A second look on definition and equivalent norms of Sobolev spaces

Joachim Naumann; Christian G. Simader

Mathematica Bohemica (1999)

  • Volume: 124, Issue: 2-3, page 315-328
  • ISSN: 0862-7959

Abstract

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Sobolev’s original definition of his spaces L m , p ( Ω ) is revisited. It only assumed that Ω ⊆ ℝ n is a domain. With elementary methods, essentially based on Poincare’s inequality for balls (or cubes), the existence of intermediate derivates of functions u ∈ L m , p ( Ω ) with respect to appropriate norms, and equivalence of these norms is proved.

How to cite

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Naumann, Joachim, and Simader, Christian G.. "A second look on definition and equivalent norms of Sobolev spaces." Mathematica Bohemica 124.2-3 (1999): 315-328. <http://eudml.org/doc/248450>.

@article{Naumann1999,
abstract = {Sobolev’s original definition of his spaces $L^\{m,p\}(\Omega )$ is revisited. It only assumed that $\Omega \subseteq \mathbb \{R\}^n$ is a domain. With elementary methods, essentially based on Poincare’s inequality for balls (or cubes), the existence of intermediate derivates of functions $u\in L^\{m,p\}(\Omega )$ with respect to appropriate norms, and equivalence of these norms is proved.},
author = {Naumann, Joachim, Simader, Christian G.},
journal = {Mathematica Bohemica},
keywords = {Sobolev spaces; Poincaré’s inequality; existence of intermediate derivates; Sobolev spaces; Poincaré’s inequality; existence of intermediate derivates},
language = {eng},
number = {2-3},
pages = {315-328},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A second look on definition and equivalent norms of Sobolev spaces},
url = {http://eudml.org/doc/248450},
volume = {124},
year = {1999},
}

TY - JOUR
AU - Naumann, Joachim
AU - Simader, Christian G.
TI - A second look on definition and equivalent norms of Sobolev spaces
JO - Mathematica Bohemica
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 124
IS - 2-3
SP - 315
EP - 328
AB - Sobolev’s original definition of his spaces $L^{m,p}(\Omega )$ is revisited. It only assumed that $\Omega \subseteq \mathbb {R}^n$ is a domain. With elementary methods, essentially based on Poincare’s inequality for balls (or cubes), the existence of intermediate derivates of functions $u\in L^{m,p}(\Omega )$ with respect to appropriate norms, and equivalence of these norms is proved.
LA - eng
KW - Sobolev spaces; Poincaré’s inequality; existence of intermediate derivates; Sobolev spaces; Poincaré’s inequality; existence of intermediate derivates
UR - http://eudml.org/doc/248450
ER -

References

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  9. Nikoľskij S. M., Approximation of Functions of Several Variables and Imbedding Theorems, Engl. transl: Springer-Verlag, Berlin 1975, Izd. Nauka, Moskva, 1969 (In Russian.). (1975) MR0374877
  10. Sobolev S. L., On some estimates related to families of functions having derivatives which are square integrable, Dokl. Akad. Nauk 1 (1936), 267-270. (In Russian.) (1936) 
  11. Sobolev S. L., On a theorem in functional analysis, Mat. Sborn. 4 (1938), 471-497 (In Russian.); Engl. transl. Amer. Math. Soc. Transl. II Ser. 34 (1963), 39-68. (1938) 
  12. Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics, 1st ed.: LGU Leningrad, 1950; 2nd ed.: NGU Novosibirsk, 1962; Зrd ed.; Izd. Nauka, Moskva 1988. (In Russian.) Engl. transl.: Amer. Math. Soc. Providence R.I, 1963; German transl: Akademie-Verlag Berlin 1964. (1950) MR0986735
  13. Triebel H., Interpolation Theory, Function Spaces, Differential Operators, (2nd ed.), J. A.Barth Verlag, Beidelberg, 1995. (1995) Zbl0830.46028MR1328645
  14. Ziemer W. P., Weakly Differentiable Functions, Springer-Verlag, New York, 1989. (1989) Zbl0692.46022MR1014685

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