Conference on Calculus of Variations in Lille - 3rd edition - July 4-6 2022

A Lagrangian description of entropy solutions of the eikonal equation

Jul 5, 2022, 4:30 PM

1h

M2 building, Cité Scientifique - Meeting room, 1st floor (Laboratoire Paul Painlevé)

Speaker

Elio Marconi (EPFL)

Description

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

Part of the work is done in collaboration with Xavier Lamy.