On the limit $ p\rightarrow∞ $ of global minimizers for a $p$-Ginzburg–Landau-type energy

Yaniv Almog; Leonid Berlyand; Dmitry Golovaty; Itai Shafrir

doi:10.1016/j.anihpc.2012.12.013

JournalsaihpcVol. 30, No. 6pp. 1159–1174

On the limit $p \to \infty$ of global minimizers for a $p$ -Ginzburg–Landau-type energy

Yaniv Almog
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
Leonid Berlyand
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
Dmitry Golovaty
Department of Theoretical and Applied Mathematics, The University of Akron, Akron, OH 44325, USA
Itai Shafrir
Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel

$On the limit $ p\rightarrow∞ $ of global minimizers for a $p$-Ginzburg–Landau-type energy cover$

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Abstract

We study the limit $p \to \infty$ of global minimizers for a $p$ -Ginzburg–Landau-type energy

E_{p} (u) = \int_{R^{2}} ∣\nabla u ∣^{p} + \frac{1}{2} (1 - ∣ u ∣^{2})^{2} .

The minimization is carried over maps on $R^{2}$ that vanish at the origin and are of degree one at infinity. We prove locally uniform convergence of the minimizers on $R^{2}$ and obtain an explicit formula for the limit on $B (0, 2)$ . Some generalizations to dimension $N ⩾ 3$ are presented as well.

Cite this article

Yaniv Almog, Leonid Berlyand, Dmitry Golovaty, Itai Shafrir, On the limit $p \to \infty$ of global minimizers for a $p$ -Ginzburg–Landau-type energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 6, pp. 1159–1174

DOI 10.1016/J.ANIHPC.2012.12.013