Fractional elliptic equations, Caccioppoli estimates and regularity

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