Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

We prove some optimal logarithmic estimates in the Hardy space H ^∞(G) with Hölder regularity, where G is the open unit disk or an annular domain of ?. These estimates extend the results established by S.Chaabane and I.Feki in the Hardy-Sobolev space H ^k,∞ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem for the Laplace operator and the Robin inverse problem.     

Growth estimates in the Hardy-Sobolev space of an annular domain with applications

We give an explicit estimate on the growth of functions in the Hardy-Sobolev space Hk,2(Gs) of an annulus. We apply this result, first, to find an upper bound on the rate of convergence of a recovery interpolation scheme in H1,2(Gs) with points located on the outer boundary of Gs. We also apply this result for the study of a geometric inverse problem, namely we derive an explicit upper bound on the area of an unknown cavity in a bounded planar domain from the difference of two electrostatic potentials measured on the boundary, when the cavity is present and when it is not.     

Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications

Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics, can be expressed by a fairly general discrete group of dilations {A ^k : k ∈ ?}, where A is a real n × n matrix with all its eigenvalues λ satisfy |λ| > 1. The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let φ: ? ⁿ ×[0,∞) → [0,∞) be an anisotropic p-growth function with p ∈ (0, 1]. The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type H _A ^φ,∞ (? ⁿ ) and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to H _A ^φ,∞ (? ⁿ ) and the boundedness of the anisotropic Calderón-Zygmund operator from H _A ^φ,∞ (? ⁿ ) to L _A ^φ,∞ (? ⁿ ). It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of H _A ^φ,∞ (? ⁿ ) and the interpolation theorem.     

Global superconvergence of simplified hybrid combinations of the Ritz–Galerkin and FEMs for elliptic equations with singularities II. Lagrange elements and Adini's elements

To solve the elliptic problems with singularities, the simplified hybrid combinations of the Ritz–Galerkin method and the finite element method (RGM–FEM) are explored, which lead to the high global superconvergence rates on the entire solution domain. Let the solution domain be split into a singular subdomain involving a singular point, and a regular subdomain where the true solution is smooth enough. In the singular subdomain, the singular particular functions are chosen to be admissible functions. In the regular subdomain either the k-order Lagrange rectangles or Adini's elements are adapted. Along their common boundary, the simplified hybrid techniques are employed to couple two different numerical methods. It is proven in this paper that the global superconvergence rates, O(h^k+3/2), on the entire domain can be achieved for k(⩾2)-order Lagrange rectangles, and that the global superconvergence rates O(h^3.5) for the Adini's elements. Numerical experiments are reported for the combinations of the Ritz–Galerkin and Adini's methods. This paper presents a development of [Z.C. Li, Computing 65 (2000) 27–44] in high accurate solutions for the general case of the Poisson problems on a polygonal domain S estimates for the Sobolev norm ∥·∥₁, given in a much more general sense than known before, cf. [P.G. Ciarlet, J.L. Lions (Eds.) Finite Element Methods (Part 1), North-Holland, Amsterdam, 1991, pp. 17–351, 501–522; SIAM J. Sci. Statist. Comput. 11 (1990) 343; J. Comput. Appl. Math. 20 (1987) 341; Numer. Math. 63 (1992) 483; L. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Springer, Berlin, 1995; Numer. Methods Partial Differential Equations 3 (1987) 65, 357].     

Estimates for Integral Means of Hyperbolically Convex Functions

We prove the Mejia-Pommerenke conjecture that the Taylor coefficients of hyperbolically convex functions in the disk behave like O(log^?2(n)/n) as n → ∞ assuming that the image of the unit disk under such functions is a domain of bounded boundary rotation. Moreover, we obtain some asymptotically sharp estimates for the integral means of the derivatives of such functions and consider an example of a hyperbolically convex function that maps the unit disk onto a domain of infinite boundary rotation.     

Error estimates of the finite element method for the exterior Helmholtz problem with a modified DtN boundary condition

A priori error estimates in the H¹- and L²-norms are established for the finite element method applied to the exterior Helmholtz problem, with modified Dirichlet-to-Neumann (MDtN) boundary condition. The error estimates include the effect of truncation of the MDtN boundary condition as well as that of discretization of the finite element method. The error estimate in the L²-norm is sharper than that obtained by the author [D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math. 200 (1) (2007) 21-31] for the truncated DtN boundary condition.     

Ginzburg–Landau minimizers with prescribed degrees: dependence on domain

We study minimizers of the Ginzburg–Landau functional in an annular type domain with holes. We assume degrees 1 and ?1 on the boundary of the annulus, degree 0 on the boundaries of the holes. Two types of qualitatively different behavior of minimizers occur, depending on the value of the H¹-capacity of the domain. We also describe the asymptotic behavior of minimizers as the coherency length tends to ∞. To cite this article: L. Berlyand, P. Mironescu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Meshless Galerkin algorithms for boundary integral equations with moving least square approximations

In this paper, we first give error estimates for the moving least square (MLS) approximation in the Hk norm in two dimensions when nodes and weight functions satisfy certain conditions. This two-dimensional error results can be applied to the surface of a three-dimensional domain. Then combining boundary integral equations (BIEs) and the MLS approximation, a meshless Galerkin algorithm, the Galerkin boundary node method (GBNM), is presented. The optimal asymptotic error estimates of the GBNM for three-dimensional BIEs are derived. Finally, taking the Dirichlet problem of Laplace equation as an example, we set up a framework for error estimates of the GBNM for boundary value problems in three dimensions.     

Estimations intérieures avec régularité optimale pour un modèle de plaques en élasticité linéaireInterior estimates with optimal regularity for a plate model in linear elasticity

We are interested in the 3d–2d passage for an asymptotically thin plate in linear elasticity. The classical approach by asymptotic expansion gives an error estimate on the displacements in H¹, assuming the volumic forces at least of regularity L² (and more for certain components). In our work we apply the regularity theory for solutions of elliptic equations. This approach gives, for a new model of Kirchhoff–Love of higher order, an error estimate in H² assuming volumic forces only in L², which is optimal. We also get some interior error estimates in W^k,p, C^k,α. To cite this article: R. Monneau, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 207–212.     

Abrahamse's interpolation theorem and Fuchsian groups

We generalize Abrahamse's interpolation theorem from the setting of a multiply connected domain to that of a more general Riemann surface. Our main result provides the scalar-valued interpolation theorem for the fixed-point subalgebra of H^∞ associated to the action of a Fuchsian group. We rely on two results from a paper of Forelli. This allows us to prove the interpolation result using duality techniques that parallel Sarason's approach to the interpolation problem for H^∞ [Donald Sarason, Generalized interpolation in H^∞, Trans. Amer. Math. Soc. 127 (1967) 179-203, MR0208383. [26]]. In this process we prove a more general distance formula, very much like Nehari's theorem, and obtain relations between the kernel function for the character automorphic Hardy spaces and the Szegö kernel for the disk. Finally, we examine our interpolation results in the context of the two simplest examples of Fuchsian groups acting on the disk.     

Sharp Weak Type Inequalities for the Dyadic Maximal Operator

We obtain sharp estimates for the localized distribution function of $\mathcal{M}\phi $ , when ? belongs to L ^p,∞ where $\mathcal{M}$ is the dyadic maximal operator. We obtain these estimates given the L ¹ and L ^q norm, q<p and certain weak-L ^p conditions.In this way we refine the known weak (1,1) type inequality for the dyadic maximal operator. As a consequence we prove that the inequality 0.1 is sharp allowing every possible value for the L ¹ and the L ^q norm for a fixed q such that 1<q<p, where ∥?∥ _p,∞ is the usual quasi norm on L ^p,∞.     

Finite Element Methods for the Stokes System in Three-Dimensional Exterior Domains

The article treats the question of how to numerically solve the Dirichlet problem for the Stokes system in the exterior of a three-dimensional bounded Lipschitz domain. In a first step, the solution of this problem is approximated by functions solving the Stokes system in a truncated domain and satisfying a suitable artificial boundary condition on the outer boundary of this truncated domain. In a second step, this new problem is approximately solved in finite element spaces related to a graded mesh as introduced by Goldstein [Math. Comp., 36, 387–404 (1981)]. The difference between this finite element approximation and the exact solution of the exterior Stokes problem is estimated in the norm of suitable unweighted L²-Sobolev spaces. These estimates are analogous to corresponding results which are known for the Poisson equation. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

On the solution of Maxwell's equations in polygonal domains

This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H² when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H², and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

A class of viscous p-Laplace equation with nonlinear sources

In this paper, we prove the global existence of solutions to the initial boundary value problem of a viscous p-Laplace equation with nonlinear sources. The asymptotic behavior of solutions as the viscous coefficient k tends to zero is also investigated. In particular, we discuss the H¹-Galerkin finite element method for our problem and establish the error estimates for two semi-discrete approximate schemes.     

Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids

This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary‐value problems in (Math. Meth. Appl. Sci. 2006; 29:2225–2244) with the help of variational methods based on duality theory from convex analysis. In the present paper, it is shown that error majorants can be derived in a different way by certain transformations of the variational inequalities that define generalized solutions. The error bounds derived by this techniques for the velocity function differ from those obtained by the variational method. These estimates involve only global constants coming from Korn‐ and Friedrichs‐type inequalities, which are not difficult to evaluate in case of Dirichlet boundary conditions. For the case of mixed boundary conditions, we also derive another form of the estimate that contains only one constant coming from the following assertion: the L² norm of a vector‐valued function from H¹(Ω) in the factor space generated by the equivalence with respect to rigid motions is bounded by the L² norm of the symmetric part of the gradient tensor. As for some ‘simple’ domains such as squares or cubes, the constants in this inequality can be found analytically (or numerically), we obtain a unified form of an error majorant for any domain that admits a decomposition into such subdomains. Copyright © 2009 John Wiley & Sons, Ltd.     

Improper colouring of (random) unit disk graphs

For any graph G, the k-improper chromatic numberχ^k(G) is the smallest number of colours used in a colouring of G such that each colour class induces a subgraph of maximum degree k. We investigate χ^k for unit disk graphs and random unit disk graphs to generalise results of McDiarmid and Reed [Colouring proximity graphs in the plane, Discrete Math. 199(1-3) (1999) 123-137], McDiarmid [Random channel assignment in the plane, Random Structures Algorithms 22(2) (2003) 187-212], and McDiarmid and Müller [On the chromatic number of random geometric graphs, submitted for publication].     

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain are given. These procedures use implicit Galerkin method in the subdomains and simple explicit flux calculation on the interdomain boundaries by integral mean method or extrapolation method to predict the inner‐boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the subdomains and across interboundaries. The explicit nature of the flux prediction induces a time‐step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. L²‐norm error estimates are derived for these procedures. Compared with the work of Dawson and Dupont [Math Comp 58 (1992), 21–35], these L²‐norm error estimates avoid the loss of H^?1/2 factor. Experimental results are presented to confirm the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Spline approximation of a non-linear Riemann–Hilbert problem†

In this article we use linear spline approximation of a non-linear Riemann–Hilbert problem on the unit disk. The boundary condition for the holomorphic function is reformulated as a non-linear singular integral equation A(u) = 0, where A : H ¹(Γ) → H ¹(Γ) is defined via a Nemytski operator. We approximate A by A ⁿ : H ¹(Γ) → H ¹(Γ) using spline collocation and show that this defines a Fredholm quasi-ruled mapping. Following the results of (A.I. ?nirel'man, The degree of quasi-ruled mapping and a nonlinear Hilbert problem, Math. USSR-Sbornik 18 (1972), pp. 373–396; M.A. Efendiev, On a property of the conjugate integral and a nonlinear Hilbert problem, Soviet Math. Dokl. 35 (1987), pp. 535–539; M.A. Efendiev, W.L. Wendland, Nonlinear Riemann–Hilbert problems for multiply connected domains, Nonlinear Anal. 27 (1996), pp. 37–58; Nonlinear Riemann–Hilbert problems without transversality. Math. Nachr. 183 (1997), pp. 73–89; Nonlinear Riemann–Hilbert problems for doubly connected domains and closed boundary data, Topol. Methods Nonlinear Anal. 17 (2001), pp. 111–124; Nonlinear Riemann–Hilbert problems with Lipschitz, continuous boundary data without transversality, Nonlinear Anal. 47 (2001), pp. 457–466; Nonlinear Riemann–Hilbert problems with Lipschitz-continuous boundary data: Doubly connected domains, Proc. Roy. Soc. London Ser. A 459 (2003), pp. 945–955.), we define a degree of mapping and show the existence of the spline solutions of the fully discrete equations A ⁿ (u) = 0, for n large enough. We conclude this article by discussing the solvability of the non-linear collocation method, where we shall need an additional uniform strong ellipticity condition for employing the spline approximation.     

On the steady viscous flow of a nonhomogeneous asymmetric fluid

We consider a boundary value problem for the system of equations describing the stationary motion of a viscous nonhomogeneous asymmetric fluid in a bounded planar domain having a C ² boundary. We use a stream-function formulation after the manner of Frolov (Math Notes 53(5–6):650–656, 1993) in which the fluid density depends on the stream-function by means of another function determined by the boundary conditions. This allows for dropping some of the equations, most notably the continuity equation. We show existence to the resulting system using Galerkin method.     

Approximation des branches de solutions d’un problème de bifurcation semi-linéaire

This Note deals with the approximation, by a finite element method, including numerical integration, of solution branches for a semilinear problem. Because of both mixed boundary conditions and geometrical properties of the domain, some of the solutions do not belong to H². So, classical results for convergence lead to poor estimates. We show how to improve such estimates with the use of weighted Sobolev spaces together with a mesh “a priori adapted” to the singularity. For the H¹ or L²-norms, we achieve optimal results, in the sense that they are as good as for H²-regular solutions.