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On the regularity of solutions to fully nonlinear elliptic equations via the Liouville property

Author: Qingbo Huang
Journal: Proc. Amer. Math. Soc. 130 (2002), 1955-1959
MSC (2000): Primary 35J60; Secondary 35B65
Published electronically: February 8, 2002
MathSciNet review: 1896027
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Abstract: We show that any $C^{1,1}$ solution to the uniformly elliptic equation $F(D^{2}u)=0$must belong to $C^{2,\alpha }$, if the equation has the Liouville property.

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  • [A] T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer-Verlag, 1982. MR 85j:58002
  • [CC] L. Caffarelli & X. Cabré, Fully nonlinear elliptic equations, AMS Colloquium Publications V. 43, AMS, Rhode Island, 1993. MR 96h:35046
  • [CFL] F. Chiarenza, M. Frasca & P. Longo, $W^{2,p}$-solvability of the Dirichlet Problem for Nondivergence Elliptic Equations with $VMO$ Coefficients, Trans. Amer. Math. Soc. 336 (1993), 841-853. MR 93f:35232
  • [GT] D. Gilbarg & N. S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Springer-Verlag, 1983. MR 86c:35035
  • [Gi] M. Giaquinta & J. Necas, On the Regularity of Weak Solutions to Nonlinear Ellptic Systems of Partial Differential Equations, J. Reine Angew. Math. 316 (1980), 140-159. MR 81m:35056
  • [H] Q. Huang, Estimates on the Generalized Morrey Spaces $L_{\phi }^{2,\lambda }$and $\text{BMO}_{\psi }$ for Linear Elliptic Systems, Indiana Univ. Math. J. 45 (1996), 397-439. MR 97i:35033
  • [K] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order, Mathematics and its Applications, Reidel, 1987. MR 88d:35005
  • [N] N. Nadirashvili, Nonclassical Solutions to Fully Nonlinear Elliptic Equations, Preprint.

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Additional Information

Qingbo Huang
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Address at time of publication: Department of Mathematics & Statistics, Wright State University, Dayton, Ohio 45435

Keywords: Fully nonlinear elliptic equation, regularity, Liouville property, VMO
Received by editor(s): September 20, 1999
Published electronically: February 8, 2002
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society