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On the nodal line of the second eigenfunction of the Laplacian in \(\mathbb{R}^ 2\). (English) Zbl 0769.58056
Let \(\Omega \subset \mathbb{R}^ 2\) be a bounded convex domain with \(C^ \infty\) boundary, let \(u_ 2\) be a nontrivial solution of the Dirichlet problem \[ \begin{cases} \Delta u_ 2 + \lambda_ 2u_ 2 = 0 &\text{in \(\Omega\)},\\u_ 2 = 0 & \text{on \(\partial\Omega\)}\end{cases} \] where \(\Delta = \sum^ 2_{i = 1}{\partial^ 2\over \partial x^ 2_ i}\) is the Laplace operator and \(\lambda_ 2\) its second eigenvalue. The nodal line \(N\) is given by \(N = \{\overline{x \in \Omega: u_ 2(x) = 0}\}\). The main result of the paper is
Theorem. The nodal line \(N\) intersects the boundary \(\partial\Omega\) at exactly two points. In particular, \(N\) does not enclose a compact subregion of \(\Omega\).
Reviewer: C.Bär (Bonn)

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J32 Boundary value problems on manifolds
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