Shape optimization for low Neumann and Steklov eigenvalues

By A. Girouard, Iosif Polterovich. 2008

59 citations recorded by the source corpus.

We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature: they are proved using methods of complex analysis. In particular, we present modernized proofs of the classical inequalities due to Szegö and Weinstock for the first nonzero Neumann and Steklov eigenvalues. We also extend the inequality for the second nonzero Neumann eigenvalue, obtained recently by Nadirashvili and the authors, to nonhomogeneous membranes with log‐subharmonic densities. In the homogeneous case, we show that this inequality is strict, which implies that the maximum of the second nonzero Neumann eigenvalue is not attained in the class of simply connected membranes of a given mass. The same is true for the second nonzero Steklov eigenvalue, as follows from our results on the Hersch–Payne–Schiffer inequalities. Copyright © 2009 John Wiley & Sons, Ltd.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/0692fc15a7dc62323a27c6a6581e466c09acad93

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In Applied Mathematics

1. G. Kokarev
2. A. Fraser
3. Iosif Polterovich
4. R. Schoen
5. David A. Sher
6. M. Li
7. A. Girouard
8. R. Petrides

In Computational Theory and Mathematics

1. G. Kokarev
2. David A. Sher
3. A. Fraser
4. R. Schoen
5. Iosif Polterovich
6. A. Girouard
7. S. Raju
8. V. Millot

In Control and Systems Engineering

1. G. Kokarev
2. Iosif Polterovich

Citations in the corpus

• The first Steklov eigenvalue, conformal geometry, and minimal surfaces (2009, background): “ and Schiffer [HPS], Kuttler and Siggilito [KS], Shamma [Sh], Edward [Ed], Escobar [E1], [E2], Brock [Br], Dittmar [D], Girouard and Polterovich [GP1]). The reader may consult Girouard and Polterovich [GP2] for a recent survey which includes this topic. The first author was partially supported by the ”
• Heat Invariants of the Steklov Problem (2013, background): “heat invariants we have computed to show that in fact M must be a sphere. Remark 1.5.4. An analogue of Theorem 1.5.3 in two dimensions is well–known. Indeed, it was shown by Weinstock ([We], see also [GP1]) that among simply connected planar domains, of a given perimeter, the first Steklov eigenvalue”
• Variational aspects of Laplace eigenvalues on Riemannian surfaces (2011, result): “ry volume measure. Then the eigenvalues λk(µg,c)coincide with the so-called Steklov eigenvalues of a metric g, representing the spectrum of the Dirichlet-to-Neumann map. We refer to the recent papers [17, 13] for the account and further references on the subject. In particular, Theorems Ak and A1 ab”
• SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “In fact, Weinstock’s proof is the simplest application of the center of mass renormalization (also known as Hersch’s lemma, see [59, 96, 45, 48]).”
• Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces (2012, unknown): “This eigenvalue problem was considered in 1902 by Steklov and since then has been studied extensively; we refer to [3] and the recent papers [16, 11] for a comprehensive list of references on the subject.”
• Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions (2016, unknown): “…(6) with Neumann boundary conditions, but with the positive weight (also called density) 0 < a ≤ m ( ) ≤ b and without a mass constraint on the weight, see [22, Theorem 1.4.2] and [3, 4]. also for See [17] a similar problem in the context of Riemannian manifolds with a mass preservation constraint.”