1D symmetry for solutions of semilinear and quasilinear elliptic equations

Farina, Alberto; Valdinoci, Enrico

doi:10.1090/S0002-9947-2010-05021-4

$1$D symmetry for solutions of semilinear and quasilinear elliptic equations
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by Alberto Farina and Enrico Valdinoci PDF

Trans. Amer. Math. Soc. 363 (2011), 579-609 Request permission

Abstract:

Several new $1$D results for solutions of possibly singular or degenerate elliptic equations, inspired by a conjecture of De Giorgi, are provided. In particular, $1$D symmetry is proven under the assumption that either the profiles at infinity are $2$D, or that one level set is a complete graph, or that the solution is minimal or, more generally, $Q$-minimal.

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