Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions (Q1417404)

Consider the family \(\mathcal F\) of all bounded open subsets of the plane having fixed area (say equal to \(1\)), and let \(\lambda_k(\Omega)\) be the \(k-\)th eigenvalue of \(\Omega\in\mathcal F\) for the Dirichlet boundary conditions. The well-known Faber-Krahn inequality asserts that \(\lambda_1(\Omega)\) is minimized by the disk, and Szegő proved that \(\lambda_2(\Omega)\) is minimized by the union of two identical disks (a disconnected set). Looking for minimizers of \(\lambda_2\) among (arbitrary) connected domains has no solution, and so it is natural to ask the following question: What is the convex plane domain \(\Omega^*\) of unit area which minimizes \(\lambda_2\)? (The minimizer \(\Omega^*\) is actually shown to exist). Numerical experiments lead \textit{A. B. Troesch} [Math. Comput. 27, 767--772 (1973; Zbl 0271.35018)] to conjecture that \(\Omega^*\) is a stadium, i.e. the convex hull of two identical tangent disks. The main result of the paper under review is to show that the above conjecture is false; more generally it is shown that the minimizer \(\Omega^*\) has no arc of circle in its boundary. In the course of the proof, the authors also prove some interesting facts about \(\Omega^*\); namely that it is at least \(C^1\) and at most \(C^2\) and that, if it is assumed to be \(C^{1,1}\) then it has two (and only two) straight lines in its boundary, and these lines are parallel; finally, under the \(C^{1,1}\) assumption the second eigenvalue is simple.