The Hölder continuous subsolution theorem for complex Hessian equations

Benali, Amel; Zeriahi, Ahmed

doi:10.5802/jep.133

The Hölder continuous subsolution theorem for complex Hessian equations
[Le théorème des sous-solutions Hölder continues pour les équations hessiennes complexes]

Benali, Amel ¹ ; Zeriahi, Ahmed 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

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 981-1007.

corrigé par Erratum to “The Hölder continuous subsolution theorem for complex Hessian equations”
[Erratum à « Le théorème des sous-solutions Hölder continues pour les équations hessiennes complexes »]

Résumé
Abstract

Soit $Ω ⋐ ℂ^{n}$ un domaine borné fortement $m$ -pseudoconvexe ( $1 \leq m \leq n$ ) et $μ$ une mesure de Borel positive de masse finie sur $Ω$ . Nous démontrons que l’équation hessienne complexe ${(d d^{c} u)}^{m} \land β^{n - m} = μ$ sur $Ω$ admet une solution Hölder continue sur $\bar{Ω}$ pour une donnée au bord Hölder continue si (et seulement si) elle admet une sous-solution Hölder continue sur $\bar{Ω}$ . 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 $\bar{Ω}$ avec valeur au bord nulle est dominée par la capacité $m$ -hessienne par rapport à $Ω$ avec un exposant explicite $τ > 1$ .

Let $Ω ⋐ ℂ^{n}$ be a bounded strongly $m$ -pseudoconvex domain ( $1 \leq m \leq n$ ) and $μ$ a positive Borel measure with finite mass on $Ω$ . We solve the Hölder continuous subsolution problem for the complex Hessian equation ${(d d^{c} u)}^{m} \land β^{n - m} = μ$ on $Ω$ . Namely, we show that this equation admits a unique Hölder continuous solution on $\bar{Ω}$ with given Hölder continuous boundary values if it admits a Hölder continuous subsolution on $\bar{Ω}$ . 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 $\bar{Ω}$ with zero boundary values is dominated by the $m$ -Hessian capacity with respect to $Ω$ with an (explicit) exponent $τ > 1$ .

Reçu le : 2020-04-17
Accepté le : 2020-06-29
Publié le : 2020-07-16

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

Affiliations des auteurs :

Benali, Amel ¹ ; Zeriahi, Ahmed ²

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     author = {Benali, Amel and Zeriahi, Ahmed},
     title = {The {H\"older} continuous subsolution theorem for complex {Hessian} equations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {981--1007},
     publisher = {Ecole polytechnique},
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Benali, Amel; Zeriahi, Ahmed. The Hölder continuous subsolution theorem for complex Hessian equations. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 981-1007. doi : 10.5802/jep.133. http://www.numdam.org/articles/10.5802/jep.133/

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