Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T03:14:23.123Z Has data issue: false hasContentIssue false

Energy Functionals and Complex Monge–Ampère Equations

Published online by Cambridge University Press:  10 February 2010

Zuoliang Hou
Affiliation:
Mathematics Department, Columbia University, 2990 Broadway, New York, NY 10027, USA ([email protected]; [email protected])
Qi Li
Affiliation:
Mathematics Department, Columbia University, 2990 Broadway, New York, NY 10027, USA ([email protected]; [email protected])

Abstract

We introduce certain energy functionals to complex Monge–Ampère equations over bounded domains with inhomogeneous boundary conditions, and use these functionals to show the convergence of solutions to certain parabolic Monge–Ampère equations.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bakelman, I. J., Variational problems and elliptic Monge–Ampère equations, J. Diff. Geom. 18(4) (1983), 669699.Google Scholar
2.Bedford, E. and Taylor, B. A., The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math. 37(1) (1976), 144.CrossRefGoogle Scholar
3.Błocki, Z., Weak solutions to the complex Hessian equation, Annales Inst. Fourier 55(5) (2005), 17351756.CrossRefGoogle Scholar
4.Caffarelli, L., Kohn, J. J., Nirenberg, L. and Spruck, J., The Dirichlet problem for nonlinear second-order elliptic equations, II, Complex Monge–Ampère, and uniformly elliptic, equations, Commun. Pure Appl. Math. 38(2) (1985), 209252.CrossRefGoogle Scholar
5.Cao, H. D., Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds, Invent. Math. 81(2) (1985), 359372.CrossRefGoogle Scholar
6.Chen, X. X. and Tian, G., Ricci flow on Kähler–Einstein surfaces, Invent. Math. 147(3) (2002), 487544.CrossRefGoogle Scholar
7.Guan, B., The Dirichlet problem for complex Monge–Ampère equations and regularity of the pluri-complex Green function, Commun. Analysis Geom. 6(4) (1998), 687703.CrossRefGoogle Scholar
8.Kołodziej, S., The complex Monge–Ampère equation, Acta Math. 180(1) (1998), 69117.CrossRefGoogle Scholar
9.Krylov, N. V., Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47(1) (1983), 75108.Google Scholar
10.Li, S.-Y., On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian, Asian J. Math. 8(1) (2004), 87106.CrossRefGoogle Scholar
11.Phong, D. H. and Sturm, J., On stability and the convergence of the Kähler–Ricci flow, J. Diff. Geom. 72(1) (2006), 149168.Google Scholar
12.Tian, G. and Zhu, X., Convergence of Kähler–Ricci flow, J. Am. Math. Soc. 20(3) (2007), 675699.(electronic).CrossRefGoogle Scholar
13.Trudinger, N. S., On the Dirichlet problem for Hessian equations, Acta Math. 175(2) (1995), 151164.CrossRefGoogle Scholar
14.Trudinger, N. S. and Wang, X.-J., Hessian measures, I, Topol. Methods Nonlin. Analysis 10(2) (1997), 225239.(dedicated to Olga Ladyzhenskaya).CrossRefGoogle Scholar
15.Trudinger, N. S. and Wang, X.-J., A Poincaré type inequality for Hessian integrals, Calc. Var. Partial Diff. Eqns 6(4) (1998), 315328.CrossRefGoogle Scholar
16.Tso, K., On a real Monge–Ampère functional, Invent. Math. 101(2) (1990), 425448.CrossRefGoogle Scholar
17.Wang, X. J., A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J. 43(1) (1994), 2554.CrossRefGoogle Scholar
18.Yau, S. T., On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Commun. Pure Appl. Math. 31(3) (1978), 339411.CrossRefGoogle Scholar