Interior Schauder estimates for fractional elliptic equations in nondivergence form

Abstract

We obtain sharp interior Schauder estimates for solutions to nonlocal Poisson problems driven by fractional powers of nondivergence form elliptic operators $(-a^{ij}(x){\partial }_{ij}{)}^s$, for $0<s<1$, in bounded domains under minimal regularity assumptions on the coefficients $a^{ij}(x)$. Solutions to the fractional problem are characterized by a local degenerate/singular extension problem. We introduce a novel notion of viscosity solutions for the extension problem and implement Caffarelli's perturbation methodology in the corresponding degenerate/singular Monge--Ampère geometry to prove Schauder estimates in the extension. This in turn implies interior Schauder estimates for solutions to the fractional nonlocal equation. Furthermore, we prove a new Hopf lemma, the interior Harnack inequality and Hölder regularity in the Monge--Ampère geometry for viscosity solutions to the extension problem.

43 pages. To appear in SIAM Journal on Mathematical Analysis