Analysis of Segregated Boundary-Domain Integral Equations for BVPs with Non-smooth Coefficient on Lipschitz Domains

TL;DR

This paper develops and analyzes boundary-domain integral equations for boundary value problems with non-smooth coefficients on Lipschitz domains, establishing their equivalence, solvability, and conditions for invertibility.

Contribution

It formulates segregated BDIEs for PDEs with limited-smoothness coefficients on Lipschitz domains and analyzes their properties, including invertibility and stabilization techniques.

Findings

BDIEs are equivalent to original BVPs.

BDIE operators are not invertible without perturbation.

Finite-dimensional perturbations can stabilize operators.

Abstract

Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable non-smooth (or limited-smoothness) coefficient on Lipschitz domains are formulated. The PDE right hand sides belong to the Sobolev (Bessel-potential) space $H^{s-2}(\Omega)$ or $\widetilde H^{s-2}(\Omega)$, $1/2<s<3/2$, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, as well as Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, however some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators.