On United Boundary-Domain Integro-Differential Equations for Variable Coefficient Dirichlet Problem with General Right-Hand Side

Abstract

he Dirichlet boundary value problem (BVP) for the linear stationary diffusion partial differential equation with a variable coefficient is considered. The PDE right-hand side belongs to the Sobolev spaces H-1(Ω), when neither classical nor canonical co-normal derivatives are well defined. Using an appropriate parametrix (Levi function) the problem is reduced to a direct boundary-domain integro-differential equation (BDIDE) or to a domain integral equation supplemented by the original boundary condition thus constituting a boundary-domain integro-differential problem (BDIDP). Solvability, solution uniqueness, and equivalence of the BDIDE/BDIDP to the original BVP are analysed in Sobolev (Bessel potential) spaces.