# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar
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