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On the range of the first two Dirichlet and nontrivial Neumann eigenvalues of the Laplacian

Antunes, Pedro R. S.; Henrot, Antoine

Proceedings of the Royal Society A, 467 (2011), 1577-1603 (pdf -

In this paper, we study the set of points in the plane defined by {(x, y) =
(l1(U), l2(U)), |U| = 1}, where (l1(U), l2(U)) are either the first two eigenvalues of the
Dirichlet–Laplacian, or the first two non-trivial eigenvalues of the Neumann–Laplacian.
We consider the case of general open sets together with the case of convex open domains.
We give some qualitative properties of these sets, show some pictures obtained through
numerical computations and state several open problems.