From Steklov to Neumann and Beyond, via Robin: The Szegő Way

2019 ◽  
Vol 72 (4) ◽  
pp. 1024-1043 ◽  
Author(s):  
Pedro Freitas ◽  
Richard S. Laugesen
Keyword(s):  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Planar Domains ◽  
Steklov Eigenvalue ◽  
Robin Laplacian ◽  
Fixed Area ◽  
Neumann Laplacian ◽  
Simply Connected ◽  
Second Eigenvalue ◽  
Simply Connected Domains

AbstractThe second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$, and $\unicode[STIX]{x1D6FC}$ lies between $-2\unicode[STIX]{x1D70B}$ and $2\unicode[STIX]{x1D70B}$. Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.

scholarly journals Principal frequency of the p-Laplacian and the inradius of Euclidean domains

2015 ◽  
Vol 07 (03) ◽  
pp. 505-511 ◽  
Author(s):  
Guillaume Poliquin
Keyword(s):  
Lower Bound ◽  
Laplace Operator ◽  
Lower Bounds ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Planar Domains ◽  
Principal Frequency ◽  
The Laplace Operator ◽  
Simply Connected ◽  
Euclidean Domains

We study the lower bounds for the principal frequency of the p-Laplacian on N-dimensional Euclidean domains. For p > N, we obtain a lower bound for the first eigenvalue of the p-Laplacian in terms of its inradius, without any assumptions on the topology of the domain. Moreover, we show that a similar lower bound can be obtained if p > N - 1 assuming the boundary is connected. This result can be viewed as a generalization of the classical bounds for the first eigenvalue of the Laplace operator on simply connected planar domains.

scholarly journals Extrema of Low Eigenvalues of the Dirichlet–Neumann Laplacian on a Disk

2010 ◽  
Vol 62 (4) ◽  
pp. 808-826
Author(s):  
Eveline Legendre
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
First Eigenvalue ◽  
Bounded Number ◽  
Eigenvalues Of The Laplacian ◽  
Neumann Laplacian ◽  
Dirichlet And Neumann Conditions ◽  
Second Eigenvalue

AbstractWe study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

THE REGION OF SOLVABILITY OF A PARAMETERIZED BOUNDARY VALUE PROBLEM CAN BE DISCONNECTED

2004 ◽  
Vol 06 (06) ◽  
pp. 901-912
Author(s):  
ANTONIO J. UREÑA
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Second Eigenvalue

A celebrated result by Amann, Ambrosetti and Mancini [1] implies the connectedness of the region of existence for some parameter-depending boundary value problems which are resonant at the first eigenvalue. The analogous thing does not hold for problems which are resonant at the second eigenvalue.

scholarly journals New features of the first eigenvalue on negatively curved spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexandru Kristály
Keyword(s):  
Transcendental Equation ◽  
First Eigenvalue ◽  
Gaussian Hypergeometric Function ◽  
Curved Spaces ◽  
The First Eigenvalue ◽  
Negatively Curved ◽  
Measure Spaces ◽  
Eigenvalue Comparison ◽  
Monotonicity Hypothesis ◽  
Simply Connected

AbstractThe paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model n-dimensional hyperbolic space, complementing the results of Borisov and Freitas (2017), Hurtado, Markvorsen and Palmer (2016) and Savo (2008); in odd dimensions, such eigenvalues appear as roots of an inductively constructed transcendental equation. We then give a synthetic proof of Cheng’s sharp eigenvalue comparison theorem in metric measure spaces satisfying a Bishop–Gromov-type volume monotonicity hypothesis. As a byproduct, we provide an example of simply connected, non-compact Finsler manifold with constant negative flag curvature whose first eigenvalue is zero; this result is in a sharp contrast with its celebrated Riemannian counterpart due to McKean (1970). Our proofs are based on specific properties of the Gaussian hypergeometric function combined with intrinsic aspects of the negatively curved smooth/non-smooth spaces.

scholarly journals Isoperimetric Upper Bound for the First Eigenvalue of Discrete Steklov Problems

2020 ◽  
Author(s):  
Hélène Perrin
Keyword(s):  
Upper Bound ◽  
Polynomial Growth ◽  
Upper Bounds ◽  
First Eigenvalue ◽  
Zero Eigenvalue ◽  
Finite Graphs ◽  
Metric Properties ◽  
The First Eigenvalue ◽  
Steklov Eigenvalue ◽  
Groups Of Polynomial Growth

AbstractWe study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also satisfies a discrete isoperimetric inequality, we show that the first non-zero Steklov eigenvalue tends to zero as the number of vertices of the graph tends to infinity. This extends recent results of Han and Hua, who obtained a similar result in the case of $$\mathbb {Z}^n$$ Z n . We obtain the result using metric properties of Cayley graphs associated to groups of polynomial growth.

scholarly journals On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains – with some numerics –

2020 ◽  
Vol 121 (1) ◽  
pp. 35-57
Author(s):  
B. Helffer ◽  
T. Hoffmann-Ostenhof ◽  
F. Jauberteau ◽  
C. Léna
Keyword(s):  
Bounded Domain ◽  
Dirichlet Laplacian ◽  
Eigenvalues Of The Laplacian ◽  
Simply Connected ◽  
Second Eigenvalue ◽  
Simply Connected Domains

We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R 2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity 3. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.

A semilinear elliptic eigenvalue problem, II. The plasma problem

1980 ◽  
Vol 87 (1-2) ◽  
pp. 83-109 ◽  
Author(s):  
Grant Keady ◽  
John Norbury
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Image Position ◽  
Plasma Problem ◽  
Simply Connected ◽  
Special Case

SynopsisThis paper continues the study of the boundary value problem, for (λ, ψ)Here δ denotes the Laplacian,kis a given positive constant, and λ1will denote the first eigenvalue for the Dirichlet problem for −δ on Ω. For λ ≦ λ1, the only solutions are those with ψ = 0. Throughout we will be interested in solutions (λ, ψ) with λ > λ1and with ψ > 0 in Ω.In the special case Ω =B(0,R) there is a branch ℱe, of explicit exact solutions which bifurcate from infinity at λ = λ1and for which the following conclusions are valid, (a) The setAψ,is simply-connected, (b) Along ℱe, ψm→k, ‖ψ‖1→ 0 and the diameter ofAψtends to zero as λ → ∞, whereHere it is shown that the above conclusions hold for other choices of Ω, and in particular, for Ω = (−a, a)×(−b, b). (Existence is settled in Part I, and elsewhere.)The results of numerical and asymptotic calculations when Ω = (−a, a)×(−b, b) are given to illustrate both the above, and some limitations in the conclusions of our analysis.

scholarly journals Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains

2021 ◽  
Author(s):  
Bruno Colbois ◽  
Alessandro Savo
Keyword(s):  
Magnetic Field ◽  
Upper Bound ◽  
Ground State ◽  
State Energy ◽  
Upper Bounds ◽  
Zero Magnetic Field ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Simply Connected ◽  
Lowest Eigenvalue

AbstractWe obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential 1-form (hence, with zero magnetic field) acting on complex functions of a planar domain $$\Omega $$ Ω , with magnetic Neumann boundary conditions. It is well known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance, the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal $$\epsilon $$ ϵ -net. In the last part, we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one.

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