Density of smooth functions in weighted Sobolev spaces with variable exponent

Surnachev, M. D.

doi:10.1134/s1064562414020045

Cited by 8 publications

(10 citation statements)

References 10 publications

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“…There is a loss of ellipticity on the set {c(x) = 0} and it has been studied at length in the context of equations involving Muckenhoupt weights; see for instance [30]. New, interesting conditions for basic properties of functions spaces related to this functional have been recently given in [6,58,67]. The functional V has also been the object of intensive interest nowadays and a huge literature has been developed on it.…”

Section: 4)mentioning

confidence: 99%

Non-autonomous functionals, borderline cases and related function classes

Baroni

Colombo

Mingione

2016

St. Petersburg Math. J.

233

116

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We consider a class of non-autonomous functionals characterised by the fact that the energy density changes its ellipticity and growth properties according to the point, and prove some regularity results for related minimisers. These results are the borderline counterpart of analogous ones previously derived for non-autonomous functionals with (p, q)-growth. We also discuss similar functionals related to Musielak-Orlicz spaces in which basic properties like density of smooth functions, boundedness of maximal and integral operators, and validity of Sobolev type inequalities naturally relate to the assumptions needed to prove regularity of minima.

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Section: 4)mentioning

confidence: 99%

Non-autonomous functionals, borderline cases and related function classes

Baroni

Colombo

Mingione

2016

St. Petersburg Math. J.

233

116

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“…Moreover, by (17) we have Remark 39 Note, that Zhikov has also provided in [20,Section 5.3] another example for H 1,ϕ(•) ( ) = W 1,ϕ(•) ( ) for the model of the weighted p-energy for p = 2. He was interested in an example, where the weight is bounded.…”

Section: Theorem 34 Letmentioning

confidence: 91%

“…We have proved that ψ(•, |b|) ∈ L r ,∞ ( ) some r > 1, which proves ψ(•, |b|) ∈ L 1 ( ). Due to (17) this implies ϕ * (•, |b|) ∈ L 1 ( ) and therefore b ∈ L ϕ * (•) ( ). Overall, we have verified Assumption 23.…”

Section: Theorem 34 Letmentioning

confidence: 92%

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New Examples on Lavrentiev Gap Using Fractals

2020

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Zhikov showed 1986 with his famous checkerboard example that functionals with variable exponents can have a Lavrentiev gap. For this example it was crucial that the exponent had a saddle point whose value was exactly the dimension. In 1997 he extended this example to the setting of the double phase potential. Again it was important that the exponents crosses the dimensional threshold. Therefore, it was conjectured that the dimensional threshold plays an important role for the Lavrentiev gap. We show that this is not the case. Using fractals we present new examples for the Lavrentiev gap and non-density of smooth functions. We apply our method to the setting of variable exponents, the double phase potential and weighted p-energy.

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“…Acknowledgements. We would like to thank M. D. Surnachev for kindly explaining the change of weight in [26] (which we adapted in Proposition 4.4) and for informing us of his generalization of Zhikov's result to more general exponents [25]. The authors acknowledge the support of the grant ERC ADG GeMeThNES.…”

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confidence: 98%

Weighted Sobolev spaces on metric measure spaces

Ambrosio

Pinamonti

Speight

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We investigate weighted Sobolev spaces on metric measure spaces (X, d, m). Denoting by ρ the weight function, we compare the space W 1,p (X, d, ρm) (which always concides with the closure H 1,p (X, d, ρm) of Lipschitz functions) with the weighted Sobolev spaces W 1,p ρ (X, d, m) and H 1,p ρ (X, d, m) defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W 1,p (X, d, ρm) = H 1,p ρ (X, d, m). We also adapt the results in [22] and in the recent paper [26] to the metric measure setting, considering appropriate conditions on ρ that ensure the equality W 1,p ρ (X, d, m) = H 1,p ρ (X, d, m).

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Density of smooth functions in weighted Sobolev spaces with variable exponent

Cited by 8 publications

References 10 publications

Non-autonomous functionals, borderline cases and related function classes

Non-autonomous functionals, borderline cases and related function classes

New Examples on Lavrentiev Gap Using Fractals

Weighted Sobolev spaces on metric measure spaces

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