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- W4301030096 abstract "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 $Omegasubset R^2$ the functional is $I_{epsilon}(u)=1/2int_{Omega} epsilon^{-1}|1-|Du|^2|^2+epsilon|D^2 u|^2$ where $u$ belongs to the subset of functions in $W^{2,2}_{0}(Omega)$ whose gradient (in the sense of trace) satisfies $Du(x)cdot eta_x=1$ where $eta_x$ is the inward pointing unit normal to $partial Omega$ at $x$. In Jabin, Otto, Perthame characterized a class of functions which includes all limits of sequences $u_nin W^{2,2}_0(Omega)$ with $I_{epsilon_n}(u_n)to 0$ as $epsilon_nto 0$. A corollary to their work is that if there exists such a sequence $(u_n)$ for a bounded domain $Omega$, then $Omega$ must be a ball and (up to change of sign) $u:=lim_{nto infty} u_n =mathrm{dist}(cdot,partialOmega)$. 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 $I_{epsilon}$ 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 $Omega=B_1(0)$ without the requiring the trace condition on $Du$." @default.
- W4301030096 created "2022-10-04" @default.
- W4301030096 creator A5087718408 @default.
- W4301030096 date "2010-04-13" @default.
- W4301030096 modified "2023-09-30" @default.
- W4301030096 title "A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity" @default.
- W4301030096 doi "https://doi.org/10.48550/arxiv.1004.2274" @default.
- W4301030096 hasPublicationYear "2010" @default.
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