The Robin Laplacian—Spectral conjectures, rectangular theorems

Richard S. Laugesen

doi:10.1063/1.5116253

On the honeycomb conjecture for Robin Laplacian eigenvalues

Communications in Contemporary Mathematics ◽

10.1142/s0219199718500074 ◽

2019 ◽

Vol 21 (02) ◽

pp. 1850007 ◽

Cited By ~ 2

Author(s):

Dorin Bucur ◽

Ilaria Fragalà

Keyword(s):

Torsional Rigidity ◽

Cluster Problem ◽

Laplacian Eigenvalues ◽

Convexity Assumption ◽

Robin Laplacian ◽

Cheeger Sets ◽

Optimal Cluster

We prove that the optimal cluster problem for the sum/the max of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity. In the specific case of the max of the first Robin eigenvalue, we are able to remove the convexity assumption on the cells.

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Counterexample to Strong Diamagnetism for the Magnetic Robin Laplacian

Mathematical Physics Analysis and Geometry ◽

10.1007/s11040-020-09350-6 ◽

2020 ◽

Vol 23 (3) ◽

Cited By ~ 2

Author(s):

Ayman Kachmar ◽

Mikael P. Sundqvist

Keyword(s):

Magnetic Field ◽

Boundary Condition ◽

Laplace Operator ◽

Unit Disc ◽

Uniform Magnetic Field ◽

Robin Boundary Condition ◽

Robin Laplacian ◽

The Laplace Operator ◽

Robin Boundary ◽

Lowest Eigenvalue

Abstract We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to $-\infty $ − ∞ .

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Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2019.48 ◽

2019 ◽

Vol 150 (6) ◽

pp. 2871-2893 ◽

Cited By ~ 1

Author(s):

Sergei A. Nazarov ◽

Nicolas Popoff ◽

Jari Taskinen

Keyword(s):

Dimension Reduction ◽

Discrete Spectrum ◽

Spectral Theory ◽

Real Axis ◽

Complex Plane ◽

Lipschitz Domain ◽

High Rate ◽

The Real ◽

Robin Laplacian ◽

Asymptotic Forms

We consider the Robin Laplacian in the domains Ω and Ωε, ε > 0, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient a is large enough, the spectrum of the problem in Ω is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain Ωε is discrete. However, our results reveal the strange behaviour of the discrete spectrum as the blunting parameter ε tends to 0: we construct asymptotic forms of the eigenvalues and detect families of ‘hardly movable’ and ‘plummeting’ ones. The first type of the eigenvalues do not leave a small neighbourhood of a point for any small ε > 0 while the second ones move at a high rate O(| ln ε|) downwards along the real axis ℝ to −∞. At the same time, any point λ ∈ ℝ is a ‘blinking eigenvalue’, i.e., it belongs to the spectrum of the problem in Ωε almost periodically in the | ln ε|-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

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