scholarly journals Spectral Stability of the $${\overline{\partial }}-$$Neumann Laplacian: Domain Perturbations

2022 ◽  
Vol 32 (2) ◽  
Author(s):  
Siqi Fu ◽  
Weixia Zhu
Keyword(s):  
Bounded Domain ◽  
Spectral Stability ◽  
Pseudoconvex Domains ◽  
Continuity Properties ◽  
Quantitative Estimates ◽  
Lower Semi ◽  
Neumann Laplacian ◽  
Domain Perturbations ◽  
Satisfy Property

AbstractWe study spectral stability of the $${\bar{\partial }}$$ ∂ ¯ -Neumann Laplacian on a bounded domain in $${\mathbb {C}}^n$$ C n when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the $${\bar{\partial }}$$ ∂ ¯ -Neumann Laplacian on bounded pseudoconvex domains in $${\mathbb {C}}^n$$ C n , lower semi-continuity properties on pseudoconvex domains that satisfy property (P), and quantitative estimates on smooth bounded pseudoconvex domains of finite D’Angelo type in $${\mathbb {C}}^n$$ C n .

scholarly journals On the Continuity of Orthogonal Sets in the Sense of Operator Orthogonality

2018 ◽  
Vol 11 (3) ◽  
pp. 793-802
Author(s):  
Mahdi Iranmanesh ◽  
M. Saeedi Khojasteh ◽  
M. K. Anwary
Keyword(s):  
Numerical Range ◽  
Alternative Definition ◽  
Normed Spaces ◽  
Operator Approach ◽  
Linear Spaces ◽  
Continuity Properties ◽  
Lower Semi

In this paper, we introduce the operator approach for orthogonality in linear spaces. In particular, we represent the concept of orthogonal vectors using an operator associated with them, in normed spaces. Moreover, we investigate some of continuity properties of this kind of orthogonality. More precisely, we show that the set valued function F(x; y) = {μ : μ ∈ C, p(x − μy, y) = 1} is upper and lower semi continuous, where p(x, y) = sup{pz1,...,zn−2 (x, y) : z1, . . . , zn−2 ∈ X} and pz1,...,zn−2 (x, y) = kPx,z1,...,zn−2,yk−1 where Px,z1,...,zn−2,y denotes the projection parallel to y from X to the subspace generated by {x, z1, . . . , zn−2}. This can be considered as an alternative definition for numerical range in linear spaces.

Limiting dynamics for stochastic FitzHugh–Nagumo equations on large domains

2019 ◽  
Vol 19 (05) ◽  
pp. 1950037 ◽  
Author(s):  
Yangrong Li ◽  
Fuzhi Li
Keyword(s):  
Bounded Domain ◽  
Unbounded Domain ◽  
Bounded Domains ◽  
Limit Set ◽  
Coupled Equations ◽  
Lower Semi ◽  
The Family ◽  
Random Attractors ◽  
Nagumo Equations ◽  
Theoretical Results

This paper is devoted to the convergence of bi-spatial random attractors as a family of bounded domains is extended to be unbounded. Some criteria in terms of expansion and restriction are provided to ensure that the unbounded-domain attractor is approximated by the family of bounded-domain attractors in both upper and lower semi-continuity senses. The theoretical results are applied to show that the stochastic FitzHugh–Nagumo coupled equations have an attractor in [Formula: see text]-times Lebesgue space irrespective of whether the domain is bounded or unbounded. Furthermore, we prove that the family of bounded-domain attractors continuously converges to the unbounded-domain attractor, and the latter can be constructed by the metric-limit set of all bounded-domain attractors.

A NOTE ON BERGMAN COMPLETENESS

2001 ◽  
Vol 12 (04) ◽  
pp. 383-392 ◽  
Author(s):  
BO-YONG CHEN
Keyword(s):  
Bounded Domain ◽  
Pseudoconvex Domain ◽  
Pseudoconvex Domains

The purpose of this note is to deal with two problems of Kobayashi: 1. Which bounded pseudoconvex domain is Bergman complete? 2. Is a bounded domain Bergman complete if it coincides with its outerhull? We verify Problem 1 for a class of pseudoconvex domains which are not necessary hyperconvex. However, we will show that the answer to Problem 2 is negative in general.

scholarly journals Spectral stability of the Neumann Laplacian

2002 ◽  
Vol 186 (2) ◽  
pp. 485-508 ◽  
Author(s):  
V.I. Burenkov ◽  
E.B. Davies
Keyword(s):  
Spectral Stability ◽  
Neumann Laplacian

The spectral radius in a locally convex algebra

1974 ◽  
Vol 76 (3) ◽  
pp. 493-496
Author(s):  
A. W. Wood
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Spectral Theory ◽  
Interesting Property ◽  
Commutative Banach Algebra ◽  
Continuity Properties ◽  
Lower Semi ◽  
Continuous Seminorm ◽  
Locally Convex ◽  
Locally Convex Algebra

In (1), Allan introduced the concept of the spectrum of an element of a locally convex algebra, and developed a spectral theory for pseudo-complete algebras. In a commutative Banach algebra the spectral radius is a continuous seminorm, and so it is natural to investigate continuity properties of the spectral radius in various classes of locally convex algebra. Continuity is too strong a condition to be expected in any general case, and an interesting property to investigate appears to be lower semi-continuity. We shall show easily that in a commutative pseudo-complete locally m-convex algebra (in the sense of (4)) the spectral radius is lower semi-continuous. We shall then exhibit a commutative complete metrizable algebra in which lower semi-continuity fails to hold.

scholarly journals Spectral stability estimates for the Dirichlet and Neumann Laplacian in rough domains

2013 ◽  
Vol 264 (9) ◽  
pp. 2097-2135 ◽  
Author(s):  
Antoine Lemenant ◽  
Emmanouil Milakis ◽  
Laura V. Spinolo
Keyword(s):  
Spectral Stability ◽  
Stability Estimates ◽  
Neumann Laplacian

scholarly journals Conformal spectral stability estimates for the Neumann Laplacian

2016 ◽  
Vol 289 (17-18) ◽  
pp. 2133-2146 ◽  
Author(s):  
V. I. Burenkov ◽  
V. Gol'dshtein ◽  
A. Ukhlov
Keyword(s):  
Spectral Stability ◽  
Stability Estimates ◽  
Neumann Laplacian
Sign in / Sign up

Export Citation Format

Share Document