Logarithmic lower bounds for Néel walls
type: Item
instance of: scholarly article
title: Logarithmic lower bounds for Néel walls
zbMATH Open document ID: 1054.78011
author: Christof Melcher
published in: Calculus of Variations and Partial Differential Equations
publication date: 2004-11-05T00:00:00Z
review text: Let \(\varepsilon\) be a positive small parameter. Let \(u_\varepsilon\) be a minimizer of the functional \({\mathcal I}_\varepsilon (m)=\varepsilon \| m\| ^2_{H^1}+\| u\| ^2_{H^{1/2}}+\| u\| ^2_{L^2}\), over the class of functions \(m=(u,v):{\mathbb R}\rightarrow {\mathbb S}^1\) satisfying \(u(0)=1\) and \(v(\pm\infty)=\pm 1\). A first result of the present paper concerns the behaviour of \(u_\varepsilon\). More precisely, it is showed that \(u_\varepsilon (x)\) behaves like \(\log x-\log \varepsilon\), provided that \(\varepsilon <x<1/\varepsilon\) and \(0<\varepsilon <1/4\). Next, it is considered the relaxed variational functional \({\mathcal I}_\varepsilon (u)=\varepsilon \| u\| ^2_{H^1}+\| u\| ^2_{H^{1/2}}+\| u\| ^2_{L^2}\). Let \(\tilde u_\varepsilon\) be the unique minimizer of \({\mathcal I}_\varepsilon (u)\) in the class of functions satisfying \(u(0)=1\). The main result of the paper asserts that \(\tilde u_\varepsilon\) is a pointwise lower bound. The idea of the proof is to deduce suitable pseudo-differential inequalities for the profiles to be compared. An essential ingredient is the symmetric convexity of comparison profiles, combined with a global maximum principle for the non-local field operator \((-\Delta)^{1/2}\).
reviewed by: Teodora-Liliana Rădulescu
Mathematics Subject Classification ID: 78A30
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 45G15
Mathematics Subject Classification ID: 49S05
zbMATH DE Number: 2113882
zbMATH Keywords: minimization problem
zbMATH Keywords: lower bound
zbMATH Keywords: Néel wall
zbMATH Keywords: variational principle
MaRDI profile type: Publication
full work available at URL: s00526-003-0253-6
OpenAlex ID: W1976874167
DOI: 10.1007/S00526-003-0253-6
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description: scientific article; zbMATH DE number 2113882