Journal of the Mathematical Society of Japan

Kiselman's principle, the Dirichlet problem for the Monge–Ampère equation, and rooftop obstacle problems


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First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data. This yields a new relation between solutions of the Dirichlet problem for the homogeneous real and complex Monge–Ampère equations and Kiselman's minimum principle. More generally, it establishes partial regularity for a Bremermann envelope whether or not it solves the Monge–Ampère equation. Second, we prove the second order regularity of the solution of the free-boundary problem for the Laplace equation with a rooftop obstacle, based on a new a priori estimate on the size of balls that lie above the non-contact set. As an application, we prove that convex- and plurisubharmonic-envelopes of rooftop obstacles have bounded second derivatives.

Article information

J. Math. Soc. Japan, Volume 68, Number 2 (2016), 773-796.

First available in Project Euclid: 15 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]
Secondary: 32W20: Complex Monge-Ampère operators

Kiselman Monge–Ampère Kähler metrics


DARVAS, Tamás; RUBINSTEIN, Yanir A. Kiselman's principle, the Dirichlet problem for the Monge–Ampère equation, and rooftop obstacle problems. J. Math. Soc. Japan 68 (2016), no. 2, 773--796. doi:10.2969/jmsj/06820773.

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  • E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math., 37 (1976), 1–44.
  • J. Benoist and J.-B. Hiriart-Urruty, What is the subdifferential of the closed convex hull of a function?, SIAM J. Math. Anal., 27 (1996), 1661–1679.
  • R. Berman, Bergman kernels and equilibrium measures for line bundles over projective manifolds, Amer. J. Math., 131 (2009), 1485–1524.
  • R. J. Berman, On the optimal regularity of weak geodesics in the space of metrics on a polarized manifold, arXiv:1405.6482.
  • R. Berman, From Monge–Ampère equations to envelopes and geodesic rays in the zero temperature limit, preprint, arXiv:1307.3008.
  • R. Berman and J. P. Demailly, Regularity of plurisubharmonic upper envelopes in big cohomology classes, In: Perspectives in analysis, geometry and topology, Progr. Math., 296, Birkhäuser/Springer, 2012, 39–66.
  • B. Berndtsson, A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry, Invent. Math., 200 (2015), 149–200.
  • Z. Błocki, A gradient estimate in the Calabi-Yau theorem, Math. Ann., 344 (2009), 317–327.
  • H.-J. Bremermann, On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex domains, Characterization of Silov boundaries, Trans. Amer. Math. Soc., 91 (1959), 246–276.
  • L. A. Caffarelli, The obstacle problem, Accademia Nazionale dei Lincei, Scuola Normale Superiore, 1998.
  • L. A. Caffarelli and S. Salsa, A geometric approach to free boundary problems, Amer. Math. Soc., 2005.
  • T. Darvas, Envelopes and Geodesics in Spaces of Kähler Potentials, arXiv:1401.7318.
  • J. P. Demailly, Complex Analytic and Differential Geometry,
  • S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, In: Northern California Symplectic Geometry Seminar (eds. Ya. Eliashberg et al.), Amer. Math. Soc., 1999, pp.,13–33.
  • S. K. Donaldson, Nahm's equations and free-boundary problems, The many facets of geometry, 71–91, Oxford Univ. Press, Oxford, 2010.
  • W. Fenchel, On conjugate convex functions, Canad. J. Math., 1 (1949), 73–77.
  • A. Griewank and P. J. Rabier, On the smoothness of convex envelopes, Trans. Amer. Math. Soc., 322 (1990), 691–709.
  • V. Guedj (Ed.), Complex Monge–Ampère equations and geodesics in the space of Kähler metrics, Lecture Notes in Math., 2038, 2012.
  • W. He, On the space of Kähler potentials, Comm. Pure Appl. Math., 68 (2015), 332–343.
  • J.-B. Hiriart-Urruty and C. Lemaréchal, Convex analysis and minimization algorithms II, Springer, 1993.
  • T. Jeffres, R. Mazzeo and Y. A. Rubinstein, Kähler–Einstein metrics with edge singularities, (with an appendix by C. Li and Y. A. Rubinstein), preprint, arXiv:1105.5216.
  • B. Kirchheim and J. Kristensen, Differentiability of convex envelopes, C. R. Acad. Sci. Paris, Ser. I, 333 (2001), 725–728.
  • S. Kolodziej, The complex Monge–Ampère equation and pluripotential theory, Mem. Amer. Math. Soc., 178 (2005), no.,840.
  • C. O. Kiselman, The partial Legendre transformation for plurisubharmonic functions, Invent. Math., 49 (1978), 137–148.
  • C. O. Kiselman, Plurisubharmonic functions and their singularities, In: Complex potential theory (eds. P. M. Gauthier et al.), Kluwer, 1994, pp.,273–323.
  • T. Mabuchi, Some symplectic geometry on compact Kähler manifolds I, Osaka J. Math., 24 (1987), 227–252.
  • S. Mandelbrojt, Sur les fonctiones convexes, C. R. Acad. Sci. Paris, 209 (1939), 977–978.
  • A. Petrosyan and T. To, Optimal regularity in rooftop-like obstacle problem, Comm. Partial Differential Equations, 35 (2010), 1292–1325.
  • A. Petrosyan, H. Shahgholian and N. Uraltseva, Regularity of free boundaries in obstacle-type problems, Amer. Math. Soc., 2012.
  • R. T. Rockafellar, Convex analysis, Princeton University Press, 1970.
  • J. Ross and D. Witt-Nyström, Analytic test configurations and geodesic rays, J. Symplectic Geom., 12 (2014), 125–169.
  • J. Ross and D. Witt-Nyström, Envelopes of positive metrics with prescribed singularities, arXiv:1210.2220.
  • Y. A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math., 218 (2008), 1526–1565.
  • Y. A. Rubinstein, Smooth and singular Kähler–Einstein metrics, preprint, Geometric and Spectral Analysis, (eds. P. Albin et al.), Contemp. Math., 630, AMS and Centre Recherches Mathematiques, 2014.
  • Y. A. Rubinstein and S. Zelditch, The Cauchy problem for the homogeneous Monge–Ampère equation, II, Legendre transform, Adv. Math., 228 (2011), 2989–3025.
  • Y. A. Rubinstein and S. Zelditch, The Cauchy problem for the homogeneous Monge–Ampère equation, III, Lifespan, preprint, arXiv:1205.4793.
  • S. Semmes, Interpolation of spaces, differential geometry and differential equations, Rev. Mat. Iberoamericana, 4 (1988), 155–176.
  • S. Semmes, Complex Monge–Ampère and symplectic manifolds, Amer. J. Math., 114 (1992), 495–550.
  • D. Wu, Kähler–Einstein metrics of negative Ricci curvature on general quasi-projective manifolds, Comm. Anal. Geom., 16 (2008), 395–435.