The Hölder continuous subsolution theorem for complex Hessian equations
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 981-1007.

Let Ω n be a bounded strongly m-pseudoconvex domain (1mn) and μ a positive Borel measure with finite mass on Ω. We solve the Hölder continuous subsolution problem for the complex Hessian equation (dd c u) m β n-m =μ on Ω. Namely, we show that this equation admits a unique Hölder continuous solution on Ω ¯ with given Hölder continuous boundary values if it admits a Hölder continuous subsolution on Ω ¯. The main step in solving the problem is to establish a new capacity estimate showing that the m-Hessian measure of a Hölder continuous m-subharmonic function on Ω ¯ with zero boundary values is dominated by the m-Hessian capacity with respect to Ω with an (explicit) exponent τ>1.

Soit Ω n un domaine borné fortement m-pseudoconvexe (1mn) et μ une mesure de Borel positive de masse finie sur Ω. Nous démontrons que l’équation hessienne complexe (dd c u) m β n-m =μ sur Ω admet une solution Hölder continue sur Ω ¯ pour une donnée au bord Hölder continue si (et seulement si) elle admet une sous-solution Hölder continue sur Ω ¯. L’étape principale dans la résolution du problème consiste à établir une nouvelle estimation capacitaire, qui montre que la mesure m-hessienne complexe d’une fonction m-sous-harmonique Hölder continue sur Ω ¯ avec valeur au bord nulle est dominée par la capacité m-hessienne par rapport à Ω avec un exposant explicite τ>1.

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DOI: 10.5802/jep.133
Classification: 31C45, 32U15, 32U40, 32W20, 35J96
Keywords: Complex Monge-Ampère equations, complex Hessian equations, Dirichlet problem, obstacle problems, maximal subextension, capacity.
Mot clés : Équations de Monge-Ampère complexes, équations hessienne complexes, problème de Dirichlet, problèmes d’obstacle, sous-extension maximale, capacités hessiennes

Amel Benali 1; Ahmed Zeriahi 2

1 University of Gabès, Faculty of Sciences of Gabès, LR17ES11, Mathematics and Applications 6072 Gabès, Tunisia
2 Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS, UPS 118 route de Narbonne, 31062 Toulouse cedex 09, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {The {H\"older} continuous subsolution theorem for complex {Hessian} equations},
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Amel Benali; Ahmed Zeriahi. The Hölder continuous subsolution theorem for complex Hessian equations. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 981-1007. doi : 10.5802/jep.133. https://jep.centre-mersenne.org/articles/10.5802/jep.133/

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