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- W2224942010 abstract "In this paper, we prove that the integral functional |$mathcal {F}[u]colon {mathrm {BV}}(Omega ;{mathbb {R}}^m)to {mathbb {R}}$| defined by [mathcal {F}[u]:=int _Omega f(x,u(x),nabla u(x)),{mathrm {d}}x+ int _Omega int _0^1 f^infty left ( x,u^theta (x),frac {{mathrm {d}}D^s u}{{mathrm {d}}|D^s u|}(x)right ) {mathrm {d}}theta ,{mathrm {d}}|D^su|(x)] is continuous over |${mathrm {BV}}(Omega ;{mathbb {R}}^m)$|, with respect to the topology of area-strict convergence, a topology in which |$({{W}}^{1,1}cap {{C}}^infty )(Omega ;{mathbb {R}}^m)$| is dense. This provides conclusive justification for the treatment of |$mathcal {F}$| as the natural extension of the functional [umapsto int _Omega f(x,u(x),nabla u(x)),{mathrm {d}}x,] defined for |$uin {{W}}^{1,1}(Omega ;{mathbb {R}}^m)$|. This result is valid for a large class of integrands satisfying |$|f(x,y,A)|leq C(1+ |y|^{d/(d-1)}+ |A|)$| and its proof makes use of Reshetnyak's Continuity Theorem combined with a lifting map |$mu [u]colon {mathrm {BV}}(Omega ;{mathbb {R}}^m)to {mathbf {M}}(Omega times {mathbb {R}}^m;{mathbb {R}}^{mtimes d})$|. 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." @default.
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- W2224942010 date "2015-06-30" @default.
- W2224942010 modified "2023-09-27" @default.
- W2224942010 title "STRICTLY CONTINUOUS EXTENSION OF FUNCTIONALS WITH LINEAR GROWTH TO THE SPACE BV" @default.
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- W2224942010 doi "https://doi.org/10.1093/qmath/hav022" @default.
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