On the equivalence of fractional-order Sobolev semi-norms

@article{Heuer2012OnTE,
  title={On the equivalence of fractional-order Sobolev semi-norms},
  author={Norbert Heuer},
  journal={arXiv: Functional Analysis},
  year={2012}
}
  • N. Heuer
  • Published 2 November 2012
  • Mathematics
  • arXiv: Functional Analysis
View PDF on arXiv
32 Citations
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32 Citations

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References

SHOWING 1-10 OF 18 REFERENCES
Polynomial approximation of functions in Sobolev spaces
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomical…
Elliptic Problems in Nonsmooth Domains
Foreword Preface 1. Sobolev spaces 2. Regular second-order elliptic boundary value problems 3. Second-order elliptic boundary value problems in convex domains 4. Second-order boundary value problems…
Fractional Poincaré inequalities for general measures
Improved Poincaré inequalities with weights
The Mathematical Theory of Finite Element Methods
Strongly Elliptic Systems and Boundary Integral Equations
Introduction 1. Abstract linear equations 2. Sobolev spaces 3. Strongly elliptic systems 4. Homogeneous distributions 5. Surface potentials 6. Boundary integral equations 7. The Laplace equation 8.…
Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods Part II. The three-dimensional case
TLDR
New local a-posteriori error indicators for the Galerkin discretization of three-dimensional boundary integral equations are introduced, based on local norms of the computable residual, and can be used for controlling the adaptive refinement.
General Properties of Interpolation Spaces
In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.
A note on polynomial approximation in Sobolev spaces
Pour des domaines etoiles on donne de nouvelles bornes sur les constantes dans les inegalites de Jackson pour les espaces de Sobolev. Pour des domaines convexes, les bornes ne dependent pas de…
An adaptive boundary element method for the exterior Stokes problem in three dimensions
TLDR
An adaptive refinement strategy for the h-version of the boundary element method with weakly singular operators on surfaces with optimal lower a priori error estimates for edge singularities on uniform and graded meshes is presented.
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