Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Sergey E Mikhailov

doi:10.1186/s13661-018-0992-0

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 Hölder-continuous coefficients 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), frac{1}{2}< s<frac{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, and the 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.

Highlights

Many applications in science and engineering can be modelled by boundary-value problems (BVPs) for partial differential equations with variable coefficients
Using a parametrix (Levi function) introduced in [20, 25] as a substitute of a fundamental solution, it is possible to reduce such a BVP to a system of boundary-domain integral equations, BDIEs
The operators associated with the lefthand sides of all the BDIE systems were analysed in the corresponding Sobolev spaces

Summary

Introduction

Many applications in science and engineering can be modelled by boundary-value problems (BVPs) for partial differential equations with variable coefficients. Theorem 2.9 (Lemma 4.3 in [27], Theorem 3.2 in [31], and Theorem 5.3 in [32]) Under the hypotheses of Definition 2.8, the generalised co-normal derivatives T±u(f±; u) are independent of (non-unique) choice of the operator γ –1, and we have the estimate. Theorem 2.13 (Theorem 3.9 in [31] and Theorem 6.6 in [32]) Under the hypotheses of Definition 2.12, the canonical co-normal derivatives T±u are independent of (non-unique) choice of the operator. Note that the canonical co-normal derivatives coincide with the classical conormal derivatives T±u = Tc±u if the latter do exist (see [32, Corollaries 6.11 and 6.14]), which is generally not the case for the generalised conormal derivatives even for smooth functions u, unless f± = A ± u is chosen.

Parametrix and potential type operators on Lipschitz domains

Segregated BDIE systems for the Dirichlet problem

BDIE formulations and equivalence to the Dirichlet problem

BDIE formulations and equivalence to the Neumann problem

Concluding remarks