We give a new proof of Brakke's partial regularity theorem up to for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard's regularity theorem in the sense that the special time-independent case reduces to Allard's theorem
We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals-Cahn-Hilliard energy functionals with perturbation parameter ! 0 þ is supported by an embedded stable minimal hypersurface which in low dimensions has no singularities and in general dimensions has possibly a closed singular set of co-dimension f 7.This result was previously known in case the critical points are local minimizers of energy, in which case the limit-interface is locally area minimizing and its (normalized) multiplicity is 1 a.e.Our theorem uses earlier work of the first author establishing stability of the limitinterface as an integral varifold, and relies on a recent general theorem of the second author for its regularity conclusions in the presence of higher multiplicity.Y.T. is partially supported by JSPS Grant-in-aid for scientific research 21340033. Both authors wish to thank Mathematisches Forschungsinstitut Oberwolfach for providing the opportunity to initiate this joint research.Brought to you by | Purdue University Libraries Authenticated Download Date | 5/25/15 7:28 PM W ju e i j þ E e i ðu e i Þ e c for some c f 1 independent of i, then either (a) u e i ! 1 or u e i ! À1 locally uniformly in W, or (b) there exists an embedded smooth stable minimal hypersurface M of W such that after passing to a subsequence of fe i g without changing notation, for each fixed s A ð0; 1Þ, the interface regions fx A W : ju e i ðxÞj < sg converge locally in Hausdor¤ distance to M; furthermore, in case (b), the interior singular set sing M 1 ðMnMÞ X W is empty for 2 e n e 7, sing M is (at most) a discrete set for n ¼ 8 and sing M has Hausdor¤ dimension at most n À 8 for n f 9. This regularity result was known for the limit-interfaces corresponding to sequences fu e i g of energy minimizers since in that case the limit-interfaces are area-minimizing and the well known regularity theory for locally area minimizing currents is applicable. The new result in this paper is that the 192 Tonegawa and Wickramasekera, Stable phase interfaces in the van der Waals-Cahn-Hilliard theory Brought to you by | Purdue University Libraries Authenticated Download Date | 5/25/15 7:28 PM Tonegawa and Wickramasekera, Stable phase interfaces in the van der Waals-Cahn-Hilliard theory Brought to you by | Purdue University Libraries Authenticated Download Date | 5/25/15 7:28 PM
We study a singular perturbation problem arising in the scalar two-phase field model. Assuming only the stability of the critical points for ε-problems, we show that the interface regions converge to a generalized stable minimal hypersurface as ε → 0. The limit has an L 2 generalized second fundamental form and the stability condition is expressed in terms of the corresponding inequalities satisfied by stable minimal hypersurfaces. We show that the limit is a finite number of lines with no intersections when the dimension of the domain is 2.
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