scholarly journals On the estimates of eigenvalues of the boundary value problem with large parameter

2015 ◽  
Vol 63 (1) ◽  
pp. 101-113 ◽  
Author(s):  
Alexey V. Filinovskiy
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Boundary Value ◽  
Dirichlet Condition ◽  
Large Parameter ◽  
Robin Condition ◽  
The Difference ◽  
The Laplace Operator

Abstract We consider the eigenvalue problem Δu + λu = 0 in Ω with Robin condition + αu = 0 on ∂Ω , where Ω ⊂ Rn , n ≥ 2 is a bounded domain and α is a real parameter. We obtain the estimates to the difference between λDk - λk(α) eigenvalue of the Laplace operator in with Dirichlet condition and the corresponding Robin eigenvalue for large positive values of for all k = 1,2,… We also show sharpness of these estimates in the power of α.

Laplacian structure mirroring surface topography in determining the gravity potential by successive approximations

2021 ◽  
Author(s):  
Petr Holota ◽  
Otakar Nesvadba
Keyword(s):  
Boundary Value Problem ◽  
Laplace Operator ◽  
Boundary Value ◽  
Successive Approximations ◽  
Gravity Potential ◽  
Geodetic Boundary Value Problem ◽  
Physical Surface ◽  
The Earth ◽  
Geodetic Boundary ◽  
The Laplace Operator

<p>Similarly as in other branches of engineering and mathematical physics, a transformation of coordinates is applied in treating the geodetic boundary value problem. It offers a possibility to use an alternative between the boundary complexity and the complexity of the coefficients of the partial differential equation governing the solution. In our case the Laplace operator has a relatively simple structure in terms of spherical or ellipsoidal coordinates which are frequently used in geodesy. However, the physical surface of the Earth and thus also the solution domain substantially differ from a sphere or an oblate ellipsoid of revolution, even if optimally fitted. The situation becomes more convenient in a system of general curvilinear coordinates such that the physical surface of the Earth is imbedded in the family of coordinate surfaces. Applying tensor calculus the Laplace operator is expressed in the new coordinates. However, its structure is more complicated in this case and in a sense it represents the topography of the physical surface of the Earth. The Green’s function method together with the method of successive approximations is used for the solution of the geodetic boundary value problem expressed in terms of the new coordinates. The structure of iteration steps is analyzed and if useful and possible, it is modified by means of integration by parts. Subsequently, the iteration steps and their convergence are discussed and interpreted, numerically as well as in terms of functional analysis.</p>

scholarly journals The stiff Neumann problem: Asymptotic specialty and “kissing” domains

2021 ◽  
pp. 1-36
Author(s):  
V. Chiadò Piat ◽  
L. D’Elia ◽  
S.A. Nazarov
Keyword(s):  
Boundary Value Problem ◽  
Neumann Problem ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Mixed Boundary ◽  
Asymptotic Series ◽  
Mixed Boundary Value Problem ◽  
Smooth Bounded Domain ◽  
Dimension 2 ◽  
The Laplace Operator

We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain Ω ⊂ R d which is divided into two subdomains: an annulus Ω 1 and a core Ω 0 . The density and the stiffness constants are of order ε − 2 m and ε − 1 in Ω 0 , while they are of order 1 in Ω 1 . Here m ∈ R is fixed and ε > 0 is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as ε → 0 for any m. In dimension 2 the case when Ω 0 touches the exterior boundary ∂ Ω and Ω 1 gets two cusps at a point O is included into consideration. The possibility to apply the same asymptotic procedure as in the “smooth” case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as x → O for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given.

scholarly journals EIGENVALUE PROBLEMS WITH WEIGHT AND VARIABLE EXPONENT FOR THE LAPLACE OPERATOR

2010 ◽  
Vol 08 (03) ◽  
pp. 235-246
Author(s):  
MIHAI MIHĂILESCU ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Variable Exponent ◽  
Eigenvalue Problems ◽  
Smooth Boundary ◽  
The Laplace Operator

This paper deals with an eigenvalue problem for the Laplace operator on a bounded domain with smooth boundary in ℝ N (N ≥ 3). We establish that there exist two positive constants λ* and λ* with λ* ≤ λ* such that any λ ∈ (0, λ*) is not an eigenvalue of the problem while any λ ∈ [λ*, ∞) is an eigenvalue of the problem.

scholarly journals ON THE SOLVABILITY OF SOME BOUNDARY VALUE PROBLEMS WITH INVOLUTION

2020 ◽  
Vol 26 (3) ◽  
pp. 7-16
Author(s):  
K. Zh. Nazarova ◽  
B. Kh. Turmetov ◽  
K. I. Usmanov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Laplace Operator ◽  
Poisson Equation ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Boundary Operator ◽  
Uniqueness Of The Solution ◽  
The Laplace Operator

This article is devoted to the study of the solvability of some boundary value problems with involution.In the space Rn, the map Sx=x is introduced. Using this mapping, a nonlocal analogue of the Laplace operator is introduced, as well as a boundary operator with an inclined derivative. Boundary-value problems are studied that generalize the well-known problem with an inclined derivative. Theorems on the existence and uniqueness of the solution of the problems under study are proved. In the Helder class, the smoothness of the solution is also studied. Using well-known statements about solutions of a boundary value problem with an inclined derivative for the classical Poisson equation, exact orders of smoothness of a solution to the problem under study are found.

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