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1997 Singular limits of scalar Ginzburg-Landau equations with multiple-well potentials
Robert L. Jerrard
Adv. Differential Equations 2(1): 1-38 (1997).

Abstract

We characterize the limiting behavior of scalar phase-field equations with infinitely many potential wells as the density of potential wells tends to infinity. An example of such a family of equations is \[ \mathcal{u}^e_t = \Delta\mathcal{u}^e - \frac1{\epsilon^{1+\alpha}}W'(\frac{\mathcal{u}^e}{\epsilon^{1-\alpha}}), \] where $W$ is a periodic function. We prove that solutions of the above equation converge to solutions of the Mean Curvature partial differential equation for a range of positive values of the parameter $\alpha$, and we also determine the limiting equation when $\alpha = 0$. We show that our techniques can be modified to apply to fully nonlinear equations and to other classes of infinite-well equations. We discuss some applications to questions of interaction between wave fronts in dynamic phase transitions.

Citation

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Robert L. Jerrard. "Singular limits of scalar Ginzburg-Landau equations with multiple-well potentials." Adv. Differential Equations 2 (1) 1 - 38, 1997.

Information

Published: 1997
First available in Project Euclid: 24 April 2013

zbMATH: 1023.35526
MathSciNet: MR1424762

Subjects:
Primary: 35Q55
Secondary: 35B40

Rights: Copyright © 1997 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.2 • No. 1 • 1997
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