Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality

Abstract

We define the fractional powers Ls=(−aij(x)∂ij)s, 0<s<1, of nondivergence form elliptic operators L=−aij(x)∂ij in bounded domains Ω⊂Rn, under minimal regularity assumptions on the coefficients aij(x) and on the boundary ∂Ω. We show that these fractional operators appear in several applications such as fractional Monge–Ampère equations, elasticity, and finance. The solution u to the nonlocal Poisson problem{(−aij(x)∂ij)su=finΩu=0on∂Ω is characterized by a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge–Ampère geometry and prove the interior Harnack inequality and Hölder estimates for solutions to the extension problem when the coefficients aij(x) are bounded, measurable functions. This in turn implies the interior Harnack inequality and Hölder estimates for solutions u to the fractional problem.