Maximizer of the first Dirichlet eigenvalue under convexity constraint

Problem's Description

Edit from 17/01/2022. The first open question described below is partially solved in the recent preprint Lamboley-Novruzi-Pierre : Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains (https://arxiv.org/abs/2109.10669) More precisely, it is proven that the free boundary is locally a polygonal line. It remains to show that the free boundary is "globally polygonal".

Consider a bounded open set $D$ in $\mathbb{R}^2$. For any open set $\Omega\subset D$, we denote $\lambda_1(\Omega)$ the first Dirichlet eigenvalue of the Laplacian, that is to say $$\lambda_1(\Omega)=\min\left\{\frac{\int_{\Omega}|\nabla u|^2dx}{\int_{\Omega}|u|^2dx}, \;u\in H^1_0(\Omega)\right\}.$$

We are interested in the shape optimization problem $$\max\left\{\lambda_1(\Omega), \;\;\Omega\textrm{ convex }\subset D, \;|\Omega|=v\right\}$$ where $|\Omega|$ denotes the area of $\Omega$ and $v\in(0,|D|)$.

Thanks to the convexity constraint, it is known that there exists $\Omega^*$ achieving the maximum, for each $v$ and $D$.

The first open question is to understand the structure of the free boundary $D\cap\partial\Omega^*$. In particular, is it a polygonal line?

Note that if $\lambda_1$ is replaced by the perimeter, then the free boundary is indeed a polygonal line.

In higher dimension, a description of the free boundary (both for the maximization of the perimeter or of $\lambda_1$) is a second open problem. In both cases though, it is known that at any point $x\in D\cap\partial\Omega^*$ where $\partial\Omega^*$ is locally of class $C^2$, the Gauss curvature must vanish at this point.

1. Article Regularity and singularities of optimal convex shapes in the plane

   • Jimmy Lamboley,
   • Arian Novruzi,
   • Michel Pierre

   Archive for Rational Mechanics and Analysis 205, 311-343, 2012fulltext

2. Chapter Regularity of optimal spectral domains

   • Jimmy Lamboley,
   • Michel Pierre

   pp. 29-77, year of publication: 2017

3. Article Optimal sets for a class of minimization problems with convex constraints

   • Chiara Bianchini,
   • Antoine Henrot

   Journal of Convex Analysis 19, 725-758, 2012

4. Article Polygons as maximizers of Dirichlet energy or First eigenvalue of Dirichlet-Laplacian among convex planar domains

   • Jimmy Lamboley,
   • Arian Novruzi,
   • Michel Pierre

   PreprintarXiv