scholarly journals On Eigenfunctions and Eigenvalues of a Nonlocal Laplace Operator with Multiple Involution

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1781
Author(s):  
Batirkhan Turmetov ◽  
Valery Karachik
Keyword(s):  
Boundary Value Problem ◽  
Unit Ball ◽  
Explicit Form ◽  
Laplace Operator ◽  
Laplace Equation ◽  
Boundary Value

We study the eigenfunctions and eigenvalues of the boundary value problem for the nonlocal Laplace equation with multiple involution. An explicit form of the eigenfunctions and eigenvalues for the unit ball are obtained. A theorem on the completeness of the eigenfunctions of the problem under consideration is proved.

scholarly journals Existence results for a nonlinear nonautonomous transmission problem via domain perturbation

2021 ◽  
pp. 1-26
Author(s):  
Matteo Dalla Riva ◽  
Riccardo Molinarolo ◽  
Paolo Musolino
Keyword(s):  
Boundary Value Problem ◽  
Laplace Equation ◽  
Boundary Value ◽  
Perturbation Parameter ◽  
Existence Results ◽  
Transmission Problem ◽  
Perforated Domain ◽  
Analytic Dependence ◽  
Domain Perturbation ◽  
Fixed Domain

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$ . First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$ .

scholarly journals The Laplace equation in the exterior of the Hankel contour and novel identities for hypergeometric functions

2013 ◽  
Vol 469 (2157) ◽  
pp. 20130081 ◽  
Author(s):  
A. S. Fokas ◽  
M. L. Glasser
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Laplace Equation ◽  
Special Functions ◽  
Boundary Value ◽  
Zeta Functions ◽  
Neumann Boundary ◽  
Neumann Boundary Value Problem ◽  
Wide Range ◽  
Definition Of

By using conformal mappings, it is possible to express the solution of certain boundary-value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identity for special functions. A convenient tool for deriving this type of identity is the so-called global relation , which has appeared recently in a wide range of boundary-value problems. As a concrete application, we analyse the Neumann boundary-value problem for the Laplace equation in the exterior of the Hankel contour, which appears in the definition of both the gamma and the Riemann zeta functions. By using the explicit solution of this problem, we derive a number of novel identities involving the hypergeometric function. Also, we point out an interesting connection between the solution of the above Neumann boundary-value problem for a particular set of Neumann data and the Riemann hypothesis.

scholarly journals Explicit solution of the jump problem for the Laplace equation and singularities at the edges

2001 ◽  
Vol 7 (1) ◽  
pp. 1-13 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Boundary Value Problem ◽  
Explicit Form ◽  
Laplace Equation ◽  
Explicit Solution ◽  
Existence Theorems ◽  
Single Layer ◽  
Normal Derivative ◽  
Jump Problem ◽  
Uniqueness And Existence ◽  
Conditions At Infinity

The boundary value problem for the Laplace equation outside several cuts in a plane is studied. The jump of the solution of the Laplace equation and the jump of its normal derivative are specified on the cuts. The problem is studied under different conditions at infinity, which lead to different uniqueness and existence theorems. The solution of this problem is constructed in the explicit form by means of single layer and angular potentials. The singularities at the ends of the cuts are investigated.

Sign in / Sign up

Export Citation Format

Share Document