Inverse problems and rigidity questions in billiard dynamics

A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.

Angular Extents and Trajectory Slopes in the Theory of Holomorphic Semigroups in the Unit Disk

We study relationships between the asymptotic behaviour of a non-elliptic semigroup of holomorphic self-maps of the unit disk and the geometry of its planar domain (the image of the Koenigs function). We establish a sufficient condition for the trajectories of the semigroup to converge to its Denjoy–Wolff point with a definite slope. We obtain as a corollary two previously known sufficient conditions.

Duality of capacities and Sobolev extendability in the plane

We reveal relations between the duality of capacities and the duality between Sobolev extendability of Jordan domains in the plane, and explain how to read the curve conditions involved in the Sobolev extendability of Jordan domains via the duality of capacities. Finally as an application, we give an alternative proof of the necessary condition for a Jordan planar domain to be $$W^{1,\,q}$$ W 1 , q -extension domain when $$2<q<\infty$$ 2 < q < ∞ .

Asymptotic behavior of $u$-capacities and singular perturbations for the Dirichlet-Laplacian

In this paper we study the asymptotic behavior of $u$-capacities of small sets and its application to the analysis of the eigenvalues of the Dirichlet-Laplacian on a bounded planar domain with a small hole. More precisely, we consider two (sufficiently regular) bounded open connected sets $\Omega$ and $\omega$ of $\mathbb{R}^2$, containing the origin. First, if $\varepsilon$ is positive and small enough and if $u$ is a function defined on $\Omega$, we compute an asymptotic expansion of the $u$-capacity $\mathrm{Cap}_\Omega(\varepsilon \omega, u)$ as $\varepsilon \to 0$. As a byproduct, we compute an asymptotic expansion for the $N$-th eigenvalues of the Dirichlet-Laplacian in the perforated set $\Omega \setminus (\varepsilon \overline{\omega})$ for $\varepsilon$ close to $0$. Such formula shows explicitly the dependence of the asymptotic expansion on the behavior of the corresponding eigenfunction near $0$ and on the shape $\omega$ of the hole.

Convexity properties of the normalized Steklov zeta function of a planar domain

We consider the zeta function \zeta_{\Omega} for the Dirichlet-to-Neumann operator of a simply connected planar domain Ω bounded by a smooth closed curve of perimeter 2\pi. We name the difference \zeta_{\Omega}-\zeta_{\mathbb{D}} the normalized Steklov zeta function of the domain Ω, where 𝔻 denotes the closed unit disk. We prove that (\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(0)\geq 0 with equality if and only if Ω is a disk. We also provide an elementary proof that, for a fixed real 𝑠 satisfying s\leq-1, the estimate (\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(s)\geq 0 holds with equality if and only if Ω is a disk. We then bring examples of domains Ω close to the unit disk where this estimate fails to be extended to the interval (0,2). Other computations related to previous works are also detailed in the remaining part of the text.