On a Modified Conjecture of De Giorgi

Röger, Matthias; Schätzle, Reiner

doi:10.1007/s00209-006-0002-6

Cited by 124 publications

(163 citation statements)

References 10 publications

(6 reference statements)

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“…for recent progress on this topic see [47], [49], [58]. We have focused on the singular limit ε → 0, but there are many other issues in the analysis of thermally-activated transitions.…”

Section: Ginzburg-landaumentioning

confidence: 99%

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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“…for recent progress on this topic see [47], [49], [58]. We have focused on the singular limit ε → 0, but there are many other issues in the analysis of thermally-activated transitions.…”

Section: Ginzburg-landaumentioning

confidence: 99%

Energy-driven pattern formation

Kohn¹

2007

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

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“…2.2) so that the limit measures obtained as the limit of (AC) have all the measure-theoretic properties satisfied by the varifold solutions constructed by Brakke in [5]. For d = 2, 3, the second author [24] noticed that one can give a very short and unified proof of [14] and [27] by utilizing the results by Röger and Schätzle [23]. We use the latter method in the present paper instead of that of Ilmanen.…”

Section: Introductionmentioning

confidence: 97%

On the existence of mean curvature ﬂow with transport term

Li¹,

Sato²,

Tonegawa³

2010

Interfaces Free Bound.

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We prove the global-in-time existence of weak solution for a hypersurface evolution problem where the velocity is the sum of the mean curvature and arbitrarily given non-smooth vector field in a suitable Sobolev space. The approximate solution is obtained by the Allen-Cahn equation with transport term. By establishing the density ratio upper bound on the phase boundary measure it is shown that the limiting surface moves with the desired velocity in the sense of Brakke.

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“…The contributions on this point [3,20,22,24,26] culminated with the proof by Röger and Schätzle [24] in space dimensions N = 2, 3 and, independently, by Nagase and Tonegawa [22] in dimension N = 2, that the result holds true for smooth sets. More precisely, given u = 1 E the characteristic function of a set E ∈ C 2 (Ω), and u ε converging to u in L 1 (Ω) with a uniform control of the approximating perimeter…”

Section: Introductionmentioning

confidence: 88%