A compactness result in the gradient theory of phase transitions

We examine the singularly perturbed variational problem E-is an element of (psi) = integral is an element of (-1) (1 - \ del psi \ (2))(2) + \ del del psi \ (2) in the plane. As epsilon --> 0, this functional favours \ del psi \ = 1 and penalizes singularities where \ del del psi \ concentrates. Our main resul t is a compactness theorem: if {E-epsilon(psi (epsilon))}(epsilon down arro w0) is uniformly bounded, then {del psi (epsilon)}(epsilon down arrow0) is compact in L-2. Thus, in the limit epsilon --> 0, psi solves the eikonal eq uation \ del psi \ = 1 almost everywhere. Our analysis uses 'entropy relati ons' and the 'div-curl lemma,' adopting Tartar's approach to the interactio n of linear differential equations and nonlinear algebraic relations.