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

Farina, Alberto

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 685-692

Résumé

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.