The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter α in the regime connecting the first nontrivial Neumann and Steklov eigenvalues, and even somewhat beyond the Steklov regime. The result is close to optimal, since the ball is not maximal when α is sufficiently large negative, and the problem admits no maximiser when α is positive. 1. Introduction and results. The Robin eigenvalue problem for the.
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