Fractional elliptic equations, Caccioppoli estimates and regularity

Abstract

Let L = −\mathrm{div}_{x}(A(x)\mathrm{∇}_{x}) be a uniformly elliptic operator in divergence form in a bounded domain Ω . We consider the fractional nonlocal equations \begin{cases} L^{s}u = f, & \text{in }\mathrm{\Omega }, \\ u = 0, & \text{on }\partial \mathrm{\Omega }, \end{cases} \quad\text{and}\quad \begin{cases} L^{s}u = f, & \text{in }\mathrm{\Omega }, \\ \partial _{A}u = 0, & \text{on }\partial \mathrm{\Omega }. \end{cases} 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.