The Hölder continuous subsolution theorem for complex Hessian equations

Amel Benali; Ahmed Zeriahi

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]

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

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 :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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/}
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Amel Benali; Ahmed Zeriahi. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.133/

Bibliographie
Cité par

[AV10] S. Alesker & M. Verbitsky - “Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds”, Israel J. Math. 176 (2010), p. 109-138 | DOI | Zbl

[BD12] R. J. Berman & J.-P. Demailly - “Regularity of plurisubharmonic upper envelopes in big cohomology classes”, in Perspectives in analysis, geometry, and topology, Progress in Math., vol. 296, Birkhäuser/Springer, New York, 2012, p. 39-66 | DOI | Zbl

[Ber19] R. J. Berman - “From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit”, Math. Z. 291 (2019) no. 1-2, p. 365-394 | DOI | Zbl

[Bre59] H. J. Bremermann - “On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex domains. Characterization of Šilov boundaries”, Trans. Amer. Math. Soc. 91 (1959), p. 246-276 | DOI | Zbl

[BT76] E. Bedford & B. A. Taylor - “The Dirichlet problem for a complex Monge-Ampère equation”, Invent. Math. 37 (1976) no. 1, p. 1-44 | DOI | Zbl

[BT82] E. Bedford & B. A. Taylor - “A new capacity for plurisubharmonic functions”, Acta Math. 149 (1982) no. 1-2, p. 1-40 | DOI | Zbl

[Bło05] Z. Błocki - “Weak solutions to the complex Hessian equation”, Ann. Inst. Fourier (Grenoble) 55 (2005) no. 5, p. 1735-1756 | DOI | Numdam | Zbl

[Ceg04] U. Cegrell - “The general definition of the complex Monge-Ampère operator”, Ann. Inst. Fourier (Grenoble) 54 (2004) no. 1, p. 159-179 | DOI | Numdam | Zbl

[Cha16a] M. Charabati - Le problème de Dirichlet pour l’équation de Monge-Ampère complexe, Ph. D. Thesis, Université de Toulouse 3, 2016 | theses.fr

[Cha16b] M. Charabati - “Modulus of continuity of solutions to complex Hessian equations”, Internat. J. Math. 27 (2016) no. 1, article ID 1650003, 24 pages | DOI | Zbl

[CIL92] M. G. Crandall, H. Ishii & P.-L. Lions - “User’s guide to viscosity solutions of second order partial differential equations”, Bull. Amer. Math. Soc. (N.S.) 27 (1992) no. 1, p. 1-67 | DOI | Zbl

[CZ19] J. Chu & B. Zhou - “Optimal regularity of plurisubharmonic envelopes on compact Hermitian manifolds”, Sci. China Math. 62 (2019) no. 2, p. 371-380 | DOI | Zbl

[DDG + 14] J.-P. Demailly, S. Dinew, V. Guedj, H. H. Pham, S. Kołodziej & A. Zeriahi - “Hölder continuous solutions to Monge-Ampère equations”, J. Eur. Math. Soc. (JEMS) 16 (2014) no. 4, p. 619-647 | DOI | Zbl

[DGZ16] S. Dinew, V. Guedj & A. Zeriahi - “Open problems in pluripotential theory”, Complex Var. Elliptic Equ. 61 (2016) no. 7, p. 902-930 | DOI | Zbl

[DK14] S. Dinew & S. Kołodziej - “A priori estimates for complex Hessian equations”, Anal. PDE 7 (2014) no. 1, p. 227-244 | DOI | Zbl

[EGZ09] P. Eyssidieux, V. Guedj & A. Zeriahi - “Singular Kähler-Einstein metrics”, J. Amer. Math. Soc. 22 (2009) no. 3, p. 607-639 | DOI | Zbl

[EGZ11] P. Eyssidieux, V. Guedj & A. Zeriahi - “Viscosity solutions to degenerate complex Monge-Ampère equations”, Comm. Pure Appl. Math. 64 (2011) no. 8, p. 1059-1094 | DOI | Zbl

[GKZ08] V. Guedj, S. Kolodziej & A. Zeriahi - “Hölder continuous solutions to Monge-Ampère equations”, Bull. London Math. Soc. 40 (2008) no. 6, p. 1070-1080 | DOI | Zbl

[GLZ19] V. Guedj, C. H. Lu & A. Zeriahi - “Plurisubharmonic envelopes and supersolutions”, J. Differential Geom. 113 (2019) no. 2, p. 273-313 | DOI | Zbl

[GZ17] V. Guedj & A. Zeriahi - Degenerate complex Monge-Ampère equations, EMS Tracts in Math., vol. 26, European Mathematical Society, Zürich, 2017 | DOI | Zbl

[KN20a] S. Kołodziej & N. C. Nguyen - “An inequality between complex hessian measures of Hölder continuous $m$ -subharmonic functions and capacity”, in Geometric analysis, Progress in Math., vol. 333, Birkhäuser, Cham, 2020, p. 157-166 | DOI | Zbl

[KN20b] S. Kołodziej & N. C. Nguyen - “A remark on the continuous subsolution problem for the complex Monge-Ampère equation”, Acta Math. Vietnam. 45 (2020) no. 1, p. 83-91 | DOI | Zbl

[Koł96] S. Kołodziej - “Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge-Ampère operator”, Ann. Polon. Math. 65 (1996) no. 1, p. 11-21 | DOI | Zbl

[Koł05] S. Kołodziej - The complex Monge-Ampère equation and pluripotential theory, Mem. Amer. Math. Soc., vol. 178, no. 840, American Mathematical Society, Providence, RI, 2005 | DOI | Zbl

[Li04] S.-Y. Li - “On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian”, Asian J. Math. 8 (2004) no. 1, p. 87-106 | DOI | Zbl

[LPT20] C. H. Lu, T. T. Phung & T. D. Tô - “Stability and Hölder continuity of solutions to complex Monge-Ampère equations on compact hermitian manifolds”, 2020 | arXiv

[Lu12] C. H. Lu - Équations hessiennes complexes, Ph. D. Thesis, Université de Toulouse 3, 2012 | theses.fr

[Lu13] C. H. Lu - “Viscosity solutions to complex Hessian equations”, J. Funct. Anal. 264 (2013) no. 6, p. 1355-1379 | DOI | Zbl

[Lu15] C. H. Lu - “A variational approach to complex Hessian equations in $ℂ^{n}$ ”, J. Math. Anal. Appl. 431 (2015) no. 1, p. 228-259 | DOI | Zbl

[Ngu12] N. C. Nguyen - “Subsolution theorem for the complex Hessian equation”, Univ. Iagel. Acta Math. 50 (2012), p. 69-88 | Zbl

[Ngu14] N. C. Nguyen - “Hölder continuous solutions to complex Hessian equations”, Potential Anal. 41 (2014) no. 3, p. 887-902 | DOI | Zbl

[Ngu18] N. C. Nguyen - “On the Hölder continuous subsolution problem for the complex Monge-Ampère equation”, Calc. Var. Partial Differential Equations 57 (2018) no. 1, article ID 8, 15 pages | DOI | Zbl

[Ngu20] N. C. Nguyen - “On the Hölder continuous subsolution problem for the complex Monge-Ampère equation, II”, Anal. PDE 13 (2020) no. 2, p. 435-453 | DOI | Zbl

[Pli14] S. Plis - “The smoothing of $m$ -subharmonic functions”, 2014 | arXiv

[SA13] A. Sadullaev & B. Abdullaev - “Capacities and Hessians in the class of $m$ -subharmonic functions”, Dokl. Akad. Nauk 448 (2013) no. 5, p. 515-517 | DOI | Zbl

[Sic81] J. Siciak - “Extremal plurisubharmonic functions in $ℂ^{n}$ ”, Ann. Polon. Math. 39 (1981), p. 175-211 | DOI | Zbl

[SW08] J. Song & B. Weinkove - “On the convergence and singularities of the $J$ -flow with applications to the Mabuchi energy”, Comm. Pure Appl. Math. 61 (2008) no. 2, p. 210-229 | DOI | Zbl

[Tos18] V. Tosatti - “Regularity of envelopes in Kähler classes”, Math. Res. Lett. 25 (2018) no. 1, p. 281-289 | DOI | Zbl

[Wal69] J. B. Walsh - “Continuity of envelopes of plurisubharmonic functions”, J. Math. Mech. 18 (1968/1969), p. 143-148 | DOI | Zbl

[Zer20] A. Zeriahi - “Remarks on the modulus of continuity of subharmonic functions” (2020), Preprint available at http://www.math.univ-toulouse.fr/~zeriahi

[ÅCK + 09] P. Åhag, U. Cegrell, S. Kołodziej, H. H. Phạm & A. Zeriahi - “Partial pluricomplex energy and integrability exponents of plurisubharmonic functions”, Adv. Math. 222 (2009) no. 6, p. 2036-2058 | DOI | Zbl

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