Scaling limits and regularity results for a class of Ginzburg–Landau systems
Robert L. Jerrard
Department of Mathematics University of Illinois, 1409 West Green Street Urbana, IL 61801 USAHalil Mete Soner
Department of Mathematics Carnegie Mellon University Pittsburgh, PA 15213, USA
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Abstract
We study a class of parabolic systems which includes the Ginzburg–Landau heat flow equation,
for , as well as some natural quasilinear generalizations for functions taking values in , .
We prove that for solutions of the general system, the limiting support as of the energy measure is a codimension manifold which evolves via mean curvature.
We also establish some local regularity results which hold uniformly in . In particular, we establish a small-energy regularity theorem for the general system, and we prove a stronger regularity result for the usual Ginzburg–Landau equation on .
Résumé
Nous étudions une classe de systèmes paraboliques qui comprennent l’equation de chaleur Ginzburg–Landau,
pour , ainsi que des généralisations quasilinéaires pour des fonctions prenant leurs valeurs dans , .
Nous prouvons que, pour les solutions du système général, le support limite (lorsque ) de mèsure d’énergie est une varieté de codimension qui évolue selon sa courbure moyenne.
Nous établissons en addition quelques resultats de regularité locale, qui sont valides uniformement en .
Cite this article
Robert L. Jerrard, Halil Mete Soner, Scaling limits and regularity results for a class of Ginzburg–Landau systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), no. 4, pp. 423–466
DOI 10.1016/S0294-1449(99)80024-9