On the growth of the eigenvalues of the Laplacian operator in a quasibounded domain

1968 ◽  
Vol 31 (5) ◽  
pp. 352-356 ◽  
Author(s):  
Colin Clark
Keyword(s):  
Laplacian Operator ◽  
Eigenvalues Of The Laplacian

Eigenvalues of the Laplacian Operator and Neighborhood Enlargement for Compact Manifolds

2020 ◽  
pp. 84-102
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Compact Manifolds ◽  
Laplacian Operator ◽  
Measure Concentration ◽  
Eigenvalues Of The Laplacian ◽  
The Laplace Operator ◽  
Concentration Phenomenon

In this investigation and under the conception of measure concentration phenomenon we found that the enlargement of the neighborhood for an n – dimensional compact Riemannian manifolds (M,g) relative to the eigenvalues λ of the Laplace operator ∆ on (M,g). And we found that r~1/√λ. تناول هذا البحث تجسيم الجوار لمتعدد الطيات المتراص في البعد n وذلك باستخدام مفهوم تركيز الحجم. كما وجدنا أن نصف القطر r لتجسيم الجوار لمتعدد الطيات يرتبط مع القيم الذاتية λ لمؤثر لابلاس Δ على متعدد الطيات. الكلمات المفتاحية: نصف قطر تجسيم الجوار، متباينات متساوي المقاييس، تركيز الحجم، مؤثر لابلاس، القيم الذاتية لمؤثر لابلاس.

scholarly journals Eigenvalues of the Laplacian with Neumann boundary conditions

1985 ◽  
Vol 26 (3) ◽  
pp. 293-309 ◽  
Author(s):  
H. P. W. Gottlieb
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Summation Formula ◽  
Laplacian Operator ◽  
Eigenvalues Of The Laplacian ◽  
Annular Sector ◽  
Simply Connected

AbstractVarious grometrical properties of a domain may be elicited from the asymptotic expansion of a spectral function of the Laplacian operator for that region with apporpriate boundary conditions. Explicit calculations, using analytical formulae for the eigenvalues, are performed for the cases fo Neumann and mixed boundary conditions, extending earlier work involving Dirichet boundary conditions. Two- and three-dimensional cases are considered. Simply-connected regions dealt with are the rectangle, annular sector, and cuboid. Evaluations are carried out for doubly-connected regions, including the narrow annulus, annular cylinder, and thin concentric spherical cavity. The main summation tool is the Poission summation formula. The calculations utilize asymptotic expansions of the zeros of the eigenvalue equations involving Bessel and related functions, in the cases of curved boundaries with radius ratio near unity. Conjectures concerning the form of the contributions due to corners, edges and vertices in the case of Neumann and mixed boundary conditions are presented.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

2021 ◽  
pp. 1-16
Author(s):  
Alexander Dabrowski
Keyword(s):  
General Type ◽  
Differential Operators ◽  
Laplacian Operator ◽  
Variational Characterization ◽  
Repeated Eigenvalues ◽  
Direct Extension ◽  
Asymptotic Formulae ◽  
Elliptic Differential Operators ◽  
Domain Perturbations ◽  
Boundary Deformation

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

scholarly journals Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abderrazak Nabti ◽  
Ahmed Alsaedi ◽  
Mokhtar Kirane ◽  
Bashir Ahmad
Keyword(s):  
Diffusion Equation ◽  
Global Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Laplacian Operator ◽  
Nonlocal Source ◽  
Time Space ◽  
Nonexistence Of Solutions ◽  
Beta 2 ◽  
Fractional Laplacian Operator

Abstract We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source $$\begin{aligned} u_{t} + (-\Delta )^{\frac{\beta }{2}} u =\bigl(1+ \vert x \vert \bigr)^{ \gamma } \int _{0}^{t} (t-s)^{\alpha -1} \vert u \vert ^{p} \bigl\Vert \nu ^{ \frac{1}{q}}(x) u \bigr\Vert _{q}^{r} \,ds \end{aligned}$$ u t + ( − Δ ) β 2 u = ( 1 + | x | ) γ ∫ 0 t ( t − s ) α − 1 | u | p ∥ ν 1 q ( x ) u ∥ q r d s for $(x,t) \in \mathbb{R}^{N}\times (0,\infty )$ ( x , t ) ∈ R N × ( 0 , ∞ ) with initial data $u(x,0)=u_{0}(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})$ u ( x , 0 ) = u 0 ( x ) ∈ L loc 1 ( R N ) , where $p,q,r>1$ p , q , r > 1 , $q(p+r)>q+r$ q ( p + r ) > q + r , $0<\gamma \leq 2 $ 0 < γ ≤ 2 , $0<\alpha <1$ 0 < α < 1 , $0<\beta \leq 2$ 0 < β ≤ 2 , $(-\Delta )^{\frac{\beta }{2}}$ ( − Δ ) β 2 stands for the fractional Laplacian operator of order β, the weight function $\nu (x)$ ν ( x ) is positive and singular at the origin, and $\Vert \cdot \Vert _{q}$ ∥ ⋅ ∥ q is the norm of $L^{q}$ L q space.

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