Minimizing sequences for nonconvex functionals, phase transitions and singular perturbations

Abstract

We study minimizers of the singularly perturbed functional I ^ε(u) = f ⁰₁ {ε⁶Ěu ^Χ_{2 Χ} + (u ^Χ₂ − 1)² + u ²} dΧ, subject to u(0) = u(1) = 0. For ε = 0 no minimizers exist and we show that for ε > 0, small, the minimizer is nearly periodic with period proportional to ε. Connections with solid-solid phase transitions in crystals are indicated.