A New Family of Boundary-Domain Integral Equations for the Dirichlet Problem of the Diffusion Equation in Inhomogeneous Media with $H^{-1}(\Omega )$ Source Term on Lipschitz Domains

Abstract

The interior Dirichlet boundary value problem for the diffusion equation in non-homogeneous media is reduced to a system of Boundary-Domain Integral Equations (BDIEs) employing the parametrix obtained in (Fresneda-Portillo, 2019) different from (Chkadua et. al 2009). We further extend the results obtained in (Fresneda-Portillo, 2019) for the mixed problem in a smooth domain with $L^2(\Omega )$ right hand side to Lipschitz domains and source term $f$ in the Sobolev space $H^{-1}(\Omega )$, where neither the classical nor the canonical co-normal derivatives are well defined. Equivalence between the system of BDIEs and the original BVP is proved along with their solvability and solution uniqueness in appropriate Sobolev spaces.