Several inviscid models in hydrodynamics and geophysics such as the incompressible Euler vorticity equations, the surface quasi-geostrophic equation and the Boussinesq equations are not known to have even local well-posedness in the corresponding borderline Sobolev spaces. Here H is referred to as a borderline Sobolev space if the L∞-norm of the gradient of the velocity is not bounded by the H-...

We deal with Orlicz-Sobolev embeddings in open subsets of R. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces ...

We study the randomized worst-case error and the randomized error of scrambled quasi–Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case s...

max(u,v)∈E |f(u)− f(v)| if p =∞. If G is connected, then the only functions f satisfying ||f ||E,p = 0 are constant functions, so || · ||E,p is a norm on each linear space of functions on VG which does not contain constants. Usually we shall consider the subspace in the space of all functions on VG given by ∑ v∈V f(v)dv = 0. The obtained normed space will be called a Sobolev space on G and will...

We develop a theory of BV and Sobolev Spaces via integration by parts formula in abstract metric spaces; the role of vector fields is played by Weaver’s metric derivations. The definition hereby given is shown to be equivalent to many others present in literature. Introduction In the last few years a great attention has been devoted to the theory of Sobolev spaces W 1,q on metric measure spaces...

with 1 ≤ p <∞. W (M,N) is equipped with the standard metric d(u, v) = ‖u− v‖W1,p . Our main concern is to determine whether or not W (M,N) is path-connected and if not what can be said about its path-connected components, i.e. its W -homotopy classes. We say that u and v are W -homotopic if there is a path u ∈ C([0, 1],W (M,N)) such that u = u and u = v. We denote by ∼p the corresponding equiva...

We investigate asymptotic behaviour of approximation numbers of Sobolev embeddings between weighted function spaces of Sobolev–Hardy–Besov type with polynomials weights. The exact estimates are proved in almost all cases. © 2005 Elsevier Inc. All rights reserved.