2006
DOI: 10.1007/s11118-006-9023-3
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The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values

Abstract: We define and study variable exponent Sobolev spaces with zero boundary values. This allows us to prove that the Dirichlet energy integral has a minimizer in the variable exponent case. Our results are based on a Poincaré-type inequality, which we prove under a certain local jump condition for the variable exponent. Mathematics Subject Classifications (2000) 46E35 · 31C45 · 35J65Key words variable exponent Sobolev space · zero boundary values · Sobolev capacity · Poincaré inequality · Dirichlet energy integral

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Cited by 128 publications
(68 citation statements)
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“…( ), as the completion of C ∞ 0 ( ) with respect to the norm · 1, p(·) . We refer to [9,12,20,32] for density results in variable exponent Sobolev spaces.…”
Section: Logarithmic Hölder Continuitymentioning
confidence: 99%
See 1 more Smart Citation
“…( ), as the completion of C ∞ 0 ( ) with respect to the norm · 1, p(·) . We refer to [9,12,20,32] for density results in variable exponent Sobolev spaces.…”
Section: Logarithmic Hölder Continuitymentioning
confidence: 99%
“…For any f ∈ W 1, p(·) ( ), there exists a unique p(·)-solution u ∈ W 1, p(·) ( ) such that u − f ∈ W 1, p(·) 0 ( ), see [16,Theorem 5.3]. It is well-known that p(·)-solutions have a locally Hölder continuous representative, see [1,2,11].…”
Section: P(·)-supersolutions and P(·)-superharmonic Functionsmentioning
confidence: 99%
“…The characterization of the trace space for generalized Sobolev functions is given in Diening and Hästö [5,6], that of the generalized Sobolev functions with zero boundary value is given in Harjulehto [17] and several boundary trace imbedding theorems for generalized Sobolev functions are obtained by Fan [12].…”
Section: Remark 23mentioning
confidence: 99%
“…(Ω), as the closure of the set of compactly supported W 1,p(·) (Ω)-functions with respect to the norm · W 1,p(·) (Ω) [20].…”
Section: Applications To Hardy's and Sobolev's Inequalitymentioning
confidence: 99%