Γ-convergence methods are used to prove homogenization results for fractional obstacle problems in periodically perforated domains. The obstacles have random sizes and shapes and their capacity scales according to a stationary ergodic process. We use a trace-like representation of fractional Sobolev norms in terms of weighted Sobolev energies established in [8], a weighted ergodic theorem and a...

The main purpose of our paper is to prove sharp Adams-type inequalities in unbounded domains of R for the Sobolev space W n m (R) for any positive integer m less than n. Our results complement those of Ruf and Sani [28] where such inequalities are only established for even integer m. Our inequalities are also a generalization of the Adams-type inequalities in the special case n = 2m = 4 proved ...

We show that any infinite-dimensional Banach (or more generally, Fréchet) space contains linear subspaces of arbitrarily high Borel complexity which admit separable complete norms giving rise to the inherited Borel structure. © 2007 Elsevier Inc. All rights reserved.

Let χλ (cf (1.1)) be the unit spectral projection operator with respect to the Laplace-Beltrami operator ∆ on a closed Riemannian manifold M . We generalize the (L2, L∞) estimate of χλ by Hörmander [3] to those of covariant derivatives of χλ Moreover we extend the (L2, Lp) estimates of χλ by Sogge [7] [8] to (L2, Sobolev Lp) estimates of χλ.

In the cases where there is no Sobolev-type or Gagliardo-Nirenberg-type fractional estimate involving $\lvert u\rvert_{W^{s,p}}$, we establish alternative estimates strong $L^p$ norms are replaced by Lorentz norms.

The solution of the Dirichlet problem relative to an elliptic system in a polyhedron has a complex singular behaviour near edges and vertices. Here, we show that this solution has a global regularity in appropriate weighted Sobolev spaces. Some useful embeddings of these spaces into classical Sobolev spaces are also established. As applications, we consider the Lamé, Stokes and Navier-Stokes sy...

We obtain an improved Sobolev inequality in Ḣ spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in Ḣ obtained in [19] using the abstract approach of di...

Our concern in this paper lies with two aspects of weighted exponential spaces connected with their role of target spaces for critical imbeddings of Sobolev spaces. We characterize weights which do not change an exponential space up to equivalence of norms. Specifically, we first prove that Lexp tα(χB) = Lexp tα(ρ) if and only if ρq ∈ Lq with some q > 1. Second, we consider the Sobolev space W ...

We study approximation properties of weighted $\mathrm{L}^2$-orthogonal projectors onto spaces polynomials bounded degree in the Euclidean unit ball, where weight is reflection-invariant form $(1-\lVert x \rVert^2)^\alpha \prod_{i=1}^d \lvert x_i \rvert^{\gamma_i}$, $\alpha, \gamma_1, \dots, \gamma_d > -1$. Said are measured Dunkl-Sobolev-type norms which same $\mathrm{L}^2$ norm used to contro...

The convergence behavior of a number of algorithms based on minimizing residual norms over Krylov subspaces, is not well understood. Residual or error bounds currently available are either too loose or depend on unknown constants which can be very large. In this paper we take another look at traditional as well as alternative ways of obtaining upper bounds on residual norms. In particular, we d...