## 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

### Keywords:

variable domain; planar domain; Dirichlet eigenvalues

PLTMG
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