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

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- by L. Caffarelli and J. Salazar PDF
- 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.## References

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## Additional Information

**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. - © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1897393