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
• C. Fresneda-Portillo,
• Zenebe Woldemicheal
C. Fresneda-Portillo
Oxford Brookes Univ
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Zenebe Woldemicheal
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## 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 \cite{carlos2} different from \cite{mikhailov1}. We further extend the results obtained in \cite{carlos2} for the mixed problem in a smooth domain with $$L^{2}(\Omega)$$ right hand side to Lipschitz domains and PDE right-hand 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.

#### Peer review status:ACCEPTED

18 Feb 2020Submitted to Mathematical Methods in the Applied Sciences
22 Feb 2020Submission Checks Completed
22 Feb 2020Assigned to Editor
22 Feb 2020Reviewer(s) Assigned
08 Jun 2020Review(s) Completed, Editorial Evaluation Pending
08 Jun 2020Editorial Decision: Revise Minor