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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: April 3, 2002
MathSciNet review: 1897393
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Abstract: In this paper, we first construct ``viscosity'' solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form

\begin{displaymath}F(D^{2} u,x) = g(x,u) \text{ on } \{\vert\nabla u\vert \ne 0\}\end{displaymath}

In fact, viscosity solutions are surprisingly weak. Since candidates for solutions are just continuous, we only require that the ``test'' polynomials $P$ (those tangent from above or below to the graph of $u$ at a point $x_{0}$) satisfy the correct inequality only if $\vert\nabla P (x_{0})\vert \ne 0$. That is, we simply disregard those test polynomials for which $\vert\nabla P (x_{0})\vert = 0$.

Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for $F = \Delta $, $g = cu$($c>0$) 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: http://dx.doi.org/10.1090/S0002-9947-02-03009-X
PII: S 0002-9947(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