TY - JOUR

T1 - A compactness result in the gradient theory of phase transitions

AU - DeSimone, Antonio

AU - Müller, Stefan

AU - Kohn, Robert V.

AU - Otto, Felix

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2001

Y1 - 2001

N2 - We examine the singularly perturbed variational problem Eε(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 {Eε(ψε)}ε↓0 is uniformly bounded, then {∇ψε}ε↓0 is compact in L2. 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.

AB - We examine the singularly perturbed variational problem Eε(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 {Eε(ψε)}ε↓0 is uniformly bounded, then {∇ψε}ε↓0 is compact in L2. 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.

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U2 - 10.1017/s030821050000113x

DO - 10.1017/s030821050000113x

M3 - Article

AN - SCOPUS:33748381840

VL - 131

SP - 833

EP - 844

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 6

ER -