On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory

Abstract

We study the interrelation between the limit L p (Ω)-Sobolev regularity s p of (classes of) functions on bounded Lipschitz domains Ω ⊆ R d , d ≥ 2, and the limit regularity α p within the corresponding adaptivity scale of Besov spaces B α τ,τ (Ω), where 1/τ = α/d + 1/p and α > 0 (p > 1 fixed). The former determines the convergence rate of uniform numerical methods, whereas the latter corresponds to the convergence rate of best N -term approximation. We show how additional information on the Besov or Triebel-Lizorkin regularity may be used to deduce upper bounds for α p in terms of s p simply by means of classical embeddings and the extension of complex interpolation to suitable classes of quasi-Banach spaces due to Kalton, Mayboroda, and Mitrea (Contemp. Math. 445). The results are applied to the Poisson equation, to the p-Poisson problem, and to the inhomogeneous stationary Stokes problem. In particular, we show that already established results on the Besov regularity for the Poisson equation are sharp.

Regularity in Sobolev and Besov Spaces for Parabolic Problems on Domains of Polyhedral Type

“…However, on general Lipschitz domains and in particular in polyhedral domains, the situation changes dramatically. On these domains, singularities at the boundary may occur that diminish the Sobolev regularity of the solution significantly [10,12,28,29,32]. However, the analysis in the above mentioned papers shows that these boundary singularities do not influence the Besov regularity too much, so that the use of adaptive algorithms for elliptic PDEs is completely justified!…”

Rate-optimal sparse approximation of compact break-of-scale embeddings

“…Again these rates are closely related to the regularity of the underlying function spaces [19]. While the smoothness of solutions with singular parts is known to be quite limited in the scale of Sobolev-Hilbert spaces [5], regularity theory shows that such functions admit higher order smoothness when derivatives are measured w.r.t. Lebesgue-norms weaker than L 2 (Ω); see, e.g., [4,10,11,14,15,22,25].…”

On the lack of interior regularity of the p‐Poisson problem with p>2

“…connected open subsets of ) are then defined by restriction such that we obtain subsets of , the topological dual of . Remark We assume that the reader is familiar with the basics of function space theory as it can be found, e.g., in the monographic series of Triebel [23–26] or in [7, Appendix A]. Anyhow, let us mention that by now various equivalent characterizations, embeddings, interpolation and duality assertions for the scale of Besov spaces are known.…”