A New Family of Boundary-Domain Integral Equations for the Dirichlet Problem of the Diffusion Equation in Inhomogeneous Media with H-1(Ω) 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 (missing citation) different from (missing citation). We further extend the results obtained in (missing citation) for the mixed problem in a smooth domain with L 2 (Ω) right hand side to Lipschitz domains and PDE right-hand in the Sobolev space H −1 (Ω), 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.

On the Existence and Uniqueness of Solution of Boundary-Domain Integral Equations for the Dirichlet Problem for the Non-Homogeneous Heat Transfer Equation defined on a 2D Unbounded Domain

“…We will now introduce the following weighted Sobolev spaces [17,18,19] which are useful when dealing with exterior problems, since constant functions are allowed to be solutions of the problem. In order to define weighted Sobolev spaces in R 2 , we will make use of the weight [1] ω 2 : Ω −→ R + given by ω 2 (x) = (1 + |x| 2 ) 1 2 ln(2 + |x| 2 ).…”

“…The parametrix-based logarithmic and remainder potential operators are respectively defined, similar to [7,19] in the 3D case for y ∈ R 2 , as…”