Title: Rigorous Evaluation of the Hadamard Derivative for Shape Optimization Studies
Abstract: This presentation introduces a newly developed computational method for rigorously evaluating the Hadamard derivative of Laplacian eigenvalues, which plays an important role in studying shape optimization problems.
To evaluate the Hadamard derivative, this method employs state-of-the-art algorithms for eigenvalues and eigenfunctions via the finite element method (Liu'2013,2015; Liu-Vejchodsky'2022), effectively handling cases of repeated or closely spaced eigenvalues.
We also present a computer-assisted proof for the optimization of Laplacian eigenvalues over triangular domains (Endo-Liu'2023), demonstrating the impact of these computational advancements in spectral geometry.
References:
1) Xuefeng Liu, A framework of verified eigenvalue bounds for self-adjoint differential operators, Applied Mathematics and Computation, Vol. 267, 2015, pp. 341-355.
2) Xuefeng Liu and Tomas Vejchodský, Fully computable a posteriori error bounds for eigenfunctions, Numer. Math. 152, 183–221 (2022).
3) Ryoki Endo, Xuefeng Liu, Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error analysis, Journal of Differential Equations, Volume 376, 2023, pp.750-772.