Advances in Differential Equations

Singular limits of scalar Ginzburg-Landau equations with multiple-well potentials

Robert L. Jerrard

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Adv. Differential Equations Volume 2, Number 1 (1997), 1-38.

Dates
First available in Project Euclid: 24 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366809227

Mathematical Reviews number (MathSciNet)
MR1424762

Zentralblatt MATH identifier
1023.35526

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35B40: Asymptotic behavior of solutions

Citation

Jerrard, Robert L. Singular limits of scalar Ginzburg-Landau equations with multiple-well potentials. Adv. Differential Equations 2 (1997), no. 1, 1--38. https://projecteuclid.org/euclid.ade/1366809227.


Export citation