Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

O Chkadua,S E Mikhailov,D Natroshvili

doi:10.1002/mma.4100

Abstract

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.

Highlights

We consider the Dirichlet boundary value problem (BVP) for a second-order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develop the generalized integral potential method based on a localized parametrix
The BVP treated in the paper is well investigated in the literature by the variational method and by the classical integral potential method, when the corresponding fundamental solution is available in explicit form (e.g. [1,2,3]) or when at least its properties are known to be good enough
We show that a solution of the problem can be represented by explicit localized parametrix-based potentials and that the corresponding localized boundary-domain integral operator (LBDIO) is invertible, which is important for analysis of convergence and stability of localized boundary-domain integral equation (LBDIE)-based numerical methods for PDEs (e.g. [6,7,8,9,10,11,12,13])

Summary

Introduction

We consider the Dirichlet boundary value problem (BVP) for a second-order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develop the generalized integral potential method based on a localized parametrix. Our goal here is to develop a localized integral potential method for general second-order strongly elliptic self-adjoint systems of partial differential equations with variable coefficients. In the references [14,15,16,17,18,19,20], the traditional and localised boundary-domain integral equation methods have been developed for the case of scalar elliptic second-order partial differential equations with variable coefficients, and here, we extend the LBDIE method to PDE systems

Boundary value problem and parametrix-based operators

Parametrix-based operators and integral identities

Symbols and invertibility of a domain operator in the half-space

Invertibility of the Dirichlet localized boundary-domain integral operator

XCk for integer k