On the Stability of Solutions to a Phase Transition Model

Abstract

In this paper, we study the functional

$$\displaystyle{J_{\epsilon }(u):= \frac{\epsilon ^{2}} {2}\int _{0}^{1}\vert u_{ x}\vert ^{2}dx +\int _{ 0}^{1}F(u)dx,\quad u \in W^{1,2}(0,1)}$$

under suitable assumptions on $F: \mathbb{R}\longrightarrow \mathbb{R}$. This functional represents the total free energy of a phase transition model. In nature, energy seeks a minimum. As such, we can see that the first integral wants the slope of u to be flat whereas the second integral is pushing towards the minimum values of F.