Electron. J. Diff. Eqns., Vol. 2006(2006), No. 124, pp. 1-12.

Convexity of level sets for solutions to nonlinear elliptic problems in convex rings

Paola Cuoghi, Paolo Salani

We find suitable assumptions for the quasi-concave envelope $u^*$ of a solution (or a subsolution) $u$ of an elliptic equation $F(x,u,\nabla u,D^2u)=0$ (possibly fully nonlinear) to be a viscosity subsolution of the same equation. We apply this result to study the convexity of level sets of solutions to elliptic Dirichlet problems in a convex ring $\Omega=\Omega_0\setminus\overline\Omega_1$.

Submitted June 23, 2005. Published October 11, 2006.
Math Subject Classifications: 35J25, 35J65.
Key Words: Elliptic equations; convexity of level sets; quasi-concave envelope.

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Paola Cuoghi
Dipt. di Matematica Pura e Applicata
Universitá degli Studi di Modena e Reggio Emilia
via Campi 213/B, 41100 Modena -Italy
email: pcuoghi@unimo.it
Paolo Salani
Dipt. di Matematica "U. Dini"
viale Morgagni 67/A, 50134 Firenze, Italy
email: salani@math.unifi.it

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