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- W3207141195 abstract "We define the fractional powers Ls=(−aij(x)∂ij)s, 0<s<1, of nondivergence form elliptic operators L=−aij(x)∂ij in bounded domains Ω⊂Rn, under minimal regularity assumptions on the coefficients aij(x) and on the boundary ∂Ω. 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{(−aij(x)∂ij)su=finΩu=0on∂Ω 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 aij(x) are bounded, measurable functions. This in turn implies the interior Harnack inequality and Hölder estimates for solutions u to the fractional problem. On définit les puissances fractionnaires Ls=(−aij(x)∂ij)s, 0<s<1, des opérateurs elliptiques sous forme non-divergence L=−aij(x)∂ij dans des domaines bornés Ω⊂Rn, sous des hypothèses de régularité minimale sur les coefficients aij(x) et à la frontière ∂Ω. Nous montrons que ces opérateurs fractionnaires apparaissent dans plusieurs applications telles que équations fractionnaires de Monge–Ampère, élasticité et finance. La solution u au problème de Poisson non local{(−aij(x)∂ij)su=fdansΩu=0au∂Ω se caractérise par un problème d'extension local dégénéré/singulier. Nous développons la méthode des paraboloïdes glissants dans la géométrie de Monge–Ampère et prouver l'inégalité intérieure de Harnack et estimations de Hölder pour les solutions à le problème d'extension lorsque les coefficients aij(x) sont des fonctions mesurables bornées. Cela implique à son tour l'inégalité intérieure de Harnack et des estimations de Hölder pour les solutions u au problème fractionnaire." @default.
- W3207141195 created "2021-10-25" @default.
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- W3207141195 date "2021-12-01" @default.
- W3207141195 modified "2023-09-27" @default.
- W3207141195 title "Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality" @default.
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- W3207141195 doi "https://doi.org/10.1016/j.matpur.2021.10.003" @default.
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