A compactness result in the gradient theory of phase transitions. (English) Zbl 0986.49009

DeSimone, Antonio; Müller, Stefan; Kohn, Robert V.; Otto, Felix

Proc. R. Soc. Edinb., Sect. A, Math. 131, No. 4, 833-844 (2001).

Summary: We examine the singularly perturbed variational problem \[ E_\varepsilon(\psi)= \int \varepsilon^{-1}(1- |\nabla\psi|^2)^2+ \varepsilon|\nabla\nabla \psi|^2 \] in the plane. As \(\varepsilon\to 0\), this functional favours \(|\nabla\psi|= 1\) and penalizes singularities where \(|\nabla\nabla\psi|\) concentrates. Our main result is a compactness theorem: If \(\{E_\varepsilon(\psi_\varepsilon)\}_{\varepsilon\downarrow 0}\) is uniformly bounded, then \(\{\nabla\psi_\varepsilon\}_{\varepsilon\downarrow 0}\) is compact in \(L^2\). Thus, in the limit \(\varepsilon\to 0\), \(\psi\) solves the eikonal equation \(|\nabla\psi|= 1\) 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.

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
74G65	Energy minimization in equilibrium problems in solid mechanics
74N99	Phase transformations in solids
76M30	Variational methods applied to problems in fluid mechanics

Keywords:

compensated compactness; Young measure; entropy; integral functional; phase transitions; singularly perturbed variational problem; eikonal equation