Logarithmic lower bounds for Néel walls
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 \((-\Delta)^{1/2}\).
scientific article; zbMATH DE number 2113882