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

Abstract

We define the fractional powers $L^s=(-a^{ij}(x){\partial }_{ij}{)}^s$, $0<s<1$, of nondivergence form elliptic operators $L=-a^{ij}(x){\partial }_{ij}$ in bounded domains $\Omega \subset {\mathbb{R}}^n$, under minimal regularity assumptions on the coefficients $a^{ij}(x)$ and on the boundary $\partial \Omega$. 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 $$\{\begin{array}{ll}(-a^{ij}(x){\partial }_{ij}{)}^{s}u=f& in\Omega \\ u=0& on\partial \Omega \end{array}$$ 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 $a^{ij}(x)$ are bounded, measurable functions. This in turn implies the interior Harnack inequality and Hölder estimates for solutions $u$ to the fractional problem.

55 pages. To appear in Journal de Mathématiques Pures et Appliquées