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

We define the fractional powers $L^s=(-a^{ij}(x){\mathrm{\partial }}_{ij}{)}^s$, $0 < s < 1$,ofnondivergenceformellipticoperators$L=-a^{ij}(x)\partial_{ij}$ in bounded domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$, under minimal regularity assumptions on the coefficients $a^{ij}(x)$ and on the boundary $\mathrm{\partial }\mathrm{\Omega }$. 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 $$\{\begin{array}{ll}(-a^{ij}(x){\mathrm{\partial }}_{ij}{)}^{s}u=f& \text{in}\mathrm{\Omega }\\ u=0& \text{on}\mathrm{\partial }\mathrm{\Omega }\end{array}$$ 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.