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Original Articles

Existence of Solutions for Jacobian and Hessian Equations Under Smallness Assumptions

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Pages 868-892 | Received 10 Dec 2012, Accepted 09 Apr 2013, Published online: 08 Jul 2014

REFERENCES

  • S. Basterrechea (in prep.) . Ecole Polytechnique Fédérale de Lausanne, PhD thesis.
  • Y. Brenier ( 1991 ). Polar factorization and monotone rearrangement of vector-valued functions . Comm. Pure Appl. Math. 44 : 375 – 417 .
  • L. Caffarelli ( 1996 ). Boundary regularity of maps with convex potentials II . Ann. of Math. 144 : 453 – 496 .
  • L. Caffarelli , L. Nirenberg , and J. Spruck ( 1984 ). Dirichlet problems for nonlinear elliptic equations I, Monge-Ampère equations . Comm. Pure Appl. Math. 37 : 369 – 402 .
  • L. Caffarelli , L. Nirenberg , and J. Spruck ( 1985 ). Dirichlet problems for nonlinear elliptic equations III, Functions of the eignevalues of the Hessian equations . Comm. Pure Appl. Math. 155 : 261 – 301 .
  • G. Csato , B. Dacorogna , and O. Kneuss ( 2012 ). The Pullback Equation for Differential Forms . Birkhaüser , New York .
  • B. Dacorogna ( 2002 ). Existence and regularity of solutions of dw = f with Dirichlet boundary conditions . In: Nonlinear Problems in Mathematical Physics and Related Topics . Kluwer/Plenum , New York , pp. 67 – 82 .
  • B. Dacorogna ( 2007 ). Direct Methods in the Calculus of Variations. , 2nd ed. , Springer-Verlag , New York .
  • B. Dacorogna ( 2009 ). Introduction to the Calculus of Variations. , 2nd ed. , Imperial College Press , London .
  • B. Dacorogna and J. Moser ( 1990 ). On a partial differential equation involving the Jacobian determinant . Ann. Inst. H. Poincaré Anal. Non Linéaire 7 : 1 – 26 .
  • P. Delanoë ( 1991 ). Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampère operator . Ann. Inst. H. Poincaré Anal. Non Linéaire 8 : 443 – 457 .
  • L. C. Evans ( 1982 ). Classical solutions of fully nonlinear, convex, second-order elliptic equations . Comm. Pure Appl. Math. 35 : 333 – 363 .
  • L. C. Evans and P. L. Lions ( 1981 ). Fully nonlinear second order elliptic equations with large zeroth order coefficient . Ann. Inst. Fourier (Grenoble) 31 : 175 – 191 .
  • D. Gilbarg and N. S. Trudinger ( 2001 ). Elliptic Partial Differential Equations of Second Order . Springer-Verlag , Berlin .
  • S. Giuffrè and G. Idone ( 2011 ). On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients . Discrete Contin. Dyn. Syst. 31 : 1347 – 1363 .
  • B. Guan (1998). The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature. Trans. Amer. Math. Soc. 350:4955–4971.
  • S. G. Krantz and H. R. Parks ( 1999 ). The Geometry of Domains in Space . Birkhäuser , Boston .
  • O. A. Ladyzhenskaya and N. N. Uraltseva ( 1968 ). Linear and Quasilinear Elliptic Equations . Academic Press , New York .
  • G. M. Lieberman and N. S. Trudinger ( 1986 ). Nonlinear oblique boundary value problems for nonlinear elliptic equations . Trans. Amer. Math. Soc. 295 : 509 – 546 .
  • A. V. Pogorelov ( 1964 ). Monge-Ampère Equations of Elliptic Type . Groningen , Noordhoff .
  • M. V. Safonov ( 1988 ). Classical solution of second-order nonlinear elliptic equations . Izv. Akad. Nauk SSSR Ser. Mat. 52 : 1272 – 1287 (Russian) [English trans. in Math. URSS-Izv. 33(1989): 567–612. .
  • N. S. Trudinger ( 1995 ). On the dirichlet problem for Hessian equations . Acta Math. 175 : 151 – 164 .
  • N. S. Trudinger and X. J. Wang ( 2008 ). Boundary regularity for the Monge-Ampère and affine maximal surface equation . Ann. Math. 167 : 993 – 1028 .
  • N. S. Trudinger and X. J. Wang ( 2008 ). The Monge-Ampère equation and its geometric applications . In: Handbook of Geometric Analysis ( L. Ji and P. Li , eds.), Advanced Lectures in Mathematics , International Press , Somerville , MA , vol. 1 , pp. 467 – 524 .
  • J. Urbas ( 1997 ). On the second boundary value problem for equations of Monge- Ampère type . J. Reine Angew. Math. 487 : 115 – 124 .
  • J. Urbas ( 1998 ). Mass Transfer Problems . Sonderforschungsbereich 256 Nichtlineare Partielle Differentialgleichungen , Bonn .
  • X. J. Wang ( 1996 ). Regularity for Monge-Ampère equation near the boundary . Analysis 16 : 101 – 107 .

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