*Published Paper*

**Inserted:** 29 may 2007

**Last Updated:** 30 mar 2010

**Journal:** Ann. Mat. Pura Appl.

**Volume:** 188

**Year:** 2009

**Abstract:**

Given a uniformly elliptic second-order operator $\A$ on a (possibly unbounded) domain $\Omega\subset\Rn$, let $(T(t))_{t\geq 0}$ be the semigroup generated by $\A$ in $L^1(\Omega)$, under homogeneous co-normal boundary conditions on $\partial\Omega$. We show that the limit as $t\to 0$ of the $L^1$-norm of the spatial gradient $D_xT(t)u_0$ 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 characterisation of $BV$ functions in terms of the short-time behaviour of $(T(t))_{t\geq 0}$ is also given.

**Keywords:**
BV functions, linear parabolic equations, semigroup theory

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