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Hölder continuous solutions to the complex Monge–Ampère equations in non-smooth pseudoconvex domains

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In this paper, we prove the Hölder continuity for solutions to the complex Monge–Ampère equations on non-smooth pseudoconvex domains of plurisubharmonic type m.

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References

  1. Åhag, P., Cegrell, U., Czyż, R., Hiep, P.H.: Monge–Ampère measures on pluripolar sets. J. Math. Pures Appl. 92, 613–627 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baracco, L., Khanh, T.V., Pinton, S., Zampieri, G.: Hölder regularity of the solution to the complex Monge–Ampère equation with \(L^p\) density. Calc. Var PDE 55, 74 (2016)

    Article  MATH  Google Scholar 

  3. Baracco, L., Khanh, T.V.: The complex Monge–Ampère equation on weakly pseudoconvex domains. C. R. Math. Acad. Sci. Paris, Ser. I 355, 411–414 (2017)

    Article  MATH  Google Scholar 

  4. Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge–Ampère operator. Invent. Math. 37, 1–44 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Błocki, Z.: On the \(L^p\)-stability for the complex Monge–Ampère operator. Michigan Math. J. 42, 269–275 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Błocki, Z.: The complex Monge–Ampère operator in hyperconvex domains. Ann. Scuola Norm. Sup. Pisa Cl. sci. 23, 721–747 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Błocki, Z.: Weak solutions to the complex Hessian equation. Ann. Inst. Fourier 55(5), 1735–1756 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caffarelli, L., Kohn, J.J., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations, II: complex Monge–Ampère, and uniformly elliptic equations. Commun.Pure Appl. Math. 38, 209–252 (1985)

    Article  MATH  Google Scholar 

  10. Cegrell, U.: Pluricomplex energy. Acta Math. 180(2), 187–217 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cegrell, U.: The general definition of the complex Monge–Ampère operator. Ann. Inst. Fourier (Grenoble) 54(1), 159–179 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cegrell, U.: A general Dirichlet problem for the complex Monge–Ampère operator. Ann. Polon. Math. 94, 131–147 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Charabati, M.: Hölder regularity for solutions to complex Monge–Ampère equations. Ann. Pol. Math. 113(2), 109–127 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Coman, D.: Domains of finite type and Hölder continuity of the Perron–Bremermann function. Proc. Am. Math. Soc. 125(12), 3569–3574 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cuong, N.N.: Hölder continuous solutions to complex Hessian equations. Potential Anal. 41(3), 887–902 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Demailly, J.P., Dinew, S., Guedj, V., Hiep, P.H., Kołodziej, S., Zeriahi, A.: Hölder continuous solutions to Monge–Ampère equations. J. Eur. Math. Soc. (JEMS) 16(4), 619–647 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guedj, V., Kołodziej, S., Zeriahi, A.: Hölder continuous solutions to the complex Monge–Ampère equations. Bull. Lond. Math. Soc. 40(6), 1070–1080 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hiep, P.H.: Hölder continuity of solutions to the Monge–Ampère equations on compact Kähler manifolds. Ann. Inst. Fourier 60(5), 1857–1869 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hong, N.X.: Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values. Complex Var. Elliptic Equ. 60(3), 429–435 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hong, N.X.: The locally \(\cal{F}\)-approximation property of bounded hyperconvex domains. J. Math. Anal. Appl. 428, 1202–1208 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hong, N.X.:Semi-continuity properties of weighted log canonical thresholds of toric plurisubharmonic functions. C. R. Acad. Sci. Paris, Ser. I (2017). doi:10.1016/j.crma.2017.04.014

  22. Hong, N.X.: Range of the complex Monge-Ampère operator on plurifinely domain. Complex Var. Elliptic Equ. (2017). doi:10.1080/17476933.2017.1325476

  23. Hong, N.X., Trao, N.X., Thuy, T.V.: Convergence in capacity of plurisubharmonic functions with given boundary values. Int. J. Math., 28(3) (2017), Article Id:1750018

  24. Hong, N.X., Viet, H.: Local property of maximal plurifinely plurisubharmonic functions. J. Math. Anal. Appl. 441, 586–592 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kołodziej, S.: Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge–Ampère operator. Ann. Pol. Math. 65(1), 11–21 (1996)

    Article  MATH  Google Scholar 

  26. Kołodziej, S.: The complex Monge–Ampère equation. Acta Math. 180(1), 69–117 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kołodziej, S.: The complex Monge–Ampère equation and pluripotential theory. Memoirs of AMS, 840 (2005)

  28. Kołodziej, S.: Hölder continuity of solutions to the complex Monge–Ampère equation with the right-hand side in \(L^p\): the case of compact Kähler manifolds. Math. Ann. 342(2), 379–386 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Khue, N.V., Hiep, P.H.: A comparison principle for the complex Monge–Ampère operator in Cegrell’s classes and applications. Trans. Am. Math. Soc. 361(10), 5539–5554 (2009)

    Article  MATH  Google Scholar 

  30. Li, S.Y.: On the existence and regularity of Dirichlet problem for complex Monge–Ampère equations on weakly pseudoconvex domains. Calc. Var PDE 20, 119–132 (2004)

    Article  MATH  Google Scholar 

  31. Trao, N.V., Viet, H., Hong, N.X.: Approximation of plurifinely plurisubharmonic functions. J. Math. Anal. Appl. 450, 1062–1075 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Simioniuc, A., Tomassini, G.: The Bremermann–Dirichlet problem for unbounded domains of \(\mathbb{C}^n\). Manuscr. Math. 126(1), 73–97 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Nguyen Xuan Hong.

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This work is finished during the first author’s post-doctoral fellowship of the Vietnam Institute for Advanced Study in Mathematics. He wishes to thank the institution for their kind hospitality and support. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2016.06. The authors would like to thank the referees for valuable remarks which lead to the improvements of the exposition of the paper.

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Hong, N.X., Van Thuy, T. Hölder continuous solutions to the complex Monge–Ampère equations in non-smooth pseudoconvex domains. Anal.Math.Phys. 8, 465–484 (2018). https://doi.org/10.1007/s13324-017-0175-7

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  • DOI: https://doi.org/10.1007/s13324-017-0175-7

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