### Abstract

We examine the singularly perturbed variational problem E_{ε}(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 {E_{ε}(ψ_{ε})}_{ε↓0} is uniformly bounded, then {∇ψ_{ε}}_{ε↓0} is compact in L^{2}. 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 language | English (US) |
---|---|

Pages (from-to) | 833-844 |

Number of pages | 12 |

Journal | Royal Society of Edinburgh - Proceedings A |

Volume | 131 |

Issue number | 6 |

State | Published - 2001 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Royal Society of Edinburgh - Proceedings A*,

*131*(6), 833-844.

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

Research output: Contribution to journal › Article

*Royal Society of Edinburgh - Proceedings A*, vol. 131, no. 6, pp. 833-844.

}

TY - JOUR

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

AU - DeSimone, Antonio

AU - Müller, Stefan

AU - Kohn, Robert

AU - Otto, Felix

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.

UR - http://www.scopus.com/inward/record.url?scp=33748381840&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748381840&partnerID=8YFLogxK

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 -