Fractional elliptic equations, Caccioppoli estimates and regularity

Abstract

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{Lsu=f,inΩ,u=0,on∂Ω,and{Lsu=f,inΩ,∂Au=0,on∂Ω. 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.