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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that any solution to the uniformly elliptic equation must belong to , if the equation has the Liouville property.

**[A]**Thierry Aubin,*Nonlinear analysis on manifolds. Monge-Ampère equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR**681859****[CC]**Luis A. Caffarelli and Xavier Cabré,*Fully nonlinear elliptic equations*, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR**1351007****[CFL]**Filippo Chiarenza, Michele Frasca, and Placido Longo,*𝑊^{2,𝑝}-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients*, Trans. Amer. Math. Soc.**336**(1993), no. 2, 841–853. MR**1088476**, 10.1090/S0002-9947-1993-1088476-1**[GT]**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190****[Gi]**M. Giaquinta and J. Nečas,*On the regularity of weak solutions to nonlinear elliptic systems of partial differential equations*, J. Reine Angew. Math.**316**(1980), 140–159. MR**581329****[H]**Qingbo Huang,*Estimates on the generalized Morrey spaces 𝐿^{2,𝜆}ᵩ and 𝐵𝑀𝑂_{𝜓} for linear elliptic systems*, Indiana Univ. Math. J.**45**(1996), no. 2, 397–439. MR**1414336**, 10.1512/iumj.1996.45.1968**[K]**N. V. Krylov,*Nonlinear elliptic and parabolic equations of the second order*, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR**901759****[N]**N. Nadirashvili,*Nonclassical Solutions to Fully Nonlinear Elliptic Equations*, Preprint.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35J60,
35B65

Retrieve articles in all journals with MSC (2000): 35J60, 35B65

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

Email:
qhuang@math.utexas.edu, qhuang@math.wright.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06503-6

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