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Let M be a connected Riemannian manifold without boundary with Ricci curvature bounded from below and such that the volume of the geodesic balls of centre x and fixed radius r > 0 have a volume bounded away from 0 uniformly with respect to x, and let À TðtÞ Á tf0 be the heat semigroup on M. We show that the total variation of the gradient of a function u A L 1 ðMÞ equals the limit of the L 1 -norm of 'TðtÞu as t ! 0. In particular, this limit is finite if and only if u is a function of bounded variation.!rrrooouuuggghhhttt ttt ooo yyyooouuu bbbyyy ||| !iii bbblll iii ooottt eeecccaaa dddeeelll (((!iii bbblll iii ooottt eeecccaaa dddeeelll )))AAAuuuttt hhheeennnttt iii cccaaattt eeeddd ||| 111777222..

We give an existence result for the evolution equation (Ru)' + Au = f in the space W = {u is an element of V\ (Ru)' is an element of V'}where V is a Banach space and R is a non-invertible operator (the equation may be partially elliptic and partially parabolic, both forward and backward) and we study the "Cauchy-Dirichlet" problem associated to this equation (indeed also for the inclusion (Ru)' + Au There Exists f). We also investigate continuous and compact embeddings of W and regularity in time of the solution. At the end we give some examples of different R

We define a homogeneous parabolic De Giorgi classes of order 2 which suits a mixed type class of evolution equations whose simplest example is\ud $\mu (x) \frac{\partial u}{\partial t} - \Delta u = 0$ where $\mu$ can be positive, null and negative, so in particular elliptic-parabolic\ud and forward-backward parabolic equations are included.\ud For functions belonging to this class we prove local boundedness and show a Harnack inequality which, as by-products, gives\ud H\"older-continuity, in particular in the interface $I$ where $\mu$ change sign, and a maximum principle