2010/04/13 by Andrew Lorent, Lorent, Andrew
Computer Science · Materials Science · Mathematics · #35J30 #49N99 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Solidification and crystal growth phenomena #math.AP #msc:35J30 #msc:49N99
paper · pdf · doi:10.48550/arxiv.1004.2274
16 pages, 1 figure
openalex publication_date 2010/04/13 · arxiv created 2011/05/16 · arxiv updated 2011/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω⊂ R2 the functional is Iε(u)=1/2∫Ω ε-1|1-|Du|2|2+ε|D2 u|2 where u belongs to the subset of functions in W2,20(Ω) whose gradient (in the sense of trace) satisfies Du(x)⋅ ηx=1 where ηx is the inward pointing unit normal to ∂ Ω at x. In Jabin, Otto, Perthame characterized a class of functions which includes all limits of sequences un∈ W2,20(Ω) with Iεn(un)→ 0 as εn→ 0. A corollary to their work is that if there exists such a sequence (un) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u:=limn→ ∞ un =dist(⋅,∂Ω). Recently we provided a quantitative generalization of this corollary over the space of convex domains using `compensated compactness' inspired calculations originating from the proof of coercivity of Iε by DeSimone, Muller, Kohn, Otto. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where Ω=B1(0) without the requiring the trace condition on Du.