Optimal stencils in Sobolev spaces

Optimal Domain Spaces in Orlicz-sobolev Embeddings

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 ...

Optimal Sobolev embeddings and Function Spaces

Sobolev Spaces

Lifting in Sobolev Spaces

for some function φ : Ω → R. The objective is to find a lifting φ “as regular as u permits.” For example, if u is continuous one may choose φ to be continuous and if u ∈ C one may also choose φ to be C. A more delicate result asserts that if u ∈ VMO (= vanishing means oscillation), then one may choose φ to be also VMO (see R. Coifman and Y. Meyer [1] and H. Brezis and L. Nirenberg [1]). In this...

Circulation integrals and critical Sobolev spaces : problems of optimal constants

We study various questions related to the best constants in the following inequalities established in [1, 2, 3]; ̨̨̨Z Γ ~ φ · ~t ̨̨̨ ≤ Cn‖∇φ‖Ln |Γ| , and ̨̨̨Z Rn ~ φ · ~ μ ̨̨̨ ≤ Cn‖∇φ‖Ln‖~ μ‖ , where Γ is a closed curve in Rn, ~ φ ∈ C∞ c (Rn; Rn) and ~ μ is a bounded measure on Rn with values into Rn such that div ~ μ = 0. In 2d the answers are rather complete and closely related to the best constants for ...