Simmetria delle soluzioni di equazioni ellittiche semilineari in \( \mathbb{R}^{N} \)

Farina, Alberto

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 10 (1999) no. 4, pp. 255-265.

Riassunto (VO)

Nella prima parte di questa Nota si dimostrano dei risultati di simmetria unidimensionale e radiale per le soluzioni di \( \Delta u + f (u) = 0 \) in \( \mathbb{R}^{N} \). Questi risultati sono legati a due congetture (De Giorgi, 1978 e Gibbons, 1994) riguardanti la classificazione delle soluzioni dell’equazione \( \Delta u + u(1 - u^{2}) = 0 \) in \( \mathbb{R}^{N} \). Si dimostra, in particolare, la seguente generalizzazione della congettura di Gibbons: se \( N > 1 \) e se l’insieme degli zeri di \( u \) è limitato nella direzione \( \nu \), allora \( u(x) = u_{0} (\nu \cdot x) \), ovvero, \( u \) è unidimensionale. Nella seconda parte si considerano le equazioni di reazione-convezione-diffusione del tipo \( a^{ij} (x) \theta_{ij} u + b^{i} (x) \theta_{i} u + f(x,u) = 0 \) in \( \mathbb{R}^{N} \) e si dimostrano dei risultati di monotonia e simmetria che, una volta combinati, conducono ad un’altra generalizzazione della congettura di Gibbons.

Abstract

In the first part of this Note we prove one-dimensional and radial symmetry results for solutions of \( \Delta u + f (u) = 0 \) in Simmetria delle soluzioni di equazioni ellittiche semilineari in \( \mathbb{R}^{N} \) . These results are connected with two conjectures (De Giorgi, 1978 and Gibbons, 1994) about the classification of solutions of the equation \( \Delta u + u(1 - u^{2}) = 0 \) in \( \mathbb{R}^{N} \). In particular we prove a stronger version of Gibbons' conjecture in any dimension \( N > 1 \), namely: if the set of zeros of \( u \) is bounded with respect to one direction, say \( \nu \) , then \( u \) is one-dimensional, i.e., \( u(x) = u_{0}(\nu \cdot x) \). In the second part we consider the reaction-convection-diffusion equations of type \( a^{ij} (x) \theta_{ij} u + b^{i} (x) \theta_{i} u + f(x,u) = 0 \) in \( \mathbb{R}^{N} \) and prove monotonicity and symmetry results which, when combined, lead to another stronger version of Gibbons’s conjecture in any dimension.

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