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):=epsilon(2)/2 integral(1)(0)vertical bar u(x)vertical bar(2)dx + integral F-1(0)(u)dx,u is an element of W-1,W-2(0,1), (1) where F : R -> 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 < alpha < 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 is an element of 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.