\(BV\) functions and parabolic initial boundary value problems on domains (Q731395)

The authors consider semigroup \((T(t))_{t\geq 0}\) generated by a uniformly elliptic second order operator \(A\) in \(L^1(\Omega)\) under homogeneous co-normal boundary conditions on \(\partial\Omega.\) It is shown that the limit as \(t\to 0\) of the \(L^1\)-norm of \(D_x T(t)u_0\) tends to the total variation of the initial datum \(u_0,\) and in particular is finite if and only if \(u_0\) belongs to \(BV(\Omega)\) or weighted \(BV\) spaces.