ON AN INHOMOGENEOUS BOUNDARY VALUE PROBLEM WITH SHIFT INTO THE REGION FOR THE CAUCHY-RIEMANN EQUATIONS WITH SPECTRAL PARAMETER

A.B Amanzholova,N.S Imanbaev,A.D Niyazymbetov

doi:10.12732/ijpam.v93i3.13

A.B Amanzholova, N.S Imanbaev + Show 1 more

Open Access

https://doi.org/10.12732/ijpam.v93i3.13

Copy DOI

Abstract

Abstract: In this paper we consider the spectral problem for the CauchyRiemann operator with Bicadze Samarskii type boundary value conditions, reduced to a singular integral equation with continuous kernel. Moreover, we characterize those spectral parameters at which the inhomogeneous boundary value problem with shift into the region for Cauchy-Riemann equations is everywhere solvable in the class of continuous functions on the unit circle.

Highlights

Boundary value problems, arising in thermal conductivity, were formulated by V.A
Contents of recent publications have led to the realization of high quality novelty of boundary value problems with shifts for the theory of partial differential equations
Boundary value problems for the generalized Cauchy-Riemann system with non smooth coefficients were investigated in [11]-[12]

Summary

Introduction

Boundary value problems, arising in thermal conductivity, were formulated by V.A. Steklov (in 1922), and in gas dynamics - by F.I. Contents of recent publications have led to the realization of high quality novelty of boundary value problems with shifts for the theory of partial differential equations. Saigo operators in the boundary condition was introduced, and questions about uniqueness and non uniqueness of solution of the problem at various functions and values of the constants, appearing in the boundary conditions, were studied. Abundance of publications, where the increasingly common situations are studied, sometimes produces the impression that the theory of boundary value problems ”with shift” has been already completed. Generality of the above questions for partial differential equations forces further to impose a number of very severe restrictions on studied operators. Figuring out the correct formulation of the problem and studying the specific properties of the solutions for the ”non-classical” equations it is convenient to begin by considering the idealized models, for example, by considering equations with constant coefficients. A pair G and S are defined , for which (b) is true

Statement of the Problem

Main Result