Localized Boundary-Domain Singular Integral Equations of Dirichlet Problem for Self-adjoint Second Order Strongly Elliptic PDE Systems

TL;DR

This paper develops a boundary-domain integral equation method for solving the Dirichlet problem for self-adjoint elliptic PDE systems, establishing operator properties and invertibility in Sobolev spaces.

Contribution

It introduces a localized parametrix-based integral approach for elliptic systems, proving operator invertibility and Fredholm properties within Boutet de Monvel algebra.

Findings

Localized boundary-domain integral operator belongs to Boutet de Monvel algebra

Established invertibility of the operator in Sobolev spaces

Proved equivalence between Dirichlet BVP and LBDIE system

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 (LBDIEs). The equivalence between the Dirichlet BVP and the corresponding LBDIE 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.