# 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

## Access Full Article

top## Abstract

top## How to cite

topNaumann, 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

top- Adams R. A., Sobolev Spaces, Academic Press, Inc, Boston, 1978, (1978) Zbl0347.46040
- Besov O. V., Il'in V. P., Nikol'skij S. M., Integral Representations of Functions and Imbedding Theorems, Engl. Transl: V.H. Winston & Sons, Washington; J. Wiley & Sons, New York, vol. I: 1978, vol. II: 1979, Izd. Nauka, Moskva, 1975. (In Russian.) (1978) MR0430771
- Burenkov V. I., Sobolev Spaces on Domains, B. G. Teubner, Stuttgart, 1998. (1998) Zbl0893.46024MR1622690
- Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, (2nd ed.), Springer-Verlag, Berlin, 1983. (1983) Zbl0562.35001MR0737190
- Kufner A., John O., Fučík S., Function Spaces, Academia, Prague, 1977. (1977) MR0482102
- Maz'ja V. G., Sobolev Spaces, Springer-Verlag, Berlin, 1985. (1985) Zbl0692.46023MR0817985
- Maz'ja V. G., Poborchij S. V., Differentiable Functions on Bad Domains, World Scientific, Singapore, 1997. (1997) MR1643072
- Nečas J., Les méthodes directes en théorie des équations elliptiques, Academia, Praha, 1967. (1967) MR0227584
- 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
- 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)
- 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)
- 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
- Triebel H., Interpolation Theory, Function Spaces, Differential Operators, (2nd ed.), J. A.Barth Verlag, Beidelberg, 1995. (1995) Zbl0830.46028MR1328645
- Ziemer W. P., Weakly Differentiable Functions, Springer-Verlag, New York, 1989. (1989) Zbl0692.46022MR1014685

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.