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

HTML articles powered by AMS MathViewer

- by L. Caffarelli and J. Salazar
- Trans. Amer. Math. Soc.
**354**(2002), 3095-3115 - DOI: https://doi.org/10.1090/S0002-9947-02-03009-X
- Published electronically: April 3, 2002
- PDF | 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

- Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman,
*Variational problems with two phases and their free boundaries*, Trans. Amer. Math. Soc.**282**(1984), no. 2, 431–461. MR**732100**, DOI 10.1090/S0002-9947-1984-0732100-6 - Luis A. Caffarelli,
*Interior a priori estimates for solutions of fully nonlinear equations*, Ann. of Math. (2)**130**(1989), no. 1, 189–213. MR**1005611**, DOI 10.2307/1971480 - Luis A. Caffarelli and Xavier Cabré,
*Fully nonlinear elliptic equations*, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR**1351007**, DOI 10.1090/coll/043 - L. Caffarelli, M. G. Crandall, M. Kocan, and A. Swięch,
*On viscosity solutions of fully nonlinear equations with measurable ingredients*, Comm. Pure Appl. Math.**49**(1996), no. 4, 365–397. MR**1376656**, DOI 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.3.CO;2-V - Luis A. Caffarelli, Lavi Karp, and Henrik Shahgholian,
*Regularity of a free boundary with application to the Pompeiu problem*, Ann. of Math. (2)**151**(2000), no. 1, 269–292. MR**1745013**, DOI 10.2307/121117 - A.-P. Calderón and A. Zygmund,
*Local properties of solutions of elliptic partial differential equations*, Studia Math.**20**(1961), 171–225. MR**136849**, DOI 10.4064/sm-20-2-181-225 - S. Jonathan Chapman,
*A mean-field model of superconducting vortices in three dimensions*, SIAM J. Appl. Math.**55**(1995), no. 5, 1259–1274. MR**1349309**, DOI 10.1137/S0036139994263665 - Charles M. Elliott, Reiner Schätzle, and Barbara E. E. Stoth,
*Viscosity solutions of a degenerate parabolic-elliptic system arising in the mean-field theory of superconductivity*, Arch. Ration. Mech. Anal.**145**(1998), no. 2, 99–127. MR**1664550**, DOI 10.1007/s002050050125 - Luis Escauriaza,
*$W^{2,n}$ a priori estimates for solutions to fully nonlinear equations*, Indiana Univ. Math. J.**42**(1993), no. 2, 413–423. MR**1237053**, DOI 10.1512/iumj.1993.42.42019 - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190**, DOI 10.1007/978-3-642-61798-0 - Hitoshi Ishii,
*On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs*, Comm. Pure Appl. Math.**42**(1989), no. 1, 15–45. MR**973743**, DOI 10.1002/cpa.3160420103 - Bernhard Kawohl,
*Rearrangements and convexity of level sets in PDE*, Lecture Notes in Mathematics, vol. 1150, Springer-Verlag, Berlin, 1985. MR**810619**, DOI 10.1007/BFb0075060 - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095**

## Bibliographic 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