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November/December 2013 Heteroclinic solutions of the prescribed curvature equation with a double-well potential
Denis Bonheurei, Franco Obersnel, Pierpaolo Omari
Differential Integral Equations 26(11/12): 1411-1428 (November/December 2013).

Abstract

We prove the existence of heteroclinic solutions of the prescribed curva\-ture equation \begin{equation*} \Big( u'/{ \sqrt{1+{u'}^2}}\Big)' = a(t)V'(u), \end{equation*} where $V$ is a double-well potential and $a$ is asymptotic to a positive periodic function. Such an equation is meaningful in the modeling theory of reaction-diffusion phenomena which feature saturation at large value of the gradient. According to numerical simulations (see [5]), the graph of the interface between the stable states of a two-phase system may exhibit discontinuities. We provide a theoretical justification of these simulations by showing that an optimal transition between the stable states arises as a minimum of the associated action functional in the space of locally bounded variation functions. In very simple cases, such an optimal transition naturally displays jumps.

Citation

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Denis Bonheurei. Franco Obersnel. Pierpaolo Omari. "Heteroclinic solutions of the prescribed curvature equation with a double-well potential." Differential Integral Equations 26 (11/12) 1411 - 1428, November/December 2013.

Information

Published: November/December 2013
First available in Project Euclid: 4 September 2013

zbMATH: 1313.34132
MathSciNet: MR3129016

Subjects:

Rights: Copyright © 2013 Khayyam Publishing, Inc.

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Vol.26 • No. 11/12 • November/December 2013
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