Xuefeng Liu, Professor, Tokyo Woman's Christian University

Aihui Zhou, Professor

Guaranteed Computation of Hadamard Shape Derivatives of Eigenvalues and Its Applications

10:00-11:00 July 16(Wednesday), N702

We investigate the Hadamard shape derivative of eigenvalues from the perspective of rigorous computation.

This study originated from our earlier work on interpolation error constants in the finite element method, developed over the past decade. Recently, we have established guaranteed computational methods for both simple and multiple eigenvalues. These rigorous techniques, which include the guaranteed computation of eigenvalues, eigenfunctions, and their shape derivatives, offer powerful tools for addressing open problems in spectral geometry. As a concrete application, we resolve the long-standing question of the simplicity of the Dirichlet Laplacian eigenvalue on a triangular domain. Furthermore, we introduce a novel “difference quotient” approach for computing eigenfunctions associated with tightly clustered eigenvalues—a classically ill-posed problem. This method enables stable and reliable computation in such challenging scenarios.

This research is conducted in collaboration with JSPS DC researcher Mr. Ryoki Endo at Niigata University.