BV functions and parabolic initial boundary value problems on domains

Given a uniformly elliptic second order operator A on a possibly unbounded domain Omega subset of R(N), let (T(t))(t >= 0) be the semigroup generated by A in L(1)(Omega), under homogeneous co-normal boundary conditions on partial derivative Omega. 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(Omega). 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.