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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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