# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Solutions of fully nonlinear elliptic equations with patches of zero gradient: Existence, regularity and convexity of level curvesHTML articles powered by AMS MathViewer

by L. Caffarelli and J. Salazar
Trans. Amer. Math. Soc. 354 (2002), 3095-3115 Request permission

## Abstract:

In this paper, we first construct “viscosity” solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form \begin{equation*}F(D^{2} u,x) = g(x,u)\ \text { on }\ \{|\nabla u| \ne 0\}\end{equation*} 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 $|\nabla P (x_{0})| \ne 0$. That is, we simply disregard those test polynomials for which $|\nabla P (x_{0})| = 0$. 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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• L. Caffarelli
• Affiliation: Department of Mathematics, University of Texas at Austin, RLM 8.100, Austin, Texas 78712-1082
• MR Author ID: 44175
• 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
• 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.