Advances in Differential Equations

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

Robert L. Jerrard

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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

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

First available in Project Euclid: 24 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Jerrard, Robert L. Singular limits of scalar Ginzburg-Landau equations with multiple-well potentials. Adv. Differential Equations 2 (1997), no. 1, 1--38.

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