One-dimensional symmetry for solutions of quasilinear equations in ${\mathbb{R}}^2$

Farina, Alberto. "One-dimensional symmetry for solutions of quasilinear equations in $\mathbb{R}^2$." Bollettino dell'Unione Matematica Italiana 6-B.3 (2003): 685-692. <http://eudml.org/doc/195614>.

@article{Farina2003,
abstract = {In this paper we consider two-dimensional quasilinear equations of the form $\text\{div\}(a(|\nabla u|) \nabla u)+ f(u)=0$ and study the properties of the solutions u with bounded and non-vanishing gradient. Under a weak assumption involving the growth of the argument of $\nabla u $(notice that $\text\{arg\}(\nabla u)$ is a well-defined real function since $|\nabla u|> 0$ on $\mathbb\{R\}^\{2\}$) we prove that $u$ is one-dimensional, i.e., $u= u(\nu \cdot x)$ for some unit vector $\nu$. As a consequence of our result we obtain that any solution $u$ having one positive derivative is one-dimensional. This result provides a proof of a conjecture of E. De Giorgi in dimension 2 in the more general context of the quasilinear equations. In particular we obtain a new and simple proof of the classical De Giorgi's conjecture.},
author = {Farina, Alberto},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {685-692},
publisher = {Unione Matematica Italiana},
title = {One-dimensional symmetry for solutions of quasilinear equations in $\mathbb\{R\}^2$},
url = {http://eudml.org/doc/195614},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Farina, Alberto
TI - One-dimensional symmetry for solutions of quasilinear equations in $\mathbb{R}^2$
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/10//
PB - Unione Matematica Italiana
VL - 6-B
IS - 3
SP - 685
EP - 692
AB - In this paper we consider two-dimensional quasilinear equations of the form $\text{div}(a(|\nabla u|) \nabla u)+ f(u)=0$ and study the properties of the solutions u with bounded and non-vanishing gradient. Under a weak assumption involving the growth of the argument of $\nabla u $(notice that $\text{arg}(\nabla u)$ is a well-defined real function since $|\nabla u|> 0$ on $\mathbb{R}^{2}$) we prove that $u$ is one-dimensional, i.e., $u= u(\nu \cdot x)$ for some unit vector $\nu$. As a consequence of our result we obtain that any solution $u$ having one positive derivative is one-dimensional. This result provides a proof of a conjecture of E. De Giorgi in dimension 2 in the more general context of the quasilinear equations. In particular we obtain a new and simple proof of the classical De Giorgi's conjecture.
LA - eng
UR - http://eudml.org/doc/195614
ER -