Fabio Paronetto

Heat semigroup and functions of bounded variation on Riemannian manifolds

Abstract 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 (T(t)) t≧0 be the heat semigroup on M. We show that the total variation of the gradient of a function u ∈ L 1(M) equals the limit of the L 1-norm of ∇T(t)u as t → 0. In particular, this limit is finite if and only if u is a function of bounded variation.

Short-time heat flow and functions of bounded variation in \mathbf{R}^N

On prouve une caracterisation des ensembles avec perimetre fini et des fonctions a variation bornee en termes du comportement du semi-groupe de la chaleur dans R N au voisinage de t = 0. On prouve aussi un resultat plus precis pour les ensembles avec frontiere assez reguliere.

On the convergence of a class of degenerate parabolic equations

Abstract In this paper we study the convergence of the Cauchy-Dirichlet problems for a sequence of parabolic operators P h = ∂ ∂t − div (a h (x,t) · D) , where the matrices of the coefficient ah(x,t) verify the following degenerate elliptic condition: λ h (x)|ζ| 2 ≤ (a h (x,t)⋯ζ,ζ)≤Lλ h (x)|ζ| 2 , being (λh)h a sequence of weights satisfying an uniform Muckenhoupts condition in h.

A remark on forward–backward parabolic equations

ABSTRACT In this note we deal with a problem regarding forward–backward parabolic equations like with r which changes sign. It is natural to have the continuity of the function . Here we show that, under very mild assumptions, one also gets the continuity of the function also in more general and abstract situations. This result turns out to be useful when dealing with De Giorgi classes for these type of equations.