Fractional elliptic equations, Caccioppoli estimates and regularity

Abstract

Let $L=-{\mathrm{div}}_x(A(x){\nabla }_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\Omega$. We consider the fractional nonlocal equations

Here $L^s$, $0<s<1$, is the fractional power of $L$ and ${\partial }_{A}u$ is the conormal derivative of $u$ with respect to the coefficients $A(x)$. We reproduce Caccioppoli type estimates that allow us to develop the regularity theory. Indeed, we prove interior and boundary Schauder regularity estimates depending on the smoothness of the coefficients $A(x)$, the right hand side $f$ and the boundary of the domain. Moreover, we establish estimates for fundamental solutions in the spirit of the classical result by Littman–Stampacchia–Weinberger and we obtain nonlocal integro-differential formulas for $L^{s}u(x)$. Essential tools in the analysis are the semigroup language approach and the extension problem.