January  2013, 33(1): 163-172. doi: 10.3934/dcds.2013.33.163

Continua of local minimizers in a quasilinear model of phase transitions

1. 

Department of Mathematics and Center N.T.I.S., University of West Bohemia, P.O. Box 314,306 14 Pilsen, Czech Republic

2. 

Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, United States

Received  July 2011 Revised  February 2012 Published  September 2012

In this paper we study critical points of the functional \begin{eqnarray*} J_{\epsilon}(u):= \frac{\epsilon^p}{p}\int_0^1|u_x|^pdx+\int_0^1F(u)dx, \; u∈w^{1,p}(0,1), \end{eqnarray*} where F:$\mathbb{R}$→$\mathbb{R}$ is assumed to be a double-well potential. This functional represents the total free energy in phase transition models. We consider a non-classical choice for $F$ modeled on $F(u)=|1-u^2|^{\alpha}$ where $1< \alpha < p$. This choice leads to the existence of multiple continua of critical points that are not present in the classical case $\alpha= p = 2$. We prove that the interior of these continua are local minimizers. The energy of these local minimizers is strictly greater than the global minimum of $J_{\epsilon}$. In particular, the existence of these continua suggests an alternative explanation for the slow dynamics observed in phase transition models.
Citation: Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163
References:
[1]

J. Carr and R. L. Pego, Metastable patterns in solutions of u t = ε2 $u_{x x}$ - f(u), Comm. Pure Appl. Math., 42 (1989), 523-576. doi: 10.1002/cpa.3160420502.

[2]

P. Drábek, A. Kufner and F. Nicolosi, "Quasilinear Elliptic Equations with Degenerations and Singularities," De Gruyter Series in Nonlinear Analysis and Applications 5, Walter de Gruyter, Berlin, New York, 1997.

[3]

P. Drábek, R. Manásevich and P. Takáč, Slow dynamics in a quasilinear model for phase transitions in one space dimension, In "Nonlinear Elliptic Partial Differential Equations," Workshop in celebration of Jean-Pierre Gossez's 65th birthday September 24, 2009, Brusels, Contemporary Mathematics Series, 540, 95-134.

[4]

P. Drábek and S. Robinson, Continua of local minimizers in a non-smooth model of phase transitions, Z. Angew. Math. Phys., 62 (2011), 609-622.

show all references

References:
[1]

J. Carr and R. L. Pego, Metastable patterns in solutions of u t = ε2 $u_{x x}$ - f(u), Comm. Pure Appl. Math., 42 (1989), 523-576. doi: 10.1002/cpa.3160420502.

[2]

P. Drábek, A. Kufner and F. Nicolosi, "Quasilinear Elliptic Equations with Degenerations and Singularities," De Gruyter Series in Nonlinear Analysis and Applications 5, Walter de Gruyter, Berlin, New York, 1997.

[3]

P. Drábek, R. Manásevich and P. Takáč, Slow dynamics in a quasilinear model for phase transitions in one space dimension, In "Nonlinear Elliptic Partial Differential Equations," Workshop in celebration of Jean-Pierre Gossez's 65th birthday September 24, 2009, Brusels, Contemporary Mathematics Series, 540, 95-134.

[4]

P. Drábek and S. Robinson, Continua of local minimizers in a non-smooth model of phase transitions, Z. Angew. Math. Phys., 62 (2011), 609-622.

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