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

TL;DR

This paper introduces a definition for fractional powers of nondivergence form elliptic operators, explores their applications in various fields, and establishes key regularity results like Harnack inequalities and Hölder estimates for solutions.

Contribution

It defines fractional elliptic operators in nondivergence form with minimal regularity assumptions and proves fundamental inequalities for solutions via a novel extension method.

Findings

Fractional operators in nondivergence form are well-defined under minimal regularity.

Solutions satisfy interior Harnack inequality and Hölder continuity.

Extension problem approach effectively yields regularity results.

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--Amp\`ere equations, elasticity, and finance. The solution $u$ to the nonlocal Poisson problem $$\begin{cases} (-a^{ij}(x) \partial_{ij})^su = f&\hbox{in}~\Omega\\ u=0&\hbox{on}~\partial\Omega \end{cases}$$ is characterized by a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge--Amp\`ere geometry and prove the interior Harnack inequality and H\"older 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 H\"older estimates for solutions $u$ to the fractional problem.