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), 30953115
MSC (2000):
Primary 35R35, 31B20
Published electronically:
April 3, 2002
MathSciNet review:
1897393
Fulltext PDF Free Access
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Abstract: In this paper, we first construct ``viscosity'' solutions (in the CrandallLions 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 AlexandroffBakelman 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 787121082
Email:
caffarel@math.utexas.edu
J. Salazar
Affiliation:
CMAF–University of Lisbon, Av. Gama Pinto 2, 1649003 Lisbon, Portugal
Email:
salazar@alf1.cii.fc.ul.pt
DOI:
http://dx.doi.org/10.1090/S000299470203009X
PII:
S 00029947(02)03009X
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
