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 \cite{KuRo}), 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.
