BV functions and parabolic initial boundary value problems on domains

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 L1(Ω), under homogeneous co-normal boundary conditions on ∂Ω. We show that the limit as t → 0 of the L1-norm of the spatial gradient DxT(t)u0 tends to the total variation of the initial datum u0, and in particular is finite if and only if u0 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.