The Eigenvalue Problems for the Laplacian on Compact Embedded Surfaces and Three Dimensional Bounded Domains

Abstract

In this paper, we show new formulas on finite element method (FEM) and three dimensional FEM. By making use of them, we give several computer visions for the eigenvalue problems of the Laplacian for bounded domains and embedded compact surfaces in the three dimensional Euclidean space. As its applications, we can examine several examples to Yau’s problem on the eigenvalues between embedded surfaces and their enclosed three dimensional bounded domains.