On Weighted Poincaré Inequalities

We are concerned with the problem of finding sharp summability conditions on the weights which render certain weighted inequalities of Poincaré - type true. The conditions we find consist of proper integral balances between the growths of the rearrangements of the weights.     

Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients

We prove extension theorems in the norms described by Stokes and Lamé operators for the three‐dimensional case with periodic boundary conditions. For the Lamé equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e. for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem. Extension theorems serve as an important tool in many applications, e.g. in domain decomposition and fictitious domain methods, and in analysis of finite element methods. We consider an application of established extension theorems to an efficient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coefficients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coefficient problem. Such preconditioner allows the use of well‐known fast algorithms for preconditioning. Under some natural assumptions on smoothness and topological properties of subdomains with small coefficients, we prove convergence of the simplest Richardson method uniform in the jump of coefficients. For the Lamé equations, the convergence is also uniform in the incompressible limit. Our preliminary numerical results for two‐dimensional diffusion problems show fast convergence uniform in the jump and in the mesh size parameter. Copyright © 2002 John Wiley & Sons, Ltd.     

Spectral problems in Sobolev‐type Banach spaces for strongly elliptic systems in Lipschitz domains

This paper is devoted to classical spectral boundary value problems for strongly elliptic second‐order systems in bounded Lipschitz domains, in general non‐self‐adjoint, namely, to questions of regularity and completeness of root functions (generalized eigenfunctions), resolvent estimates, and summability of Fourier series with respect to the root functions by the Abel–Lidskii method in Sobolev‐type spaces. These questions are not difficult in the Hilbert spaces of the type , and important results in this case are well‐known, but our aim is to extend the results to Banach spaces with in a neighborhood of (1, 2). We also touch upon some spectral problems on Lipschitz boundaries. Tools from interpolation theory of operators are used, especially the Shneiberg theorem.     

Reduction theorems for Sobolev embeddings into the spaces of Hölder,Morrey and Campanato type

Let X be a rearrangement‐invariant Banach function space on Q where Q is a cube in and let be the Sobolev space of real‐valued weakly differentiable functions f satisfying . We establish a reduction theorem for an embedding of the Sobolev space into spaces of Campanato, Morrey and Hölder type. As a result we obtain a new characterization of such embeddings in terms of boundedness of a certain one‐dimensional integral operator on representation spaces.