On Some New Mathematical and Computational Aspects on Iso-Spectral PDE Problems in 2-D and 3-D

Institute of Software Chinese Academy of Sciences

2016-11-30 ~ 2016-11-30


Science Building No.1 1479


“Can one hear the shape of a drum?”that is, if you know the frequencies at which a drum vibrates, can you determine it's shape?The most famous of Inverse Problem related to Mathamatical-Physical Equation was presneted by Mark Kac in 1966. In mathematics, two different operators are called isospectral if they are supported to have the same sets of eigenvalues. Two domains ara called to be Iso-spectral if they are not Iso-metric and there are same spectral for a given PDE operator, such as Laplacian, Helmholz, Schrödinger. Numerical mathematician do eigenvalue computation or take it as an Inverse PDE eigen-problem. Physical scientists even do spectral experiments to verify the iso-spectral in various fields. Mathematically, the earliest examples of non-isometric billiards were found in 1964 by Milnor in 16-Dimension lattices. However, 2-D solution was appeared 30 years later by Gordon in Invent Math. 1992. So far, essentially only 17 families in 2-D of examples that say no to Kac’s question were constructed in a 40 year period. How about in real 3-D case? "Can you hear the shape of a cavity?" S. Moorhead from University of Oxford presents the question in his thesis (August 2012). However, they failed with given some 3-D region pictures only. In this talk ,we will focus on the PDE iso-spectral problems in non-convex triangle domains in 2-D and propose a new way to construct some new non-convex tetrahendron combination in 3-D to be isospectral. Some PDE Eigen- computation approach, such as domain decomposition, Nodal-Line eigen-function, Transplantation Matrix, Adjacency Matrix and Piece-wise smooth approximations are considered.