On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach Academic Article uri icon

abstract

  • In this article we investigate, via numerical computations, the intersection properties of the nodal set of the eigenfunctions of the Laplace- Beltrami operator for smooth surfaces in 3 (the nodal set of a continuous function is the set of those points at which the function vanishes). First, we briey discuss the numerical solution of the eigenvalue/eigenfunction problem for the Laplace-Beltrami operator on bounded surfaces of 3, and then consider some specific surfaces and visualize how the nodal lines intersect (or not) depending of the symmetries verified by the surface. After validating our com-putational methodology with the surface of a ring torus, we will investigate a simple surface without symmetry and observe that in that case the nodal set of the computed eigenfunctions consists of non intersecting lines, suggesting some conjecture. We observe also that for the above symmetry-free surface, the number of connected components of the nodal set varies non-monotonically with the rank of the associated eigenvalue (assuming that the eigenvalues are ordered by increasing value).

published proceedings

  • Communications on Pure & Applied Analysis

author list (cited authors)

  • Bonito, A., & Glowinski, R.

citation count

  • 4

complete list of authors

  • Bonito, Andrea||Glowinski, Roland

publication date

  • January 2014