zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$. (English) Zbl 0995.46023
The well-known Sobolev embedding theorem in ${\bbfR}^N$, $$ W^k_p (\Omega) \hookrightarrow L_q (\Omega)$$ is extended to cases with variable integrability: $$ W^k_{p(x)} (\Omega) \hookrightarrow L_{q(x)} (\Omega).$$ This applies also to compactness assertions if the domain $\Omega$ is bounded.

MSC:
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
WorldCat.org
Full Text: DOI
References:
[1] Adams, R. A.: Sobolev spaces. (1975) · Zbl 0314.46030
[2] Cianchi, A.: A sharp embedding theorem for Orlicz--Sobolev spaces. Indiana univ. Math. J. 45, 39-65 (1996) · Zbl 0860.46022
[3] Donaldson, T. K.; Trudinger, N. S.: Orlicz--Sobolev spaces and imbedding theorems. J. funct. Anal. 8, 52-75 (1971) · Zbl 0216.15702
[4] Fan, X. L.: The regularity of Lagrangians $f(x,{\xi})=|{\xi}|{\alpha}(x)$ with hölder exponents ${\alpha}$(x). Acta math. Sinica (N.S.) 12, 113-120 (1996) · Zbl 0874.49031
[5] Fan, X. L.; Zhao, D.: A class of de giorgi type and hölder continuity. Nonlinear anal. 36, 295-318 (1999) · Zbl 0927.46022
[6] Fan, X. L.; Zhao, D.: On the generalized Orlicz--Sobolev space wk,$p(x)({\Omega})$. J. gansu edu. College 12, 1-6 (1998)
[7] Hudzik, H.: On generalized Orlicz--Sobolev space. Funct. approx. Comment. math. 4, 37-51 (1976) · Zbl 0355.46012
[8] Hudzik, H.: On continuity of the imbedding operation from wm1k({$\Omega$}) into wm2k({$\Omega$}). Funct. approx. Comment. math. 6, 111-118 (1978) · Zbl 0385.46015
[9] Marcellini, P.: Regularity and existence of solutions of elliptic equations with p,q-growth conditions. J. differential equations 90, 1-30 (1991) · Zbl 0724.35043
[10] Musielak, J.: Orlicz spaces and modular spaces. Lecture notes in mathematics 1034 (1983) · Zbl 0557.46020
[11] Natanson, I. P.: Theory of functions of a real variable. (1950)
[12] Zhao, D.; Qiang, W. J.; Fan, X. L.: On the generalized Orlicz spaces $Lp(x)({\Omega})$. J. gansu sci. 9, 1-7 (1997)
[13] Zhikov, V.: Averaging of functionals in the calculus of variations and elasticity. Math. USSR izy. 29, 33-66 (1987) · Zbl 0599.49031
[14] Zhikov, V.: On passing to the limit in nonlinear variational problem. Mat. sb. 183, 47-84 (1992) · Zbl 0767.35021