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

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

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

Accepted:
Published online:
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
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author = {Amel Benali and Ahmed Zeriahi},
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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