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Archive for Rational Mechanics and Analysis

, Volume 229, Issue 2, pp 547–567 | Cite as

On the Second Boundary Value Problem for Monge–Ampère Type Equations and Geometric Optics

  • Feida Jiang
  • Neil S. Trudinger
Article

Abstract

In this paper, we prove the existence of classical solutions to second boundary value problems for generated prescribed Jacobian equations, as recently developed by the second author, thereby obtaining extensions of classical solvability of optimal transportation problems to problems arising in near field geometric optics. Our results depend in particular on a priori second derivative estimates recently established by the authors under weak co-dimension one convexity hypotheses on the associated matrix functions with respect to the gradient variables, (A3w). We also avoid domain deformations by using the convexity theory of generating functions to construct unique initial solutions for our homotopy family, thereby enabling application of the degree theory for nonlinear oblique boundary value problems.

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References

  1. 1.
    Fitzpatrick, P., Pejsachowicz, J.: An extension of the Leray-Schauder degree for fully nonlinear elliptic problems. Proc. Symp. Pure Math. 45, 425–439 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Fitzpatrick, P., Pejsachowicz, J.: Orientation and the Leray-Schauder Theory for Fully Nonlinear Elliptic Boundary Value Problems. Memoirs of the American Mathematical Society. vol. 10, No. 483, American Mathmatics Society, Providence (1993)Google Scholar
  3. 3.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equation of Second Order. Springer, Berlin (2001)zbMATHGoogle Scholar
  4. 4.
    Gutiérrez, C.E., Tournier, F.: Regularity for the near field parallel refractor and reflector problems. Calc. Var. Partial Differ. Equ. 45, 917–949 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Guan, P., Wang, X.-J.: On a Monge-Ampère equation arising in geometric optics. J. Differ. Geom. 48, 205–223 (1998)CrossRefzbMATHGoogle Scholar
  6. 6.
    Guillen, N., Kitagawa, J.: Pointwise inequalities in geometric optics and other generated Jacobian equations. Commun. Pure Appl. Math. 70, 1146–1220 (2017)CrossRefzbMATHGoogle Scholar
  7. 7.
    Jiang, F., Trudinger, N.S.: On Pogorelov estimates in optimal transportation and geometric optics. Bull. Math. Sci. 4, 407–431 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jiang, F., Trudinger, N.S.: Oblique boundary value problems for augmented Hessian equations II. Nonlinear Anal. 154, 148–173 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Karakhanyan, A.L.: Existence and regularity of the reflector surfaces in \(\mathbb{R}^{n+1}\). Arch. Ration. Mech. Anal. 213, 833–885 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Karakhanyan, A.L.: An inverse problem for the refractive surfaces with parallel lighting. SIAM J. Math. Anal. 48, 740–784 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li, Y.Y.: Degree theory for second order nonlinear elliptic operators and its applications. Commun. Partial Differ. Equ. 14, 1541–1578 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, Y.Y., Liu, J., Nguyen, L.: A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions. J. Fixed Point Theory Appl. 19, 853–876 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lieberman, G.M., Trudinger, N.S.: Nonlinear oblique boundary value problems for nonlinear elliptic equations. Trans. Am. Math. Soc. 295, 509–546 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liu, J., Trudinger, N.S.: On classical solutions of near field reflection problems. Discrete Contin. Dyn. Syst. 36, 895–916 (2016)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Loeper, G.: On the regularity of solutions of optimal transportation problems. Acta Math. 202, 241–283 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ma, X.-N., Trudinger, N.S., Wang, X.-J.: Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177, 151–183 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Trudinger, N.S.: Recent developments in elliptic partial differential equations of Monge-Ampère type. ICM. Madrid 3, 291–302 (2006)zbMATHGoogle Scholar
  18. 18.
    Trudinger, N.S.: On the prescribed Jacobian equation. Proceedings of Internationl Conference on for the 25th Anniversary of Viscosity Solutions, Gakuto International Series, Mathematical Sciences and Applications 20, 243–255, 2008Google Scholar
  19. 19.
    Trudinger, N.S.: On the local theory of prescribed Jacobian equations. Discrete Contin. Dyn. Syst. 34, 1663–1681 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Trudinger, N.S.: On the local theory of prescribed Jacobian equations revisited (in preparation)Google Scholar
  21. 21.
    Trudinger, N.S., Wang, X.-J.: On convexity notions in optimal transportation. Preprint, 2008Google Scholar
  22. 22.
    Trudinger, N.S., Wang, X.-J.: On strict convexity and continuous differentiability of potential functions in optimal transportation. Arch. Ration. Mech. Anal. 192, 403–418 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Trudinger, N.S., Wang, X.-J.: On the second boundary value problem for Monge–Ampère type equations and optimal transportation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. VIII, 143–174 (2009)Google Scholar
  24. 24.
    von Nessi, G.T.: On the second boundary value problem for a class of modified-Hessian equations. Commun. Partial Differ. Equ. 35, 745–785 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsNanjing University of Information Science and TechnologyNanjingPeople’s Republic of China
  2. 2.Centre for Mathematics and Its ApplicationsThe Australian National UniversityCanberraAustralia
  3. 3.School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia

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