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05/09/2022, 09:45 — 10:30 — Anfiteatro Abreu Faro
Giorgio Fusco, Università degli Studi dellʼAquila

On the fine structure of minimizers in the Allen-Cahn theory of phase transitions

Let $u^\epsilon$ be a minimizer of the Allen-Cahn energy subjected to Dirichlet condition: \[J_\Omega^\epsilon(u^\epsilon)=\min_{u\vert_{\partial\Omega}=v_0^\epsilon}\int_\Omega\Big(\frac{\epsilon}{2}\vert\nabla u\vert^2+\frac{1}{\epsilon}W(u)\Big)dx,\] where $\Omega\subset\mathbb{R}^n$ is a smooth domain, $\epsilon\gt 0$ a small parameter, $v_0^\epsilon:\partial\Omega\rightarrow\mathbb{R}^m$ the boundary datum and $W:\mathbb{R}^m\rightarrow\mathbb{R}$ a smooth nonnegative potential that vanishes on a finite set: \[A=\{a_1,\ldots,a_N\}=\{W=0\}.\] The zeros $a_1,\ldots,a_N$ of $W$ represent equally preferred phases.

For $\delta\gt 0$ small we study the diffuse interface \[\mathscr{I}^{\epsilon,\delta}=\{x\in\bar{\Omega}:\min_{a\in A}\vert u^\epsilon(x)-a\vert\gt \delta\}.\]

We give sufficient conditions on $\Omega$ and $v_0^\epsilon$ ensuring the connectivity of the diffuse interface.

Then we restrict to two space dimensions and show that one can associate a sort of spine to the diffuse interface: a connected network $\mathscr{G}$ which is contained in $\mathscr{I}^{\epsilon,\delta}$ and divides $\Omega$ in $N$ connected subsets corresponding to the $N$ phases. The network $\mathscr{G}$ has a minimality property: it minimizes an energy, a weighted length, that, under a further condition, yields sharp lower bounds for the energy of minimizers and allows for a quantitative description of $\mathscr{G}$.

Ver também

GF-Lisbon-22.pdf