{\nabla }w{\mathrm{\mid }}^p-fw)dx+{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\frac{\mathrm{\mid }w(x)-w(y){\mathrm{\mid }}^q}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+sq}}dxdy,$$ where $s\in (0,1)$, $1<p\le sq$, and $f\in L^{\mathrm{\infty }}(\mathrm{\Omega })$. This result complements the work of De Filippis and Minigione in \cite{DFM}, thereby completing the proof of $C^{1,\alpha }$ regularity for all $p,q\in (1,\mathrm{\infty })$ and $s\in (0,1)$ with locally bounded source term.