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On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two

Dolbeault, Jean; Monneau, Régis (2003), On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 2, 1, p. 181-197

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Article accepté pour publication ou publié
External document link
http://www.numdam.org/item?id=ASNSP_2003_5_2_1_181_0
Date
2003
Journal name
Annali della Scuola Normale Superiore di Pisa. Classe di Scienze
Volume
2
Number
1
Publisher
Scuola Normale Superiore di Pisa
Pages
181-197
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Author(s)
Dolbeault, Jean cc
Monneau, Régis
Abstract (EN)
In this paper we establish a Liouville type theorem for fully nonlinear elliptic equations related to a conjecture of De Giorgi in R2. We prove that if the level lines of a solution have bounded curvature, then these level lines are straight lines. As a consequence, the solution is one-dimensional. The method also provides a result on free boundary problems of Serrin type.
Subjects / Keywords
Fully nonlinear elliptic equations; semilinear elliptic equations; Quasi-linear elliptic equations; One-dimensional symmetry; Conjecture of De Giorgi; Liouville Theorem; Bernstein’s Problem; Serrin’s Problem