Some remarks on a conjecture of De Giorgi
type: Item
instance of: scholarly article
title: Some remarks on a conjecture of De Giorgi
zbMATH Open document ID: 0938.35057
author: Alberto Farina
published in: Calculus of Variations and Partial Differential Equations
publication date: 2000-06-22T00:00:00Z
review text: This article concerns one-dimensional solutions to the \(p\)-Laplacian equation \[ \text{div} \bigl(|\nabla u|^{p-2}\nabla u\bigr)= f'(u)\text{ in }\mathbb{R}^n,\tag{1} \] where \(n\geq 2\), \(p\geq 2\), \(f\in C^2(\mathbb{R})\), \(f\) is nonnegative with (at least) two zeros \(z_\pm\), \(z_-<z_+\), \(f'(t)\) has exactly one zero in \((z_-,z_+)\), and \(f(t)=O (|t-z|^p)\) near any zero \(z\) of \(f\). For a bounded domain \(\Omega\) in \(\mathbb{R}^{n-1}\), define \(S_\Omega\) to be the set of all functions \(v\in W^{1,p}_{\text{loc}}(\mathbb{R} \times\Omega)\) such that \(v(x)\to z_{\pm}\) as \(x_1\to \pm\infty\) uniformly. The main theorem states that a solution \(u\in W^{1,p}_{\text{loc}}(\mathbb{R}^n)\cap L^\infty (\mathbb{R}^n)\) to (1) obtained as a minimizer for an energy functional associated with (1) over \(S_\Omega\) must be a one-dimensional function of type \(u(x)=g(x_1+a)\) for some constant \(a\in\mathbb{R}\) and a unique function \(g\in C^3(\mathbb{R})\) with \(g(x_1)\in(z_-,z_+)\), \(g'(x_1)>0\) for all \(x_1\in\mathbb{R}\). The proof employs results/techniques of \textit{L. Caffarelli}, \textit{N. Garofalo}, and \textit{F. Segala} [Commun. Pure Appl. Math. 47, 1457-1473 (1994; Zbl 0819.35016)] and \textit{G. Carbou} [Ann. Inst. H. Poincaré, Anal. Non Linéaire 12, 305-318 (1995; Zbl 0835.35045)]. The first theorem of the present type was obtained by Modica and Mortola in the case \(n=2\), \(p=2\), \(4f(t)= (t^2-1)^2\) (i.e., scalar Ginzburg-Landau equation) [\textit{L. Modica}, \textit{St. Mortola}, Boll. Unione Mat. Ital., IV. Ser. B17, 614-622 (1980; Zbl 0448.35044)], confirming a conjecture of De Giorgi in 1978.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 1299953
Wikidata QID: Q123278575
reviewed by: Charles A. Swanson
MaRDI profile type: Publication
full work available at URL: s005260050124
OpenAlex ID: W2055930776
DOI: 10.1007/S005260050124
Recommended article: Some remarks related to De Giorgi’s conjecture
Recommended article: On a conjecture of De Giorgi and some related problems
Recommended article: On some problems related to De Giorgi's conjecture
Recommended article: On De Giorgi’s conjecture and beyond
Recommended article: On a modified conjecture of De Giorgi
Recommended article: A Note on Giuga's Conjecture
Recommended article: On De Giorgi's conjecture: recent progress and open problems
Recommended article: On Giuga's conjecture
Recommended article: On a conjecture of Degos
Recommended article: Some remarks on Garay's conjecture
description: scientific article; zbMATH DE number 1299953