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- W4287547609 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 $Omegasubsetmathbb{R}^n$, under minimal regularity assumptions on the coefficients $a^{ij}(x)$ and on the boundary $partialOmega$. 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 $$begin{cases} (-a^{ij}(x) partial_{ij})^su = f&hbox{in}~Omega u=0&hbox{on}~partialOmega end{cases}$$ 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 Holder 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 Holder estimates for solutions $u$ to the fractional problem." @default.
- W4287547609 created "2022-07-25" @default.
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- W4287547609 date "2020-12-29" @default.
- W4287547609 modified "2023-09-24" @default.
- W4287547609 title "Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality" @default.
- W4287547609 hasPublicationYear "2020" @default.
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