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

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
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