scholarly journals Electromagnetic Steklov eigenvalues: approximation analysis

2020 ◽  
Author(s):  
Martin Halla
Keyword(s):  
Eigenvalue Problem ◽  
Operator Function ◽  
Projection Operators ◽  
Compact Perturbation ◽  
Open Questions ◽  
Galerkin Approximations ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Approximation Analysis ◽  
Steklov Eigenvalues

We continue the work of [Camano, Lackner, Monk, SIAM J.\ Math.\ Anal., Vol.\ 49, No.\ 6, pp.\ 4376-4401 (2017)] on electromagnetic Steklov eigenvalues. The authors recognized that in general the eigenvalues due not correspond to the spectrum of a compact operator and hence proposed a modified eigenvalue problem with the desired properties. The present article considers the original and the modified electromagnetic Steklov eigenvalue problem. We cast the problems as eigenvalue problem for a holomorphic operator function $A(\cdot)$. We construct a ``test function operator function'' $T(\cdot)$ so that $A(\lambda)$ is weakly $T(\lambda)$-coercive for all suitable $\lambda$, i.e.\ $T(\lambda)^*A(\lambda)$ is a compact perturbation of a coercive operator. The construction of $T(\cdot)$ relies on a suitable decomposition of the function space into subspaces and an apt sign change on each subspace. For the approximation analysis, we apply the framework of T-compatible Galerkin approximations. For the modified problem, we prove that convenient commuting projection operators imply T-compatibility and hence convergence. For the original problem, we require the projection operators to satisfy an additional commutator property involving the tangential trace. The existence and construction of such projection operators remain open questions.

Steklov eigenvalue problem with a-harmonic solutions and variable exponents

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Belhadj Karim ◽  
Abdellah Zerouali ◽  
Omar Chakrone
Keyword(s):  
Eigenvalue Problem ◽  
Smooth Boundary ◽  
Normal Vector ◽  
Variable Exponents ◽  
Variational Technique ◽  
Unit Normal Vector ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Outward Unit ◽  
Outward Unit Normal Vector

AbstractUsing the Ljusternik–Schnirelmann principle and a new variational technique, we prove that the following Steklov eigenvalue problem has infinitely many positive eigenvalue sequences:\left\{\begin{aligned} &\displaystyle\operatorname{div}(a(x,\nabla u))=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle a(x,\nabla u)\cdot\nu=\lambda m(x)|u|^{p(x)-2}u&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}}{(N\geq 2)} is a bounded domain of smooth boundary {\partial\Omega} and ν is the outward unit normal vector on {\partial\Omega}. The functions {m\in L^{\infty}(\partial\Omega)}, {p\colon\overline{\Omega}\mapsto\mathbb{R}} and {a\colon\overline{\Omega}\times\mathbb{R}^{N}\mapsto\mathbb{R}^{N}} satisfy appropriate conditions.

scholarly journals The Steklov Problem on Differential Forms

2019 ◽  
Vol 71 (2) ◽  
pp. 417-435
Author(s):  
Mikhail A. Karpukhin
Keyword(s):  
Eigenvalue Problem ◽  
Spectral Properties ◽  
Differential Forms ◽  
Type Inequality ◽  
Slight Modification ◽  
Dimensional Manifold ◽  
Hodge Laplacian ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Dirichlet To Neumann Map

AbstractIn this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\unicode[STIX]{x039B}$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of $\unicode[STIX]{x039B}$ and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\unicode[STIX]{x039B}$ are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$-forms on the boundary of a $2p+2$-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.

scholarly journals On a positive solution for $(p,q)$-Laplace equation with Nonlinear

2019 ◽  
Vol 38 (4) ◽  
pp. 219-133
Author(s):  
Abdellah Zerouali ◽  
Belhadj Karim ◽  
Omar Chakrone ◽  
Abdelmajid Boukhsas
Keyword(s):  
Eigenvalue Problem ◽  
Positive Solution ◽  
Laplace Equation ◽  
Existence Results ◽  
Indefinite Weight ◽  
Principal Eigenvalues ◽  
Steklov Eigenvalue ◽  
Indefinite Weights ◽  
Steklov Eigenvalue Problem

In the presentp aper, we study the existence and non-existence results of a positive solution for the Steklov eigenvalue problem driven by nonhomogeneous operator $(p,q)$-Laplacian with indefinite weights. We also prove that in the case where $\mu>0$ and with $1<q<p<\infty$ the results are completely different from those for the usua lSteklov eigenvalue problem involving the $p$-Laplacian with indefinite weight, which is retrieved when $\mu=0$. Precisely, we show that when $\mu>0$ there exists an interval of principal eigenvalues for our Steklov eigenvalue problem.

On a fourth order Steklov eigenvalue problem

Analysis ◽  
2005 ◽  
Vol 25 (4) ◽  
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

scholarly journals Conforming Finite Element Approximations for a Fourth-Order Steklov Eigenvalue Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Hai Bi ◽  
Shixian Ren ◽  
Yidu Yang
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Fourth Order ◽  
Finite Element Approximation ◽  
Continuous Operator ◽  
Element Approximation ◽  
Morley Element ◽  
Steklov Eigenvalue ◽  
Conforming Finite Element ◽  
Steklov Eigenvalue Problem

This paper characterizes the spectrum of a fourth-order Steklov eigenvalue problem by using the spectral theory of completely continuous operator. The conforming finite element approximation for this problem is analyzed, and the error estimate is given. Finally, the bounds for Steklov eigenvalues on the square domain are provided by Bogner-Fox-Schmit element and Morley element.

A Posteriori Error Estimates for a Steklov Eigenvalue Problem

2012 ◽  
Vol 557-559 ◽  
pp. 2081-2086 ◽  
Author(s):  
Ling Ling Sun ◽  
Yi Du Yang
Keyword(s):  
Eigenvalue Problem ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Error Estimator ◽  
Element Approximation ◽  
A Posteriori ◽  
Validity And Reliability ◽  
A Posteriori Error ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem

This paper discusses the finite element approximation for a Steklov eigenvalue problem. Based on the work of Armentano and Padra ( Appl Numer Math 58 (2008) 593-601), an a posteriori error estimator is provided and its validity and reliability are proved theoretically. Finally, numerical experiments with Matlab program are carried out to confirm the theoretical analysis.

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