Speaker
Description
We consider the behaviour as $\varepsilon \to 0^+$ of the following family of functionals introduced by P. Aviles and Y. Giga:
$$
F_\varepsilon(u,\Omega):= \int_{\Omega} \left( \varepsilon |\nabla^2 u|^2 + \frac{1}{\varepsilon}\left|1-|\nabla u|^2\right|^2\right)dx, \qquad \mbox{where }\Omega\subset \mathbb{R}^2.
$$
Functions with equi-bounded energy as $\varepsilon \to 0$ are pre-compact in $L^1(\Omega)$ and all the limits belong to the class of the so called 'entropy solutions' of the eikonal equation $|\nabla u| =1$ in $\Omega$.
We introduce a Lagrangian description of these solutions and we investigate their fine properties.
As a corollary we obtain that if $\Omega$ is an ellipse, then minimizers of $F_\varepsilon(\cdot, \Omega)$ in the space
$$
\left\{ u\in W^{2,2}(\Omega) : u=0 \mbox{ and } \frac{\partial u}{\partial n} = -1 \mbox{ at }\partial \Omega\right\}
$$
converge to $u_\ast := \mathrm{dist}(\cdot, \partial \Omega)$.
Moreover we get a sharp quantitative version of the result in Jabin–Otto–Perthame (2002), stating that the only bounded simply connected domain $\Omega$ admitting zero energy states with Dirichlet boundary conditions is the disk.
Part of the work is done in collaboration with Xavier Lamy.
