Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility

Numerical Methods Partial

Natroshvili

2010

Self Cite

Segregated direct boundary-domain integral equation (BDIE) systems associated with mixed, Dirichlet and Neumann boundary value problems (BVPs) for a scalar "Laplace" PDE with variable coefficient are formulated and analysed for domains with interior cuts (cracks). The main results established in the paper are the BDIE equivalence to the original BVPs and invertibility of the BDIE operators in the corresponding Sobolev spaces.

Section: Segregated Bdies For the Problem (Mc)mentioning

confidence: 95%

Section: σ) Solves One Of the Bdie Systems (Mc11) (Mc12) (Mc21) Or mentioning

confidence: 99%

“…possesses a weak singularity of type O(|x − y| −2 ) for small |x − y|, see [11,2]. Further we introduce parametrix-based surface potential operators…”

Section: Surface and Volume Potentials And The Third Green Identitymentioning

confidence: 99%

“…Mapping and jump properties of operators (3.4)-(3.9) in Sobolev spaces are given in [2] (see also the Appendix below). Particularly, by Theorems A.3 and B.1 of the Appendix, for…”

Section: Surface and Volume Potentials And The Third Green Identitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analysis of segregated boundary‐domain integral equations for variable‐coefficient problems with cracks

Chkadua

Numerical Methods Partial

Natroshvili

2010

Self Cite

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

Chkadua

Math Methods in App Sciences

Natroshvili

2016

Self Cite

The paper deals with the three-dimensional Dirichlet boundary value problem (BVP) for a second-order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary-domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary-domain integral equation system is studied. We establish that the obtained localized boundary-domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener-Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces.

On the mixed problem for the semilinear Darcy‐Forchheimer‐Brinkman PDE system in Besov spaces on creased Lipschitz domains

Gutt

Kohr

Math Methods in App Sciences

et al. 2017

The purpose of this paper is to study the mixed Dirichlet-Neumann boundary value problem for the semilinear Darcy-Forchheimer-Brinkman system in L p -based Besov spaces on a bounded Lipschitz domain in R 3 , with p in a neighborhood of 2. This system is obtained by adding the semilinear term |u|u to the linear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and nontangential traces, as well as between the weak canonical conormal derivatives and the nontangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well-posedness results for the Dirichlet and Neumann problems in L p -based Besov spaces on bounded Lipschitz domains in R n (n ≥ 3) are also presented. Then, using integral potential operators, we show the well-posedness in L 2 -based Sobolev spaces for the mixed problem of Dirichlet-Neumann type for the linear Brinkman system on a bounded Lipschitz domain in R n (n ≥ 3). Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann-to-Dirichlet operator, we extend the well-posedness property to some L p -based Sobolev spaces. Next, we use the well-posedness result in the linear case combined with a fixed point theorem to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkman system in L p -based Besov spaces, with p ∈ (2 − , 2 + ) and some parameter > 0. KEYWORDSSemilinear Darcy-Forchheimer-Brinkman system, mixed Dirichlet-Neumann problem, L p -based Besov spaces, layer potential operators, Neumann-to-Dirichlet operator, existence and uniqueness INTRODUCTIONBoundary integral methods are a powerful tool to investigate linear elliptic boundary value problems that appear in various areas of science and engineering (see, eg, previous studies 1-5 ). Among many valuable contributions in the field, we mention the well-posedness result of the Dirichlet problem for the Stokes system in Lipschitz domains in R n (n ≥ 3) with boundary data in L 2 -based Sobolev spaces, which have been obtained by Fabes, Kenig, and Verchota 6 by using a layer potential analysis. Also, Mitrea and Wright 7 obtained the well-posedness results for Dirichlet, 7780Neumann, and transmission problems for the Stokes system on arbitrary Lipschitz domains in R n (n ≥ 2), with data in Sobolev and Besov-Triebel-Lizorkin spaces. By using a boundary integral method, Mitrea and Taylor 5 obtained well-posedness results for the Dirichlet problem for the Stokes system on arbitrary Lipschitz domains on a compact Riemannian manifold, with boundary data in L 2 . Their results extended the results of Fabes, Kenig, and Verchota 6 from the Euclidean setting to the case of compact Riemannian manifolds. Continuing the study of Mitrea and Taylor 5 , Dindos and Mitrea 3 developed a layer potential analysis to obtain existence and uniqueness results for the Poiss...