A compactness result in the gradient theory of phase transitions

We examine the singularly perturbed variational problem E ε(ψ) = ∫ ε -1(1 - |∇ψ| 2) 2 + ε|∇∇ψ| 2 in the plane. As ε → 0, this functional favours |∇ψ| = 1 and penalizes singularities where |∇∇ψ| concentrates. Our main result is a compactness theorem: if {E ε(ψ ε)} ε↓0 is uniformly bounded, then {∇ψ ε} ε↓0 is compact in L 2. Thus, in the limit ε → 0, ψ solves the eikonal equation |∇ψ| = 1 almost everywhere. Our analysis uses 'entropy relations' and the 'div-curl lemma,' adopting Tartar's approach to the interaction of linear differential equations and nonlinear algebraic relations.