Gradient regularity in mixed local and nonlocal problems

Abstract

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