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- W3088366946 abstract "This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary Hölder regularity under proper geometric conditions. “Unified” means that our method is applicable for the Laplace equation, linear elliptic equations in divergence and non-divergence form, fully nonlinear elliptic equations, the p−Laplace equations and the fractional Laplace equations etc. In addition, these geometric conditions are quite general. In particular, for local equations, the measure of the complement of the domain near the boundary point concerned could be zero. The key observation in the method is that the strong maximum principle implies a decay for the solution, then a scaling argument leads to the Hölder regularity. Moreover, we also give a geometric condition, which guarantees the solvability of the Dirichlet problem for the Laplace equation. The geometric meaning of this condition is more apparent than that of the Wiener criterion. Dans cet article, nous étudies la relation entre les propriétés géométriques des frontière et la régularité frontière des solutions d'équations elliptiques. Nous prouvons par une nouvelle méthode unifiée la régularité höldérienne ponctuelle dans des conditions géométriques appropriées. Unifié signifie que notre méthode est applicable à l'équation de Laplace, aux équations elliptiques linéaires sous forme de divergence et de non-divergence, aux équations elliptiques entièrement non linéaires, aux équations p-Laplace et aux équations fractionnelles de Laplace, etc. En outre, ces conditions géométriques sont assez générales. En particulier, pour les équations locales, la mesure du complément du domaine près du point frontière concerné pourrait être nulle. Une observation clé de notre méthode est que le principe du maximum fort implique une décroissance de la solution, puis un argument d'échelle nous conduit à la régularité de Hölder. De plus, nous donnons également une condition géométrique, qui garantit la solvabilité du problème de Dirichlet pour l'équation de Laplace. La signification géométrique de cette condition est plus apparente que celle du critère de Wiener." @default.
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- W3088366946 date "2020-11-01" @default.
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- W3088366946 title "Boundary Hölder regularity for elliptic equations" @default.
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- W3088366946 doi "https://doi.org/10.1016/j.matpur.2020.09.012" @default.
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