Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality
| Autor: | Stinga, P. R., Vaughan, M. |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 55 pages. To appear in Journal de Math\'ematiques Pures et Appliqu\'ees |
| Databáze: | arXiv |
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