Solutions of fully nonlinear elliptic equations with patches of zero gradient: Existence, regularity and convexity of level curves

Authors:
L. Caffarelli and J. Salazar

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3095-3115

MSC (2000):
Primary 35R35, 31B20

DOI:
https://doi.org/10.1090/S0002-9947-02-03009-X

Published electronically:
April 3, 2002

MathSciNet review:
1897393

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we first construct ``viscosity'' solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form

In fact, viscosity solutions are surprisingly weak. Since candidates for solutions are just continuous, we only require that the ``test'' polynomials (those tangent from above or below to the graph of at a point ) satisfy the correct inequality only if . That is, we simply disregard those test polynomials for which .

Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for , () and constant boundary conditions on a convex domain, to prove that there is only one convex patch.

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Additional Information

**L. Caffarelli**

Affiliation:
Department of Mathematics, University of Texas at Austin, RLM 8.100, Austin, Texas 78712-1082

Email:
caffarel@math.utexas.edu

**J. Salazar**

Affiliation:
CMAF–University of Lisbon, Av. Gama Pinto 2, 1649-003 Lisbon, Portugal

Email:
salazar@alf1.cii.fc.ul.pt

DOI:
https://doi.org/10.1090/S0002-9947-02-03009-X

Keywords:
Viscosity solutions,
free boundary problems,
regularity

Received by editor(s):
May 15, 2000

Published electronically:
April 3, 2002

Additional Notes:
L. Caffarelli was supported by NSF grant DMS 9714758.

J. Salazar was partially supported by FCT Praxis/2/2.1/MAT/124/94, and also thanks the Mathematics Department of the University of Texas at Austin for its warm hospitality.

Article copyright:
© Copyright 2002
American Mathematical Society