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

Abstract

The paper deals with the three dimensional Dirichlet boundary value problem (BVP) for a second order strongly elliptic self-adjoint system of partial di erential equations in the divergence form with variable coe cients 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.