The first nodal line of a convex planar domain. (English) Zbl 0780.35068

Jerison, David

Int. Math. Res. Not. 1991, No. 1, 1-5 (1991).

The author announces the proof of a conjecture about the first nodal line of a planar region in the case of a long, thin convex set. Let \(B(z,r)=\{\zeta\in\mathbb{C}\): \(|\zeta-z|<r\}\), \(\text{inradius}(\Omega)= \max\{r\): \(B(z,r)\subset\Omega\) for some \(z\}\).
Theorem: Let \(\Omega\) be a convex, open subset of the plane. There is an absolute constant \(C\) such that, if \(\text{diameter}(\Omega)/\text{inradius}(\Omega)\geq C\), then the nodal line for the second Dirichlet eigenfunction for \(\Omega\) touches the boundary. In other words, if \(\Delta u=-\lambda u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), and \(\lambda\) is the second eigenvalue for the Dirichlet problem in \(\Omega\), then \(\Gamma=\{z\in\Omega\): \(u(z)=0\}\) satisfies \(\overline {\Gamma}\cap \partial\Omega\neq \emptyset\).

MSC:

35P05	General topics in linear spectral theory for PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
49Q99	Manifolds and measure-geometric topics

Keywords:

first nodal line; planar region; long, thin convex set; second Dirichlet eigenfunction