Continua of local minimizers in a non-smooth model of phase transitions

In this paper, we study critical points of the functional $$J_{\epsilon}(u):=\frac{\epsilon^2}{2} \int\limits_0^1|u_x|^2{\rm {d}}x+\int\limits_0^1F(u){\rm {d}}x, u \,\in W^{1,2}{(0,1)}, \quad \quad \quad(1)$$ where ${F : \mathbb {R}\rightarrow \mathbb {R}}$ is assumed to be a double-well potential. This functional represents the total free energy in models of phase transition and allows for the study of interesting phenomena such as slow dynamics. In particular, we consider a non-classical choice for F modeled on ${F(u)=|1-u^2|^{\alpha}}$ where 1?<????<?2. The discontinuity in F??? at ±1 leads to the existence of multiple continua of critical points that are not present in the classical case ${F \in C^2}$ . We prove that the interior points of these continua are local minima. The energy of these local minimizers is strictly greater than the global minimum of ${J_{\epsilon}}$ . In particular, the existence of these continua leads to an alternative explanation for the slow dynamics observed in phase transition models.