Fractional elliptic equations, Caccioppoli estimates and regularity

Let L = −divₗ (A (x) ∇ₗ) be a uniformly elliptic operator in divergence form in a bounded domain Ω. We consider the fractional nonlocal equations cases L^su = f, & in, \\ u = 0, & on, cases cases L^su = f, & in, \\ ₀u = 0, & on. cases Here L^s, 0 < s < 1, is the fractional power of L and ₀u 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^su (x). Essential tools in the analysis are the semigroup language approach and the extension problem.