Mathematics Colloquia and Seminars

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This talk will be concerned with the lowest Dirichlet eigenvalues of the Laplace operator on some arbitrary bounded domain $\Omega\subset\mathbb R^n$. More precisely, I will give a survey on isoperimetric inequalities which state that certain functions of these eigenvalues are maximized or minimized on certain domains (typically on balls). One example is the Rayleigh-Faber-Krahn inequality, which states that the fundamental eigenvalue is minimized on balls among all domains of the same volume. Another example is the ratio of the second and the first eigenvalue, which is maximized on balls according to the Payne-P\'olya-Weinberger inequality. I will present recent generalizations of this statement, e.g. to Schr\"odinger operators or to the Laplace operator on a sphere or in hyperbolic space.