Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains
The authors consider wave and Schrödinger equations on a bounded open connected subset \(\Omega\) of a Riemannian manifold, subject to Dirichlet, Neumann, or Robin boundary conditions. For a measurable subset \(\omega\subseteq \Omega\) and \(T>0\) an observability inequality for the pair \((\omega, T)\) guarantees that the total energy of solutions can be estimated by the energy localized in \(\omega\times (0,T)\). The paper is concerned with the optimal location of \(\omega\) among all subsets of \(\Omega\) with measure \(|\omega| = L|\Omega|\) for fixed \(L\in (0,1)\). This question is reformulated in terms of maximizing the obervability constant. For practical reasons, the authors consider maximizing an average over random initial data or asymptotically in time. It turns out that these constants can be formulated as optimal points of a functional measuring the concentration of eigenfunctions. Since the original problem lacks convexity properties, under suitable quantum ergodicity assumptions on \(\Omega\) a no-gap result between the original problem and its convexified version is proved. Furthermore, the optimal value is computed in this case. The paper is complemented with various applications such as the 1D case and also numerical simulations confirming the theoretical results.
scientific article; zbMATH DE number 6578173