Nonlinear degenerate elliptic equations and axially symmetric problems

We consider semilinear elliptic equations with a principal part degenerating on a boundary hyperplane. Weak existence, uniqueness and regularity of solutions are established by variational methods and by reduction to uniformly elliptic equations. An important application arises in the mathematical treatment of the rotating star problem in general relativity, where the axial symmetry admits the reduction of one of the Einstein equations to a problem of the above form on a meridian half plane. Received February 12, 1997 / Accepted May 15, 1997     

On best error bounds for approximation by piecewise polynomial functions

Summary An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.The work presented in this paper was supported by the ERDA Mathematics and Computing Laboratory, Courant Institute of Mathematical Sciences, New York University, under Contract E(11-1)-3077 with the Energy Research and Development Administration.     

Residual type a posteriori error estimates for elliptic obstacle problems

Summary. A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed. Received June 19, 1998 / Published online December 6, 1999     

The similarity between the best approximation and the best copositive approximation

In this paper the author studies the copositive approximation in C(?) by elements of finite dimensional Chebyshev subspaces in the general case when ? is any totally ordered compact space. He studies the similarity between me behavior of the ordinary best approximation and the behavior pf the copositive best approximation. At the end of this paper, the author isolates many cases at which the two behaviors are the same.     

A posteriori error estimates for optimal control problems governed by parabolic equations

Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem. Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002 RID="*" ID="*" Supported by EPSRC research grant GR/R31980     

Compact,convex sets in R n and a new banach lattice. II.- Application: An approximation problem.

We develop a calculus structure in the Banach lattice introduced in a preceding paper, having in mind an approximation problem appearing in non-smooth optimization. We show that the essential results depend as much on the order structure as on the analytical one.     

Finite element approximation of a fourth order nonlinear degenerate parabolic equation

Summary. We consider a fully practical finite element approximation of the fourth order nonlinear degenerate parabolic equation where generically for any given . An iterative scheme for solving the resulting nonlinear discrete system is analysed. In addition to showing well-posedness of our approximation, we prove convergence in one space dimension. Finally some numerical experiments are presented. Received July 29, 1997     

Approximation by boolean sums of positive linear operators III: Estimates for some numerical approximation schemes

In the present note we intröduce and investigate certain sequences of discrete positive linear operators and Boolean sum modifications of them. The mappings considered are obtained by discretizing a class of transformed convolution-type operators using Gaussian quadrature of appropriate order. For our operators and their modifications we prove pointwise Jackson-type theorems involving the first and second order moduli of smoothness, thus providing new and elegant proofs of earlier results by Timan, Telyakowskii, Gopengauz and DeVore. Due to their discrete structure, optimal order of approximation and ease of computation, the operators appear to be useful for numerical approximation. In an intermediate step we solve an old problem in Approximation Theory; its importance was only recently emphasized in a paper of Butzer.     

Wavelet decompositions of L 2-functionals

Based on distribution-theoretical definitions of L ₂ and Sobolev spaces given by Werner in [P. Werner (1970). A distribution-theoretical approach to certain Lebesgue and Sobolev spaces. J. Math. Anal. Appl., 29, 19–78.] real interpolation, Besov type spaces and approximation spaces with respect to multiresolution approximations are considered. The key for the investigation are generalized moduli of smoothness introduced by Haf in [H. Haf (1992). On the approximation of functionals in Sobolev spaces by singular integrals. Applicable Analysis, 45, 295–308.]. Those moduli of smoothness allow to connect the concept of L ₂-functionals with more recent developments in multiscale analysis, see e.g. [W. Dahmen (1995). Multiscale analysis, approximation, and interpolation spaces. In: C.K. Chui and L.L. Schumaker (Eds.), Approximation Theory VIII, Vol. 2: Wavelets and Multilevel Approximation, pp. 47–88.]. In particular, we derive wavelet characterizations for the Sobolev spaces introduced by Werner and establish stable wavelet decompositions of L ₂-functionals. Generalizations to more general spaces of functionals and applications are also mentioned.     

Minimally supported error representations and approximation by the constants

Summary. Distribution theory is used to construct minimally supported Peano kernel type representations for linear functionals such as the error in multivariate Hermite interpolation. The simplest case is that of representing the error in approximation to f by the constant polynomial f(a) in terms of integrals of the first order derivatives of f. This is discussed in detail. Here it is shown that suprisingly there exist many representations which are not minimally supported, and involve the integration of first order derivatives over multidimensional regions. The distance of smooth functions from the constants in the uniform norm is estimated using our representations for the error. Received June 30, 1997 / Revised version received April 6, 1999 / Published online February 17, 2000     

Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces

Recently, in the article [LW], the authors use the notion of polynomials in metric spaces of homogeneous type (in the sense of Coifman-Weiss) to prove a relationship between high order Poincaré inequalities and representation formulas involving fractional integrals of high order, assuming only that is a doubling measure and that geodesics exist. Motivated by this and by recent work in [H], [FHK], [KS] and [FLW] about first order Sobolev spaces in metric spaces, we define Sobolev spaces of high order in such metric spaces . We prove that several definitions are equivalent if functions of polynomial type exist. In the case of stratified groups, where polynomials do exist, we show that our spaces are equivalent to the Sobolev spaces defined by Folland and Stein in [FS]. Our results also give some alternate definitions of Sobolev spaces in the classical Euclidean case. Received: 10 February 1999 / Published online: 1 February 2002     

Regularity theorems and optimal error estimates for linear parabolic Cauchy problems

Summary We prove some regularity results for the solution of a linear abstract Cauchy problem of parabolic type. As an application, we study the approximation of the solution by means of an implicit-Euler discretization in time, which is stable with respect to a wide class of Galerkin approximation methods in space. The error is evaluated in norms of typeL ²(0, ,L ²) andL ²(0, ,V)(H ₀₀ ^1/2 (0, ,H)+H ¹(0, ,V)), whereVHV are Hilbert spaces (the embeddings are supposed to be dense and continuous). We prove error estimates which are optimal with respect to the regularity assumptions on the right-hand side of the equation.The author was supported by G.N.A.F.A. and I.A.N. of C.N.R. and by M.P.I.     

On estimates of approximation numbers and best bilinear approximation

We obtain estimates of approximation numbers of integral operators, with the kernels belonging to Sobolev classes or classes of functions with bounded mixed derivatives. Along with the estimates of approximation numbers, we also obtain estimates of best bilinear approximation of such kernels.Communicated by Charles A. Micchelli.     

A numerical approach to degenerate parabolic equations

Summary. In this work we propose a numerical approach to solve some kind of degenerate parabolic equations. The underlying idea is based on the maximum principle. More precisely, we locally perturb the (initial and boundary) data instead of the nonlinear diffusion coefficients, so that the resulting problem is not degenerate. The efficiency of this method is shown analytically as well as numerically. The numerical experiments show that this new approach is comparable with the existing ones. Received January 20, 1999 / Revised version received February 28, 2000 / Published online July 25, 2001     

A posteriori error estimates and maximal regularity for approximations of fully nonlinear parabolic problems in Banach spaces

A posteriori error estimates are provided for discretizations in time of abstract nonlinear parabolic problems u′ = F(u), by the backward Euler method in the maximal regularity framework of Banach spaces. The estimates are of conditional type, i.e., are valid under assumptions on the approximate solution, and the proofs are based on appropriate fixed point arguments. Work partially supported by Grant HPMD-CT-200100121, Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion-Crete, Greece, by a Spain–Greece Research Collaboration grant jointly funded by the Ministry of Education and Science, Spain, and the General Secretariat of Research and Technology, Greece and a Pytrhagoras–EPEAEK II grant.     

Hessian estimates in Orlicz spaces for fourth-order parabolic systems in non-smooth domains

We establish the global Hessian estimate in Orlicz spaces for a fourth-order parabolic system with discontinuous tensor coefficients in a non-smooth domain under the assumptions that the coefficients have small weak BMO semi-norms, the boundary of a domain is δ-Reifenberg flat for δ>0 small and the given Young function satisfies some moderate growth condition. As a corollary we obtain an optimal global W2,p regularity for such a system.     

A regularity theory for variational integrals with L\ln L-Growth

In the present paper we study regularity for local minimizers of the convex variational integral defined on certain classes of vector–valued functions . The underlying energy spaces are natural from the point of view of existence theory. We then show that local minimizers are of class apart from a closed singular set with vanishing Lebesgue measure, provided . For twodimensional problems we obtain smoothness in the interior of . Received June 21, 1996 / In revised form December 2, 1996 / Accepted December 17, 1996