Differential and Integral Equations

Heteroclinic solutions of the prescribed curvature equation with a double-well potential

Denis Bonheurei, Franco Obersnel, and Pierpaolo Omari

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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.

Article information

Source
Differential Integral Equations, Volume 26, Number 11/12 (2013), 1411-1428.

Dates
First available in Project Euclid: 4 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1378327433

Mathematical Reviews number (MathSciNet)
MR3129016

Zentralblatt MATH identifier
1313.34132

Subjects
Primary: 34C37: Homoclinic and heteroclinic solutions 37C29: Homoclinic and heteroclinic orbits 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 70K44: Homoclinic and heteroclinic trajectories 35J93: Quasilinear elliptic equations with mean curvature operator 76B45: Capillarity (surface tension) [See also 76D45] 76D45: Capillarity (surface tension) [See also 76B45]

Citation

Bonheurei, Denis; Obersnel, Franco; Omari, Pierpaolo. Heteroclinic solutions of the prescribed curvature equation with a double-well potential. Differential Integral Equations 26 (2013), no. 11/12, 1411--1428. https://projecteuclid.org/euclid.die/1378327433


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