The first nodal set of a convex domain. (English) Zbl 0837.31005

Jerison, David

Fefferman, Charles (ed.) et al., Essays on Fourier analysis in honor of Elias M. Stein. Proceedings of the Princeton conference on harmonic analysis held at Princeton Univ., Princeton, NJ, USA, May 13-17, 1991 in honor of Elias M. Stein’s 60th birthday. Princeton, NJ: Princeton Univ. Press. Princeton Math. Ser. 42, 225-249 (1995).

In 1991 the author announced a theorem which asserts that the first nodal set of a long, thin convex domain touches the boundary [Int. Math. Res. Notes 1991, No. 1, 1-5 (1991; Zbl 0780.35068)]. The assertion was proved by A. Melas [On the nodal line of the second eigenfunction of the Laplacian in \(\mathbb{R}^2\), J. Diff. Geom. 35, No. 1, 255-263 (1992; Zbl 0769.58056)]for all smooth convex domains in \(\mathbb{R}^2\). In the present paper the result is generalized to convex domains in \(\mathbb{R}^n\).
For the entire collection see [Zbl 0810.00019].

Reviewer: M.Dont (Praha)

MSC:

31B99	Higher-dimensional potential theory
35P05	General topics in linear spectral theory for PDEs
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

nodal sets; smooth convex domains

Citations:

Zbl 0780.35068; Zbl 0769.58056