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

Math Methods in App Sciences

Wendland

2021

The aim of this paper is to develop a layer potential theory in L 2 -based weighted Sobolev spaces on Lipschitz bounded and exterior domains of R n , n ≥ 3, for the anisotropic Stokes system with L ∞ viscosity tensor coefficient satisfying an ellipticity condition for symmetric matrices with zero matrix trace. To do this, we explore equivalent mixed variational formulations and prove the well-posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of R n , with the given data in L 2 -based weighted Sobolev spaces. These results are used to define the volume (Newtonian) and layer potentials and to obtain their properties. Then, we analyze the well-posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system with L ∞ symmetrically elliptic tensor coefficient by representing their solutions in terms of the obtained volume and layer potentials.

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confidence: 99%

Layer potential theory for the anisotropic Stokes system with variable L_∞ symmetrically elliptic tensor coefficient

Kohr

Math Methods in App Sciences

Wendland

2021

“…In [9,10], the localized boundary-domain singular integral equations (LBDSIE) approach is developed for the Dirichlet, Neumann, Robin and mixed boundary value problems for a scalar elliptic second-order PDE with variable coefficients (see also [8]). In [11], the LBDSIE system approach was used to investigate the Dirichlet problem for a self-adjoint second-order strongly elliptic system of PDEs with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

Localized boundary-domain singular integral equations of the Robin type problem for self-adjoint second-order strongly elliptic PDE systems

Chkadua

Georgian Mathematical Journal

Natroshvili

2020

The paper deals with the three-dimensional Robin type boundary-value problem (BVP) for a second-order strongly elliptic system of partial differential equations in the divergence form with variable coefficients. The problem is studied by the localized parametrix based potential method. By using Green’s representation formula and properties of the localized layer and volume potentials, the BVP under consideration is reduced to the a system of localized boundary-domain singular integral equations (LBDSIE). The equivalence between the original boundary value problem and the corresponding LBDSIE system is established. The matrix operator generated by the LBDSIE system belongs to the Boutet de Monvel algebra. With the help of the Vishik–Eskin theory based on the Wiener–Hopf factorization method, the Fredholm properties of the corresponding localized boundary-domain singular integral operator are investigated and its invertibility in appropriate function spaces is proved.

“…Equivalence of BDIEs to the boundary problems and invertibility of BDIE operators in L 2 and L pbased Sobolev spaces have been analyzed in these works. Localized boundary-domain integral equations based on a harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients have been also developed, see [19] and the references therein.Brewster et al in [11] used a variational approach to show well-posedness results for Dirichlet, Neumann and mixed problems for higher order divergence-form elliptic equations with L ∞ coefficients in locally (ǫ, δ)-domains and in Besov and Bessel potential spaces. Sayas and Selgas in [54] developed a variational approach for the constant-coefficient Stokes layer potentials, by using the technique of Nédélec [51].…”

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confidence: 99%

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confidence: 99%

Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier–Stokes systems with L_∞ strongly elliptic coefficient tensor

Kohr

Complex Variables and Elliptic Equations

Wendland

2019

We obtain well-posedness results in Lp-based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier-Stokes systems with L∞ strongly elliptic coefficient tensor, in complementary Lipschitz domains of R n , n ≥ 3. The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces two linear transmission problems for the anisotropic Stokes system to equivalent mixed variational formulations with data in Lp-based weighted Sobolev and Besov spaces. We show that such a mixed variational formulation is well-posed in the space H 1 p (R n ) n × Lp(R n ), n ≥ 3, for any p in an open interval containing 2. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system. Various mapping properties of these operators are also obtained. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of the nonlinear transmission problem for the anisotropic Stokes and Navier-Stokes systems in Lp-based weighted Sobolev spaces, whenever the given data are small enough.for an unsteady exterior Stokes problem). The authors in [34] obtained mapping properties of the constant-coefficient Stokes and Brinkman layer potential operators in standard and weighted Sobolev spaces by exploiting results of singular integral operators (see also [35,36]).The methods of layer potential theory play also a significant role in the study of elliptic boundary problems with variable coefficients. Mitrea and Taylor [49, Theorem 7.1] used the technique of layer potentials to prove the well-posedness of the Dirichlet problem for the Stokes system in L p -spaces on arbitrary Lipschitz domains in a compact Riemannian manifold. Dindos and Mitrea [24, Theorems 5.1, 5.6, 7.1,7.3] used a boundary integral approach to show well-posedness results in Sobolev and Besov spaces for Poisson problems of Dirichlet type for the Stokes and Navier-Stokes systems with smooth coefficients in Lipschitz domains on compact Riemannian manifolds. A layer potential analysis of pseudodifferential operators of Agmon-Douglis-Nirenberg type in Lipschitz domains on compact Riemannian manifolds has been developed in [39]. The authors in [37] used a layer potential approach and a fixed point theorem to show well-posedness of transmission problems for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems with smooth coefficients in Lipschitz domains on compact Riemannian manifolds. Choi and Lee [21] proved the well-posedness in Sobolev spaces for the Dirichlet problem for the Stokes system with non-smooth coefficients in a Lipschitz domain Ω ⊂ R n (n ≥ 3) with a small Lipschitz constant when the coefficients have vanishing mean oscillations (VMO) with respect to all variables. Choi and Yang [22] established existence and pointwise bound of the fundamental solution for the Stokes system with measurable coefficients in the spac...