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<Emphasis Type="Italic">BV</Emphasis> functions and parabolic initial boundary value problems on domains

Authors:	Luciana Angiuli Jr" target="_blank">Michele MirandaJr Diego Pallara Fabio Paronetto

Institution:	(1) Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, P. O. Box 193, 73100 Lecce, Italy;(2) Dipartimento di Matematica, Università di Ferrara, Via Machiavelli, 35, 44100 Ferrara, Italy

Abstract:	Given a uniformly elliptic second order operator ${\mathcal{A}}$ on a possibly unbounded domain ${\Omega\,\subset\,\mathbb {R}^N}$ , let (T(t))_t≥0 be the semigroup generated by ${\mathcal{A}}$ in L ¹(Ω), under homogeneous co-normal boundary conditions on ∂Ω. We show that the limit as t → 0 of the L ¹-norm of the spatial gradient D _x T(t)u ₀ tends to the total variation of the initial datum u ₀, and in particular is finite if and only if u ₀ belongs to BV(Ω). This result is true also for weighted BV spaces. A further characterization of BV functions in terms of the short-time behaviour of (T(t))_t≥0 is also given.

Keywords:	Linear parabolic equations BV functions Semigroup theory
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