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The stadium does not minimize ${\lambda }_{2}$ amongst convex plane domains. (Le stade ne minimise pas ${\lambda }_{2}$ parmi les ouverts convexes du plan.) (French) Zbl 1011.35100
Summary: We are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the “stadium”, convex hull of two identical tangent disks (Troesch conjecture). We refute this conjecture. Nevertheless, we prove that the optimum has two parallel segments in its boundary.
##### MSC:
 35P05 General topics in linear spectral theory of PDE 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation