Topic description: Laplace operator is a fundamental differential operator in multi-variable calculus and mathematical physics. This second order linear differential operator models the important diffusion phenomenon which occurs in many physical, chemical and biological processes. The Laplacian, as a symmetric operator, has infinitely many real-valued eigenvalues which tend to infinity as a sequence. However the relation between the geometry (shape) of the spatial region and eigenvalues is not completely clear. Courant and Hilbert have shown that, under Dirichlet boundary condition, the principal eigenvalue is smaller for larger domain. People have assumed that this is also true for other boundary conditions like no-flux boundary condition. But Ni and Wang have shown in a recent paper that this is not the case, and the complete picture is rather delicate.
Courant, R.; Hilbert, D. Methods of mathematical physics. Vol. II: Partial differential equations. (Vol. II by R. Courant.) John Wiley & Sons, New York-London, 1962.
Ni, Wei-Ming; Wang, Xuefeng, On the first positive Neumann eigenvalue. Discrete Contin. Dyn. Syst. 17 (2007), no. 1, 1–19.
Research opportunities: The purpose of this project is to estimate eigenvalues of Laplacian for regions in two-dimensional space, and try to give a criterion of the relation between the eigenvalues and domain geometry based on numerical investigation.
Suggested prerequisites: Math 302, Math 441-442, CS 141
Contact: Junping Shi