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Range of the first two eigenvalues of the Laplacian. (English) Zbl 0816.35097

Summary: For each planar domain \(D\) of unit area, the first two Dirichlet eigenvalues of \(-\Delta\) on \(D\) determine a point \((\lambda_ 1 (D), \lambda_ 2 (D))\) in the \((\lambda_ 1, \lambda_ 2)\) plane. As \(D\) varies over all such domains, this point varies over a set \({\mathcal R}\) which we determine. Its boundary consists of two semi-infinite straight lines and a curve connecting their endpoints. This curve is found numerically. We also show how to minimize the \(n\)-th eigenvalue when the minimizing domain is disconnected. For \(n=3\) we show that the minimizer domain is connected and that \(\lambda_ 3\) is a local minimum for \(D\) a circular disc.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Software:

PLTMG
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