Gradient regularity in mixed local and nonlocal problems

Abstract Minimizers of functionals the type $$\begin{aligned} w\mapsto \int _{\Omega }[|Dw|^{p}-fw]\,\textrm{d}x+\int _{{\mathbb {R}}^{n}}\int {R}}^{n}}\frac{|w(x)-w(y)|^{\gamma }}{|x-y|^{n+s\gamma }}\,\textrm{d}x\,\textrm{d}y\end{aligned}$$ w ↦ ∫ Ω [ | D p - f ] d x + R n ( ) y γ s with $$p, \gamma>1>s >0$$ , > 1 0 and $$p> s\gamma $$ , are locally $$C^{1, \alpha }$$ C α -regular in $$\Omega globally Hölder continuous.