Spectral Approximation of a Boundary Condition for an Eigenvalue Problem

1995 ◽  
Vol 32 (4) ◽  
pp. 1263-1279 ◽  
Author(s):  
Anne-Sophie Bonnet-Bendhia ◽  
Nabil Gmati
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Spectral Approximation

scholarly journals Remarks on a nonlinear nonlocal operator in Orlicz spaces

2019 ◽  
Vol 9 (1) ◽  
pp. 305-326 ◽  
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 $f\left( u \right)=\lambda \psi \left( u \right).$

The regularity and stability of solutions to semilinear fourth-order elliptic problems with negative exponents

2016 ◽  
Vol 146 (1) ◽  
pp. 195-212 ◽  
Author(s):  
Baishun Lai
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Eigenvalue Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Entire Solution ◽  
Extremal Solution ◽  
Stability Of Solutions ◽  
The Stability ◽  
Image Position ◽  
Nonlinear Eigenvalue

We examine the regularity of the extremal solution of the nonlinear eigenvalue problemon a general bounded domainΩin ℝN, with Navier boundary conditionu= Δuon ∂Ω. Firstly, we prove the extremal solution is smooth for anyp> 1 andN⩽ 4, which improves the result of Guo and Wei (Discrete Contin. Dynam. Syst.A34(2014), 2561–2580). Secondly, ifp= 3,N= 3, we prove that any radial weak solution of this nonlinear eigenvalue problem is smooth in the caseΩ= 𝔹, which completes the result of Dávilaet al. (Math. Annalen348(2009), 143–193). Finally, we also consider the stability of the entire solution of Δ2u= –l/upin ℝNwithu> 0.

Expansions in terms of sets of functions with complex eigenvalues

1948 ◽  
Vol 44 (2) ◽  
pp. 242-250 ◽  
Author(s):  
R. Peierls
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Real Function ◽  
Liouville Problem ◽  
Nuclear Collisions ◽  
Complex Eigenvalues ◽  
Sturm Liouville ◽  
Image Position ◽  
Unusual Type

In the following I discuss the properties, in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type.The problem arises in the theory of nuclear collisions, and for our present purpose we take it in the simplified formwhere 0 ≤ x ≤ 1. V(x) is a given real function, which we assume to be integrable and to remain between the bounds ± M, and W is an eigenvalue. The eigenfunction ψ(x) is subject to the boundary conditionsand

Riser-Soil Interaction: Local Dynamics at TDP and a Discussion on the Eigenvalue and the VIV Problems

2005 ◽  
Vol 128 (1) ◽  
pp. 39-55 ◽  
Author(s):  
Celso P. Pesce ◽  
Clóvis de A. Martins ◽  
Lauro M. Y. da Silveira
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Flexural Rigidity ◽  
Nonlinear Eigenvalue Problem ◽  
Dynamic Tension ◽  
Soil Stiffness ◽  
Steel Catenary Riser ◽  
Type Boundary ◽  
Touchdown Point ◽  
Type Boundary Condition

The eigenvalue problem of risers is of utmost importance, particularly if vortex-induced vibration (VIV) is concerned. Design procedures rely on the determination of eigenvalues and eigenmodes. Natural frequencies are not too sensitive to the proper consideration of boundary condition, within a certain extent where dynamics at the touchdown area (TDA) may be modeled as dominated by the dynamics of the suspended part. However, eigenmodes may be strongly affected in this region because, strictly speaking, this is a nonlinear one-side (contact-type) boundary condition. Actually, we shall consider a nonlinear eigenvalue problem. Locally, at TDA, riser flexural rigidity and soil interaction play important roles and may affect the dynamic curvature. Extending and merging former analytical solutions on touchdown point (TDP) dynamics and on the eigenvalue problem, obtained through asymptotic and perturbation methods, the present work critically address soil and bending stiffness effects a little further. As far as linear soil stiffness and planar dynamics hypotheses may be considered valid, it is shown that penetration in the soil is small and that its effect does not change significantly the bending loading that is mainly caused by the cyclic excursion of the TDP and corresponding dynamic tension. A comparison of the analytical results with a full nonlinear time-domain simulation shows a remarkable agreement for a typical steel catenary riser. The WKB approximation for the eigenvalue problem gives good estimates for TDP excursion. As the dynamic tension caused solely by VIV is very small, the merged analytical solution may be used as a first estimate of the curvature variation at TDP in the cases of current perpendicular to the “riser plane.”

scholarly journals Spectral approximation theorems for bounded linear operators

1973 ◽  
Vol 8 (2) ◽  
pp. 279-287 ◽  
Author(s):  
W.S. Lo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Eigenvalue Problem ◽  
Linear Operators ◽  
Spectral Approximation ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Approximation Theorems ◽  
Generalized Eigenvectors

In this paper we present some approximation theorems for the eigenvalue problem of a compact linear operator defined on a Banach space. In particular we examine: criteria for the existence and convergence of approximate eigenvectors and generalized eigenvectors; relations between the dimensions of the eigenmanifolds and generalized eigenmanifolds of the operator and those of the approximate operators.

Riser-Soil Interaction: Local Dynamics at TDP and a Discussion on the Eigenvalue Problem

2004 ◽  
Author(s):  
C. P. Pesce ◽  
C. A. Martins
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Perturbation Methods ◽  
Flexural Rigidity ◽  
Nonlinear Eigenvalue Problem ◽  
Design Procedures ◽  
Type Boundary ◽  
Type Boundary Condition ◽  
Proper Consideration

The eigenvalue problem of risers is of outmost importance, particularly if vortex-induced vibration (VIV) is concerned. Design procedures rely on the determination of eigenvalues and eigenmodes. Natural frequencies are not too sensitive to the proper consideration of boundary condition, within a certain extent where dynamics at the Touch down Area (TDA) may be modeled as dominated by the suspended part dynamics. Nevertheless, eigenmodes may be strongly affected in this region, as, strictly speaking, this is a nonlinear one-sided (contact type) boundary condition. Actually, we should consider a nonlinear eigenvalue problem. Locally, at TDA, riser flexural rigidity and soil interaction play important roles, affecting the dynamic curvature. Extending and jointing together former analytical solutions on TDP dynamics and on the eigenvalue problem, obtained through asymptotic and perturbation methods, the present work critically address soil and bending stiffness effects a little further.

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