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Zbl 1011.35100
Henrot, Antoine; Oudet, Edouard
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. Abridged English version)
[J] C. R. Acad. Sci., Paris, Sér. I, Math. 332, No.5, 417-422 (2001). ISSN 0764-4442

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 2000:
*35P05 General spectral theory of PDE
35J05 Laplace equation, etc.

Keywords: second eigenvalue of the Laplacian; Dirichlet boundary conditions

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