Fractional elliptic equations, Caccioppoli estimates and regularity

Let $L=-{\mathrm{div}}_x(A(x){\mathrm{\nabla }}_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\mathrm{\Omega }$. We consider the fractional nonlocal equations $$\{\begin{array}{ll}{L}^{s}u=f,& \text{in}\mathrm{\Omega },\\ u=0,& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\text{and}\{\begin{array}{ll}{L}^{s}u=f,& \text{in}\mathrm{\Omega },\\ {\mathrm{\partial }}_{A}u=0,& \text{on}\mathrm{\partial }\mathrm{\Omega }\mathrm{.}\end{array}$$ Here $L^s$, $0<s<1$, is the fractional power of $L$ and ${\mathrm{\partial }}_{A}u$ 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.