Advertisement

On Monge–Ampère Type Equations and Applications

  • Cristian E. GutiérrezEmail author
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

This chapter contains an overview of recent results on the Monge–Ampère equation and its linearization. We also describe the reflector problem and regularity results for weak solutions.

Keywords

Weak Solution Type Equation Borel Measure Regularity Theory Distribution Function Estimate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Caf90]
    L. A. Caffarelli, Interior W 2,p estimates for solutions of the Monge–Ampère equation, Ann. of Math. 131 (1990), 135–150.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [Caf91]
    L. A. Caffarelli, Some regularity properties of solutions of Monge–Ampère equation, Comm. Pure Appl. Math. 44 (1991), 965–969.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [CG96]
    L. A. Caffarelli and C. E. Gutiérrez, Real analysis related to the Monge-Ampère equation, Trans. Amer. Math. Soc. 348 (1996), no. 3, 1075–1092.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [CG97]
    L. A. Caffarelli and C. E. Gutiérrez, Properties of the solutions of the linearized Monge–Ampère equation, Amer. J. Math. 119 (1997), no. 2, 423–465.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [CGH08]
    L. A. Caffarelli, C. E. Gutiérrez, and Q. Huang, On the regularity of reflector antennas, Ann. of Math. 167 (2008), 299–323.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [CO94]
    L. A. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics, Preprint, 1994.Google Scholar
  7. [Gio57]
    E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3 (1957), no. 3, 25–43.MathSciNetGoogle Scholar
  8. [Gut01]
    C. E. Gutiérrez, The Monge–Ampère equation, Birkhäuser, Boston, MA, 2001.zbMATHGoogle Scholar
  9. [GW98]
    P. Guan and X.-J. Wang, On a Monge–Ampère equation arising from optics, J. Differential Geometry 48 (1998), 205–222.zbMATHMathSciNetGoogle Scholar
  10. [KS81]
    N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izv. 16 (1981), no. 1, 151–164.CrossRefzbMATHGoogle Scholar
  11. [Mos61]
    J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [Nas58]
    J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [OG03]
    V. Oliker and T. Glimm, Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem, J. of Math. Sciences 117 (2003), 4096–4108.CrossRefMathSciNetGoogle Scholar
  14. [Pog71]
    A. V. Pogorelov, On the regularity of generalized solutions of the equation <$>{\rm det} \left ({{\partial^2 u} \over {\partial x^i \partial x^j}} \right ) = \phi(x^1, \ldots, x^n) > 0<$>, Soviet Math. Dokl. 12 (1971), no. 5, 1436–1440.zbMATHMathSciNetGoogle Scholar
  15. [TW00]
    N. S. Trudinger and X.-J. Wang, The Bernstein problem for affine maximal hypersurfaces, Invent. Math. 140 (2000), 399–422.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [Wan95]
    Some counterexamples to the regularity of Monge–Ampère equations, Proc. Amer. Math. Soc. 123 (1995), no. 3, 841–845.Google Scholar
  17. [Wan96]
    X.-J. Wang, On the design of a reflector antenna, Inverse Problems 12 (1996), 351–375.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [Wan04]
    X.-J. Wang, On the design of a reflector antenna II, Calc. Var. Partial Differential Equations 20 (2004), no. 3, 329–341.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2010

Authors and Affiliations

  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA

Personalised recommendations