A compactness result in the gradient theory of phase transitions

Antonio DeSimone, Stefan Müller, Robert Kohn, Felix Otto

Research output: Contribution to journalArticle

Abstract

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 {Eεε)}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.

Original languageEnglish (US)
Pages (from-to)833-844
Number of pages12
JournalRoyal Society of Edinburgh - Proceedings A
Volume131
Issue number6
StatePublished - 2001

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Compactness
Differential equations
Entropy
Phase Transition
Phase transitions
Gradient
Eikonal Equation
Singularly Perturbed Problem
Curl
Variational Problem
Linear differential equation
Lemma
Singularity
Interaction
Theorem

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

A compactness result in the gradient theory of phase transitions. / DeSimone, Antonio; Müller, Stefan; Kohn, Robert; Otto, Felix.

In: Royal Society of Edinburgh - Proceedings A, Vol. 131, No. 6, 2001, p. 833-844.

Research output: Contribution to journalArticle

DeSimone, Antonio ; Müller, Stefan ; Kohn, Robert ; Otto, Felix. / A compactness result in the gradient theory of phase transitions. In: Royal Society of Edinburgh - Proceedings A. 2001 ; Vol. 131, No. 6. pp. 833-844.
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