Shape optimization for low Neumann and Steklov eigenvalues

DOI10.1002/mma.1222zbMath1186.35121arXiv0811.2617OpenAlexW2964153620MaRDI QIDQ5187924

Iosif Polterovich, Alexandre Girouard

Publication date: 9 March 2010

Published in: Mathematical Methods in the Applied Sciences (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0811.2617

zbMATH Keywords

eigenvalueshape optimizationLaplacianNeumann problemcomplex analysisSteklov problem

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Estimates of eigenvalues in context of PDEs (35P15) Boundary value problems in the complex plane (30E25)

Related Items (25)

The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander's rediscovered manuscript ⋮ Two balls maximize the third Neumann eigenvalue in hyperbolic space ⋮ Trace and inverse trace of Steklov eigenvalues ⋮ A combined finite element and Bayesian optimization framework for shape optimization in spectral geometry ⋮ Computational Methods for Extremal Steklov Problems ⋮ On the Steklov spectrum of covering spaces and total spaces ⋮ Weyl's law for the Steklov problem on surfaces with rough boundary ⋮ Maximizers beyond the hemisphere for the second Neumann eigenvalue ⋮ Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces ⋮ Estimates for higher Steklov eigenvalues ⋮ Variational aspects of Laplace eigenvalues on Riemannian surfaces ⋮ Optimization of Steklov-Neumann eigenvalues ⋮ The Steklov Problem on Differential Forms ⋮ Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces ⋮ From Steklov to Neumann and Beyond, via Robin: The Szegő Way ⋮ Well-posedness of Hersch-Szegő's center of mass by hyperbolic energy minimization ⋮ Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions ⋮ The Steklov spectrum on moving domains ⋮ Optimal Shapes Maximizing the Steklov Eigenvalues ⋮ Numerical studies of the Steklov eigenvalue problem via conformal mappings ⋮ Isoperimetric control of the Steklov spectrum ⋮ Trace and inverse trace of Steklov eigenvalues. II. ⋮ Steklov eigenvalues and quasiconformal maps of simply connected planar domains ⋮ Asymptotic behaviour of the Steklov spectrum on dumbbell domains ⋮ Heat invariants of the Steklov problem

Cites Work

• The essential spectrum of Neumann Laplacians on some bounded singular domains
• Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space
• Extremal eigenvalues of the Laplacian in a conformal class of metrics: the `conformal spectrum'
• Isoperimetric inequality for the second eigenvalue of a sphere.
• Maximization of the second positive Neumann eigenvalue for planar domains
• Über das Stekloffsche Eigenwertproblem: Isoperimetrische Ungleichungen für symmetrische Gebiete
• Extreme principles and isoperimetric inequalities for some mixed problems of Stekloff's type
• Sums of reciprocal Stekloff eigenvalues
• Some isoperimetric inequalities with application to the Stekloff problem
• A Note on the Generalized Dumbbell Problem
• Isoperimetric Inequality for Some Eigenvalues of an Inhomogeneous, Free Membrane
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