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

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.