The plasmonic eigenvalue problem

A plasmon of a bounded domain Ω ⊂ ℝn is a non-trivial bounded harmonic function on ℝn\∂Ω which is continuous at ∂Ω and whose exterior and interior normal derivatives at ∂Ω have a constant ratio. We call this ratio a plasmonic eigenvalue of Ω. Plasmons arise in the description of electromagnetic waves hitting a metallic particle Ω. We investigate these eigenvalues and prove that they form a sequence of numbers converging to one. Also, we prove regularity of plasmons, derive a variational characterization, and prove a second-order perturbation formula. The problem can be reformulated in terms of Dirichlet–Neumann operators, and as a side result, we derive a formula for the shape derivative of these operators.

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An eigenvalue problem for generalized Laplacian in Orlicz—Sobolev spaces

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500027505 ◽

1999 ◽

Vol 129 (1) ◽

pp. 153-163 ◽

Cited By ~ 18

Author(s):

Vesa Mustonen ◽

Matti Tienari

Keyword(s):

Weak Solution ◽

Continuous Function ◽

Eigenvalue Problem ◽

Bounded Domain ◽

Sobolev Spaces ◽

Minimization Problem ◽

Lagrange Equation ◽

Image Position ◽

Generalized Laplacian

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problemwhere Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equationfor some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

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A result on the bifurcation from the principal eigenvalue of theAp-Laplacian

Abstract and Applied Analysis ◽

10.1155/s108533759700033x ◽

1997 ◽

Vol 2 (3-4) ◽

pp. 185-195 ◽

Cited By ~ 1

Author(s):

P. Drábek ◽

A. Elkhalil ◽

A. Touzani

Keyword(s):

Eigenvalue Problem ◽

Bounded Domain ◽

Bifurcation Point ◽

Principal Eigenvalue ◽

Bifurcation Problem

We study the following bifurcation problem in any bounded domainΩinℝN:{Apu:=−∑i,j=1N∂∂xi[(∑m,k=1Namk(x)∂u∂xm∂u∂xk)p−22aij(x)∂u∂xj]= λg(x)|u|p−2u+f(x,u,λ),u∈W01,p(Ω).. We prove that the principal eigenvalueλ1of the eigenvalue problem{Apu=λg(x)|u|p−2u,u∈W01,p(Ω),is a bifurcation point of the problem mentioned above.

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Remarks on a nonlinear nonlocal operator in Orlicz spaces

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0002 ◽

2019 ◽

Vol 9 (1) ◽

pp. 305-326 ◽

Cited By ~ 1

Author(s):

Ernesto Correa ◽

Arturo de Pablo

Keyword(s):

Boundary Condition ◽

Eigenvalue Problem ◽

Bounded Domain ◽

Elliptic Problem ◽

Sobolev Inequality ◽

Orlicz Spaces ◽

Integral Operators ◽

Laplacian Operator ◽

Nonlocal Operator ◽

Generalized Eigenvalue

Abstract We study integral operators $\mathcal{L}u\left( \chi \right)=\int{_{_{\mathbb{R}}\mathbb{N}}\psi \left( u\left( x \right)-u\left( y \right) \right)J\left( x-y \right)dy}$of the type of the fractional p-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel J considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem $\mathcal{L}u=f$in a bounded domain $\Omega ,$and boundary condition u ≡ 0 on ${{\Omega }^{c}};$both cases f = f(x) and f = f(u) are considred, including the generalized eigenvalue problem $f\left( u \right)=\lambda \psi \left( u \right).$

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Eigenvalues for a combination between local and nonlocal p-Laplacians

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0074 ◽

2019 ◽

Vol 22 (5) ◽

pp. 1414-1436 ◽

Cited By ~ 1

Author(s):

Leandro M. Del Pezzo ◽

Raúl Ferreira ◽

Julio D. Rossi

Keyword(s):

Eigenvalue Problem ◽

Bounded Domain ◽

Limit Problem ◽

First Eigenvalue ◽

Order Zero ◽

Dirichlet Eigenvalue ◽

The First Eigenvalue ◽

Homogeneous Operator ◽

Dirichlet Eigenvalue Problem

Abstract In this paper we study the Dirichlet eigenvalue problem $$\begin{array}{} \displaystyle -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \text{ in } \Omega,\quad u=0 \, \text{ in } \, \Omega^c=\mathbb{R}^N\setminus\Omega. \end{array}$$ Here Ω is a bounded domain in ℝN, Δpu is the standard local p-Laplacian and ΔJ,pu is a nonlocal p-homogeneous operator of order zero. We show that the first eigenvalue (that is isolated and simple) satisfies $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$(λ1)1/p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$ up, and find the limit problem that is satisfied in the limit.

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On a fourth order Steklov eigenvalue problem

Analysis ◽

10.1524/anly.2005.25.4.315 ◽

2005 ◽

Vol 25 (4) ◽

Cited By ~ 22

Author(s):

Alberto Ferrero ◽

Filippo Gazzola ◽

Tobias Weth

Keyword(s):

Eigenvalue Problem ◽

Bounded Domain ◽

Fourth Order ◽

Steklov Eigenvalue ◽

Steklov Eigenvalue Problem

SummaryWe study the spectrum of a biharmonic Steklov eigenvalue problem in a bounded domain of R

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An Inverse Eigenvalue Problem for an Arbitrary, Multiply Connected, Bounded Domain in $R^3 $ with Impedance Boundary Conditions

SIAM Journal on Applied Mathematics ◽

10.1137/0152041 ◽

1992 ◽

Vol 52 (3) ◽

pp. 725-729 ◽

Cited By ~ 12

Author(s):

E. M. E. Zayed

Keyword(s):

Eigenvalue Problem ◽

Boundary Conditions ◽

Bounded Domain ◽

Inverse Eigenvalue Problem ◽

Impedance Boundary ◽

Multiply Connected ◽

Impedance Boundary Conditions ◽

Inverse Eigenvalue

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Variational characterization of eigenvalues of a non-symmetric eigenvalue problem governing elastoacoustic vibrations

Applications of Mathematics ◽

10.1007/s10492-014-0037-7 ◽

2014 ◽

Vol 59 (1) ◽

pp. 1-13 ◽

Cited By ~ 2

Author(s):

Markus Stammberger ◽

Heinrich Voss

Keyword(s):

Eigenvalue Problem ◽

Variational Characterization ◽

Symmetric Eigenvalue Problem

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On the Variational Eigenvalues Which Are Not of Ljusternik-Schnirelmann Type

Abstract and Applied Analysis ◽

10.1155/2012/434631 ◽

2012 ◽

Vol 2012 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Pavel Drábek

Keyword(s):

Eigenvalue Problem ◽

Eigenvalue Problems ◽

Nonlinear Eigenvalue Problem ◽

Linear Case ◽

Minimax Principle ◽

Variational Characterization ◽

Nonlinear Eigenvalue

We discuss nonlinear homogeneous eigenvalue problems and the variational characterization of their eigenvalues. We focus on the Ljusternik-Schnirelmann method, present one possible alternative to this method and compare it with the Courant-Fischer minimax principle in the linear case. At the end we present a special nonlinear eigenvalue problem possessing an eigenvalue which allows the variational characterization but is not of Ljusternik-Schnirelmann type.

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