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In an extremal eigenvalue problem, one considers a family of eigenvalue problems, each with discrete spectra, and extremizes a chosen eigenvalue over the family. In this chapter, we consider eigenvalue problems defined on Riemannian manifolds and extremize over the metric structure. For example, we consider the problem of maximizing the principal Laplace–Beltrami eigenvalue over a family of closed surfaces of fixed volume. Computational approaches to such extremal geometric eigenvalue problems present new computational challenges and require novel numerical tools, such as the parameterization of conformal classes and the development of accurate and efficient methods to solve eigenvalue problems on domains with nontrivial genus and boundary. We highlight recent progress on computational approaches for extremal geometric eigenvalue problems, including (i) maximizing Laplace–Beltrami eigenvalues on closed surfaces and (ii) maximizing Steklov eigenvalues on surfaces with boundary.
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This content will become publicly available on May 13, 2026
Inverse Iteration for the Laplace Eigenvalue Problem With Robin and Mixed Boundary Conditions
We apply the method of inverse iteration to the Laplace eigenvalue problem with Robin and mixed Dirichlet-Neumann boundary conditions, respectively. For each problem, we prove convergence of the iterates to a non-trivial principal eigenfunction and show that the corresponding Rayleigh quotients converge to the principal eigenvalue. We also propose a related iterative method for an eigenvalue problem arising from a model for optimal insulation and provide some partial results.
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- Award ID(s):
- 2246611
- PAR ID:
- 10618242
- Publisher / Repository:
- CSU Open Journals
- Date Published:
- Journal Name:
- PUMP journal of undergraduate research
- Volume:
- 8
- ISSN:
- 2576-3725
- Page Range / eLocation ID:
- 173 to 194
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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