Layer potentials and Poisson problems for the nonsmooth coefficient Brinkman system in Sobolev and Besov spaces
type: Item
instance of: scholarly article
title: Layer potentials and Poisson problems for the nonsmooth coefficient Brinkman system in Sobolev and Besov spaces
zbMATH Open document ID: 1447.35139
author: Mirela Kohr
author: W. L. Wendland
published in: Journal of Mathematical Fluid Mechanics
publication date: 2018-12-21T00:00:00Z
review text: The authors consider the Poisson problem with Dirichlet, Neumann and mixed boundary conditions for the Brinkman system with measurable coefficients in \(L^p\)-based Sobolev and Besov spaces in a Lipschitz subset of a compact Riemaniann manifold with no boundary of dimension \(m\geq 2\). The authors first treat the case with homogeneous boundary conditions and show that the problem is well posed for \(p=2\) and deduce that the problem is well posed for \(p\) close to \(p=2\) by exploiting a stability result for isomorphisms in function space interpolation scales. Then the authors consider the case with nonhomogeneous boundary conditions. In the specific case \(p=2\) the authors can also prove a layer potential representation for the solutions by exploiting layer potentials that can be introduced as solutions of certain boundary value problems.
reviewed by: Massimo Lanza de Cristoforis
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 6996689
zbMATH Keywords: Brinkman system with \(L^\infty \) coefficients
zbMATH Keywords: boundary value problems
zbMATH Keywords: well-posedness
zbMATH Keywords: variational approach
zbMATH Keywords: Newtonian and layer potentials
MaRDI profile type: Publication
full work available at URL: s00021-018-0394-1
OpenAlex ID: W2890843548
cites work: Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains
cites work: Q5409465
cites work: Topological methods in hydrodynamics
cites work: The finite element method with Lagrangian multipliers
cites work: Layer potentials for general linear elliptic systems
cites work: Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces
cites work: Boundary Integral Solvers for an Evolutionary Exterior Stokes Problem
cites work: Q4112938
cites work: Extending Sobolev functions with partially vanishing traces from locally \(({\epsilon},{\delta})\)-domains and applications to mixed boundary problems
cites work: Q4105995
cites work: Mixed and Hybrid Finite Element Methods
cites work: Mixed boundary value problems for the Stokes system
cites work: A Riemannian version of Korn's inequality
cites work: Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient. I: Equivalence and invertibility
cites work: Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients
cites work: The Green function for the Stokes system with measurable coefficients
cites work: Fundamental solutions for stationary Stokes systems with measurable coefficients
cites work: On Korn's inequalities on a surface
cites work: Boundary Integral Operators on Lipschitz Domains: Elementary Results
cites work: On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains
cites work: On the inequalities of Babuška-Aziz, Friedrichs and Horgan-Payne
cites work: The stationary Navier-Stokes system in nonsmooth manifolds: the Poisson problem in Lipschitz and \(C^{1}\) domains
cites work: The Kato square root problem for mixed boundary conditions
cites work: Theory and practice of finite elements.
cites work: The Dirichlet problem for the Stokes system on Lipschitz domains
cites work: Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains
cites work: An Introduction to the Mathematical Theory of the Navier-Stokes Equations
cites work: Coupling of mixed finite elements and boundary elements for linear and nonlinear elliptic problems
cites work: Finite Element Methods for Navier-Stokes Equations
cites work: A \(W^{1,p}\)-estimate for solutions to mixed boundary value problems for second order elliptic differential equations
cites work: The equations of elastostatics in a Riemannian manifold
cites work: Elliptic and parabolic regularity for second- order divergence operators with mixed boundary conditions
cites work: The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients
cites work: The method of layer potentials inLpand endpoint spaces for elliptic operators withL∞coefficients
cites work: Boundary integral equations
cites work: The inhomogeneous Dirichlet problem in Lipschitz domains
cites work: Quasiconformal mappings and extendability of functions in Sobolev spaces
cites work: Q4347092
cites work: On the Navier-Stokes equations on manifolds with curvature
cites work: Integral potential method for a transmission problem with Lipschitz interface in \(\mathbb{R}^{3}\) for the Stokes and Darcy-Forchheimer-Brinkman PDE systems
cites work: Nonlinear Neumann-transmission problems for Stokes and Brinkman equations on Euclidean Lipschitz domains
cites work: Poisson problems for semilinear Brinkman systems on Lipschitz domains in \(\mathbb{R}^n\)
cites work: Transmission problems for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems in Lipschitz domains on compact Riemannian manifolds
cites work: Layer Potential Analysis for Pseudodifferential Matrix Operators in Lipschitz Domains on Compact Riemannian Manifolds: Applications to Pseudodifferential Brinkman Operators
cites work: Q4827397
cites work: Variational approach for the Stokes and Navier-Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds
cites work: Introduction to differentiable manifolds.
cites work: Static Elasticity in a Riemannian Manifold
cites work: Lp estimates of solutions tomixed boundary value problems for the Stokes system in polyhedral domains
cites work: Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks
cites work: Q4953888
cites work: Bounded solutions of the Dirichlet problem for the Stokes resolvent system
cites work: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
cites work: Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains
cites work: The Hodge-Laplacian. Boundary value problems on Riemannian manifolds
cites work: The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains
cites work: The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains
cites work: The regularity of the Stokes operator and the Fujita-Kato approach to the Navier-Stokes initial value problem in Lipschitz domains
cites work: Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem

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Layer potentials and Poisson problems for the nonsmooth coefficient Brinkman system in Sobolev and Besov spaces (DOI)
DOI: 10.1007/S00021-018-0394-1