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...

In this work we discuss the generalized treatment of the deformable registration problem in Sobolev spaces. We extend previous approaches in two points: 1) by employing a general energy model which includes a regularization term, and 2) by changing the notion of distance in the Sobolev space by problem-dependent Riemannian metrics. The actual choice of the metric is such that it has a precondit...

Lipschitz domains. Our presentations here will almost exclusively be for bounded Lipschitz domains. Roughly speaking, a domain (a connected open set) Ω ⊂ R is called a Lipschitz domain if its boundary ∂Ω can be locally represented by Lipschitz continuous function; namely for any x ∈ ∂Ω, there exists a neighborhood of x, G ⊂ R, such that G ∩ ∂Ω is the graph of a Lipschitz continuous function und...

We consider fractional Sobolev spaces with dominating mixed derivatives and prove some generalizations of Trudinger’s limiting imbedding theorem.

We prove dimension-free imbedding theorems for Sobolev spaces using extrapolation means and the Gross logarithmic inequality.

We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix $A$ into Sobolev spaces, with focus on influence embedding behaviour. For a large range parameters, we derive sharp characterizations embeddings.

The theory of nonlinear weakly hyperbolic equations was developed during the last decade in an astonishing way. Today we have a good overview about assumptions which guarantee local well posedness in spaces of smooth functions (C∞, Gevrey). But the situation is completely unclear in the case of Sobolev spaces. Examples from the linear theory show that in opposite to the strictly hyperbolic case...

1. A confusing example 2. Natural function spaces 3. Topology on limits C∞(S1) 4. Distributions (generalized functions) 5. Invariant integration, periodicization 6. Hilbert space theory of Fourier series 7. Completeness in L(S) 8. Sobolev lemma, Sobolev imbedding 9. C∞(S1) = limC(S) = limHs(S) 10. Distributions, generalized functions, again 11. The confusing example explained 12. Appendix: prod...

This paper is the first in a series devoted to the approximation theory of the p-version of the finite element method in three dimensions. In this paper, we introduce the Jacobi-weighted Besov and Sobolev spaces in the three-dimensional setting and analyze the approximability of functions in the framework of these spaces. In particular, the Jacobi-weighted Besov and Sobolev spaces with three di...