Maximizer of the first Dirichlet eigenvalue under convexity constraint

Problem's Description

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 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 an 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

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