Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory

By John E. Hutchinson, Yoshihiro Tonegawa. 2000

219 citations recorded by the source corpus.

Subfields: Atmospheric Science, Condensed Matter Physics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/dd51efbbcf3cbb47d275f56ddffd45301108bf55

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In Atmospheric Science

In Condensed Matter Physics

In Materials Chemistry

Citations in the corpus

MULTIPLE-END SOLUTIONS TO THE ALLEN-CAHN EQUATION IN R2 (2010, unknown): “Our results apply as well in the most general setting but the presentation becomes somewhat cumbersome.”
A phase field formulation of the Willmore problem (2005, background): “The asymptotic behaviour of minimizers of (3) and (6) has been studied in [9,10], and that of general critical points in [11].”
On Stable Critical Points for a Singular Perturbation Problem (2003, background): “For general critical points, Hutchinson and the author showed that the interfaces of any critical points with finite energy and with or without volume constraint converge to a locally constant mean curvature hypersurface with possible integer multiplicities [8].”
Min-max for phase transitions and the existence of embedded minimal hypersurfaces (2015, methodology): “[15, 38, 40] Suppose that Assumptions A and B from 3.”
Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces. (2009, result): “A similar result holds for critical points not necessarily minimizers, see [26].”
The Toda System and Clustering Interfaces in the Allen–Cahn equation (2008, background): “In [17,35,40] the authors have addressed the issue of understanding asymptotic behavior of interfaces for general families of solutions of (1.”
Gamma-convergence of gradient flows on Hilbert and metric spaces and applications (2011, unknown): “Then for a subsequence, as ε → 0, με → μ for almost all t, where from [39, 19], μ are integer rectifiable and have a (generalized) mean curvature.”
The Allen–Cahn equation on closed manifolds (2016, background): “rsurface on M, i.e. a critical point of the area functional. Several formal instances of this statement exist in the literature depending on the variational characteristics of the solutions, e.g. see [46, 58, 49, 33, 61]. To state a particular case which is in the interest of the present work, remem”
On a Modified Conjecture of De Giorgi (2006, unknown): “The major challenge to derive these results is the control of the so called discrepancy measures , which is much more delicate here than in [HT00], [T02] and which requires a careful analysis and some additional arguments.”