Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three

We consider the unique global solvability of initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole Euclidean space for dimension larger than three. The following sufficient condition is known: initial data is sufficiently small in some weighted Sobolev spaces for the whole space case; the generalized Fourier transform of the initial data is sufficiently small in some weighted Sobolev spaces for the exterior domain case. The purpose of this paper is to give sufficient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation. Copyright © 2004 John Wiley & Sons, Ltd.     

The Dirichlet problem for non‐divergence parabolic equations with discontinuous in time coefficients

We consider the Dirichlet problem for non‐divergence parabolic equation with discontinuous in t coefficients in a half space. The main result is weighted coercive estimates of solutions in anisotropic Sobolev spaces. We give an application of this result to linear and quasi‐linear parabolic equations in a bounded domain. In particular, if the boundary is of class C^1,δ , δ ∈ [0, 1], then we present a coercive estimate of solutions in weighted anisotropic Sobolev spaces, where the weight is a power of the distance to the boundary (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the periodic KdV equation in weighted Sobolev spaces

We prove well-posedness results for the initial value problem of the periodic KdV equation as well as Kam type results in classes of high regularity solutions. More precisely, we consider the problem in weighted Sobolev spaces, which comprise classical Sobolev spaces, Gevrey spaces, and analytic spaces. We show that the initial value problem is well posed in all spaces with subexponential decay of Fourier coefficients, and ‘almost well posed’ in spaces with exponential decay of Fourier coefficients.     

On the first stationary boundary-value problem of elasticity in weighted Sobolev spaces in exterior domains of R3

This paper solves the first boundary-value problem of elasticity both in interior and exterior domains ofR ³. The equations are set in weighted Sobolev spaces for exterior domains that describe the decay of the functions at infinity. The results established include existence, regularity, and convergence of iterations of the solution.This research was supported by the Rashi Foundation.     

Integral potential method for a transmission problem with Lipschitz interface in \mathbb{R}^{3} for the Stokes and Darcy–Forchheimer–Brinkman PDE systems

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the nonlinear Darcy–Forchheimer–Brinkman system and the linear Stokes system in two complementary Lipschitz domains in \({\mathbb{R}^{3}}\), one of them is a bounded Lipschitz domain \({\Omega}\) with connected boundary, and the other one is the exterior Lipschitz domain \({\mathbb{R}^{3} \setminus \overline{\Omega }}\). We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces.     

Graded mesh refinement and error estimates of higher order for DGFE solutions of elliptic boundary value problems in polygons

Error estimates for DGFE solutions are well investigated if one assumes that the exact solution is sufficiently regular. In this article, we consider a Dirichlet and a mixed boundary value problem for a linear elliptic equation in a polygon. It is well known that the first derivatives of the solutions develop singularities near reentrant corner points or points where the boundary conditions change. On the basis of the regularity results formulated in Sobolev–Slobodetskii spaces and weighted spaces of Kondratiev type, we prove error estimates of higher order for DGFE solutions using a suitable graded mesh refinement near boundary singular points. The main tools are as follows: regularity investigation for the exact solution relying on general results for elliptic boundary value problems, error analysis for the interpolation in Sobolev–Slobodetskii spaces, and error estimates for DGFE solutions on special graded refined meshes combined with estimates in weighted Sobolev spaces. Our main result is that there exist a local grading of the mesh and a piecewise interpolation by polynoms of higher degree such that we will get the same order O (h^α) of approximation as in the smooth case. © 2011 Wiley Periodicals, Inc. Numer Mehods Partial Differential Eq, 2012     

Interface boundary value problems of Robin‐transmission type for the Stokes and Brinkman systems on n‐dimensional Lipschitz domains: applications

In this paper, we describe a layer potential analysis in order to show an existence result for an interface boundary value problem of Robin‐transmission type for the Stokes and Brinkman systems on Lipschitz domains in Euclidean setting, when the given boundary data belong to some L^p or Sobolev spaces associated to such domains. Applications related to an exterior three‐dimensional Stokes flow past two concentric porous spheres with stress jump conditions on the fluid‐porous interface are also considered. Copyright © 2013 John Wiley & Sons, Ltd.     

Partial Fourier approximation of the Lamé equations in axisymmetric domains

In this paper, we study the partial Fourier method for treating the Lamé equations in three‐dimensional axisymmetric domains subjected to non‐axisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement û , the body force f̂ ϵ (L₂)³ and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three‐dimensional boundary value problem to an infinite sequence of two‐dimensional boundary value problems, whose solutions û _n (n = 0, 1, 2,…) are the Fourier coefficients of û . This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients û _n are proved, which are important for further numerical treatment, e.g. by the finite‐element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two‐dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients û _n and of the error of the partial Fourier approximation are given. Copyright © 1999 John Wiley & Sons, Ltd.     

An implicit–explicit residual error estimator for the coupling of dual‐mixed finite elements and boundary elements in elastostatics

We consider the coupling of dual‐mixed finite elements and boundary elements to solve a mixed Dirichlet–Neumann problem of plane elasticity. We derive an a‐posteriori error estimate that is based on the solution of local Dirichlet problems and on a residual term defined on the coupling interface. The general error estimate does not make use of any special finite element or boundary element spaces. Here the residual term is given in a negative order Sobolev norm. In practical applications, where a certain boundary element subspace is used, this norm can be estimated by weighted local L²‐norms. Copyright © 2001 John Wiley & Sons, Ltd.     

Lp estimates of solutions tomixed boundary value problems for the Stokes system in polyhedral domains

A mixed boundary value problem for the Stokes system in a polyhedral domain is considered. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescribed on the faces of the polyhedron. The authors prove the existence of solutions in (weighted and non‐weighted) L_p Sobolev spaces and obtain regularity assertions for weak solutions. The results are based on point estimates of Green's matrix. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the existence of incompressible current–vortex sheets: study of a linearized free boundary value problem

We study the initial boundary value problem resulting from the linearization of the equations of ideal incompressible magnetohydrodynamics and the jump conditions on the hypersurface of tangential discontinuity (current–vortex sheet) about an unsteady piecewise smooth solution. Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the so‐called loss of derivatives in the normal direction to the boundary takes place even for the constant coefficients linearized problem, for the variable coefficients problem and non‐planar current–vortex sheets the natural functional setting is provided by the anisotropic weighted Sobolev space W₂^1,σ. The result of this paper is a necessary step to prove the local in time existence of solutions of the original non‐linear free boundary value problem. The uniqueness of the regular solution of this problem follows already from the a priori estimate we obtain for the linearized problem. Copyright © 2005 John Wiley & Sons, Ltd.     

The dirichlet problem in lipschitz domains for higher order elliptic systems with rough coefficients

We study the Dirichlet problem, in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order with bounded, complex-valued coefficients. A sharp corollary of our main solvability result is that the operator of this problem performs an isomorphism between weighted Sobolev spaces when its coefficients and the unit normal of the boundary belong to the space VMO.     

A new system of boundary integral equations for plates with free edges

We present a new system of boundary integral equations for the free-edge plate. This system expresses an abstract symmetric variational problem posed on the boundary of the plate. In order to obtain this abstract problem, we must set the exterior boundary value problem corresponding to the free-edge plate in a framework of weighted Sobolev spaces. Finally, we take care of the hypersingular kernels appearing in our system of BIEs, by using an abstract technique of integration by parts.     

Nonlocal Venttsel' diffusion in fractal‐type domains: Regularity results and numerical approximation

We study a nonlocal Venttsel' problem in a nonconvex bounded domain with a Koch‐type boundary. Regularity results of the strict solution are proved in weighted Sobolev spaces. The numerical approximation of the problem is carried out, and optimal a priori error estimates are obtained.     

Ellipticity on Manifolds With Edges and Boundary

Ellipticity of a manifold with edges and boundary is connected to boundary and edge conditions that complete corresponding operators to Fredholm operators between weighted Sobolev spaces. We study a new parameter-dependent calculus of elliptic operators, where the interior symbols have specific properties on the boundary. We construct elliptic operators with a prescribed number of edge conditions and obtain isomorphisms in the scale of edge Sobolev spaces. Supported by the Chinese-German Cooperation Program ‘Partial Differential Equations’, NSFC of China and DFG of Germany.     

Pointwise estimates for Green's kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone

The paper deals with a mixed boundary value problem for the Stokes system in a polyhedral cone. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescribed on the sides of the polyhedron. The authors obtain regularity results for weak solutions in weighted L ₂ Sobolev spaces and prove point estimates of Green's matrix. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local Solutions of Weakly Parabolic Quasilinear Differential Equations

We consider a quasilinear parabolic boundary value problem, the elliptic part of which degenerates near the boundary. In order to solve this problem, we approximate it by a system of linear degenerate elliptic boundary value problems by means of semidiscretization with respect to time. We use the theory of degenerate elliptic operators and weighted Sobolev spaces to find a priori estimates for the solutions of the approximating problems. These solutions converge to a local solution, if the step size of the time-discretization goes to zero. It is worth pointing out that we do not require any growth conditions on the nonlinear coefficients and right-hand side, since we lire able to prove L∞ - estimates.     

Jacobi Polynomials,Weighted Sobolev Spaces and Approximation Results of Some Singularities

In this paper, we give some polynomial approximation results in a class of weighted Sobolev spaces, which are related to the Jacobi operator. We further give some embeddings of those weighted Sobolev spaces into usual ones and into spaces of continuous functions, in order to use the above approximation results in the p‐version (or the spectral method) of some finite or boundary element methods. Finally, two typical examples of the polynomial approximation of some singularities of boundary value problems in polygonal or polyhedral domains are presented.     

Weighted Lp‐theory for vector potential operators in three‐dimensional exterior domains

In the present paper, we study the vector potential problem in exterior domains of . Our approach is based on the use of weighted spaces in order to describe the behavior of functions at infinity. As a first step of the investigation, we prove important results on the Laplace equation in exterior domains with Dirichlet or Neumann boundary conditions. As a consequence of the obtained results on the vector potential problem, we establish useful results on weighted Sobolev inequalities and Helmholtz decompositions of weighted spaces. Copyright © 2015 John Wiley & Sons, Ltd.     

Dual finite element analysis of three-dimensional axisymmetric elliptic problems. Part I

This paper consists of two parts in which we propose two types of pure dual finite element (FE) models of a three-dimensional (3D) axisymmetric elliptic problem with mixed boundary conditions. Using cylindrical coordinates and weighted Sobolev spaces, a dual 3D problem is transformed into a 2D problem and finite element spaces of divergence-free vector functions are constructed with the help of a stream function. In part I of the paper a priori and a posteriori error estimates are derived for the first type of the FE model. © 1993 John Wiley & Sons, Inc.