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Strictly continuous extension of functionals with linear growth to the space BV

dc.creatorRindler, Filip
dc.creatorShaw, Giles
dc.date.accessioned2018-11-24T23:18:14Z
dc.date.available2015-06-24T10:19:32Z
dc.date.available2018-11-24T23:18:14Z
dc.date.issued2015-06-24
dc.identifierhttps://www.repository.cam.ac.uk/handle/1810/248659
dc.identifier.urihttp://repository.aust.edu.ng/xmlui/handle/123456789/3236
dc.description.abstractIn this paper, we prove that the integral functional F[u]: BV(Ω;ℝ^m) → ℝ defined by F[u] := ∫_Ω f(x,u(x),∇u(x))dx + ∫_Ω ∫_1^0 f^∞ (x, u^θ (x), (d D^s u)/(d |D^s u|) (x) is continuous over BV(Ω;ℝ^m), with respect to the topology of area-strict convergence, a topology in which (W^(1,1) ∩ C^∞)(Ω;ℝ^m) is dense. This provides conclusive justification for the treatment of F as the natural extension of the functional u ↦ ∫_Ω f(x,u(x),∇u(x))dx, defined for u ∈ W^(1,1) (Ω;ℝ^m). This result is valid for a large class of integrands satisfying |f(x,y,A)| ≤ C(1+ |y|^(d/(d−1)) + |A|) and its proof makes use of Reshetnyak's Continuity Theorem combined with a lifting map μ[u]: BV(Ω;ℝ^m) → M(Ω × ℝ^m; ℝ^(m×d)). To obtain the theorem in the case where f exhibits d/(d−1) growth in the y variable, an embedding result from the theory of concentration-compactness is also employed.
dc.description.abstractSee full abstract at http://dx.doi.org/10.1093/qmath/hav022
dc.languageen
dc.publisherOxford University Press
dc.publisherThe Quarterly Journal of Mathematics
dc.rightshttp://creativecommons.org/licenses/by/2.0/uk/
dc.rightsAttribution 2.0 UK: England & Wales
dc.titleStrictly continuous extension of functionals with linear growth to the space BV
dc.typeArticle


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