Let be a bounded strongly -pseudoconvex domain () and a positive Borel measure with finite mass on . We solve the Hölder continuous subsolution problem for the complex Hessian equation on . Namely, we show that this equation admits a unique Hölder continuous solution on with given Hölder continuous boundary values if it admits a Hölder continuous subsolution on . The main step in solving the problem is to establish a new capacity estimate showing that the -Hessian measure of a Hölder continuous -subharmonic function on with zero boundary values is dominated by the -Hessian capacity with respect to with an (explicit) exponent .
Soit un domaine borné fortement -pseudoconvexe () et une mesure de Borel positive de masse finie sur . Nous démontrons que l’équation hessienne complexe sur admet une solution Hölder continue sur pour une donnée au bord Hölder continue si (et seulement si) elle admet une sous-solution Hölder continue sur . L’étape principale dans la résolution du problème consiste à établir une nouvelle estimation capacitaire, qui montre que la mesure -hessienne complexe d’une fonction -sous-harmonique Hölder continue sur avec valeur au bord nulle est dominée par la capacité -hessienne par rapport à avec un exposant explicite .
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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
@article{JEP_2020__7__981_0, author = {Amel Benali and Ahmed Zeriahi}, 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}, volume = {7}, year = {2020}, doi = {10.5802/jep.133}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.133/} }
TY - JOUR AU - Amel Benali AU - Ahmed Zeriahi TI - The Hölder continuous subsolution theorem for complex Hessian equations JO - Journal de l’École polytechnique — Mathématiques PY - 2020 SP - 981 EP - 1007 VL - 7 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.133/ DO - 10.5802/jep.133 LA - en ID - JEP_2020__7__981_0 ER -
%0 Journal Article %A Amel Benali %A Ahmed Zeriahi %T The Hölder continuous subsolution theorem for complex Hessian equations %J Journal de l’École polytechnique — Mathématiques %D 2020 %P 981-1007 %V 7 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.133/ %R 10.5802/jep.133 %G en %F JEP_2020__7__981_0
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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