Action minimization and sharp‐interface limits for the stochastic Allen‐Cahn equation

Kohn, Robert V.; Otto, Félix; Reznikoff, Maria G.; Vanden‐Eijnden, Eric

doi:10.1002/cpa.20144

Cited by 77 publications

(110 citation statements)

References 32 publications

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“…For problems with the complexity of micromagnetics or martensitic phase transformation this question remains open. However for the simpler case of a scalar Ginzburg-Landau model there has been some progress [42], [43].…”

Section: Ginzburg-landaumentioning

confidence: 99%

“…In two space dimensions the problem has been studied numerically in [24] and analytically in [42], but a complete analysis is still lacking. The anticipated answer is similar to the one-dimensional case, except that (i) if the seeds are points rather than lines then their nucleation cost is negligible; and (ii) if a boundary moves via motion by mean curvature then its propagation cost is negligible.…”

Section: Ginzburg-landaumentioning

confidence: 99%

“…We shall explain in Section 4.1 how large deviation theory leads to the minimization of action rather than energy. Then, in Section 4.2, we discuss the singular limit of action minimization for the Ginzburg-Landau functional [42].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Energy-driven pattern formation

Kohn¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. Many physical systems can be modelled by nonconvex variational problems regularized by higher-order terms. Examples include martensitic phase transformation, micromagnetics, and the Ginzburg-Landau model of nucleation. We are interested in the singular limit, when the coefficient of the higher-order term tends to zero. Our attention is on the internal structure of walls, and the character of microstructure when it forms. We also study the pathways of thermally-activated transitions, modeled via the minimization of action rather than energy. Our viewpoint is variational, focusing on matching upper and lower bounds. Mathematics Subject Classification (2000). Primary 49-02, 82-02; Secondary 74N15, 82B24, 82D40.

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Section: Ginzburg-landaumentioning

confidence: 99%

Section: Ginzburg-landaumentioning

confidence: 99%

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Energy-driven pattern formation

Kohn¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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“…• The upper bound necessary for the Gamma-convergence was formally proved [17] by the construction of good 'recovery sequences'.…”

Section: Introductionmentioning

confidence: 99%

“…• The lower bound was proved in [17] for sequences (u ε ) ε>0 such that the associated 'energy measures' have equipartitioned energy and single multiplicity as ε → 0.…”

Section: Introductionmentioning

confidence: 99%

The Allen–Cahn action functional in higher dimensions

Mugnai¹,

Röger²

2008

Interfaces Free Bound.

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The Allen-Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen-Cahn equation. Formal calculations suggest a reduced action functional in the sharp interface limit. We prove the corresponding lower bound in two and three space dimensions. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the limit of diffuse surface area measures and introduce a generalized velocity and generalized reduced action functional in a class of evolving measures.

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Quantitative Differentiation: A General Formulation

Cheeger

2012

Comm Pure Appl Math

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Let dx denote Lebesgue measure on R n . With respect to the measure C D r 1 dr dx on the collection of balls B r .x/ R n , the subcollection of balls B r .x/ B 1 .0/ has infinite measure. Let f W B 1 .0/ ! R have bounded gradient, jrf j Ä 1. For any B r .x/ B 1 .0/ there is a natural scale-invariant quantity that measures the deviation of f j B r .x/ from being an affine linear function. The most basic case of quantitative differentiation (due to Peter Jones) asserts that for all > 0, the measure of the collection of balls on which the deviation from linearity is is finite and controlled by , independent of the particular function f . The purpose of this paper is to explain the sense in which this model case is actually a particular instance of a general phenomenon that is present in many different geometric/analytic contexts. Essentially, in each case that fits the framework, to prove the relevant quantitative differentiation theorem, it suffices to verify that the family of relative defects is monotone and coercive. We indicate one recent application to theoretical computer science and others to partial regularity theory in geometric analysis and nonlinear PDEs.

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Action minimization and sharp‐interface limits for the stochastic Allen‐Cahn equation

Cited by 77 publications

References 32 publications

Energy-driven pattern formation

Energy-driven pattern formation

The Allen–Cahn action functional in higher dimensions

Quantitative Differentiation: A General Formulation

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