Energy Solutions to One-Dimensional Singular Parabolic Problems with $${ BV}$$ Data are Viscosity Solutions

Abstract

We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in BV , which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga-Giga.

The planar Least Gradient problem in convex domains

“…We start by showing the following results that will be needed later in the construction of the solution. In fact, this a result borrowed from [12]. Lemma 4.1.…”

The planar Least Gradient problem in convex domains: the discontinuous case

“…We want to approximate f by continuous functions from below and above. A prototype of such approximation is given in [19], but here we require special properties of the approximating sequences. The technique we use requires that ∂Ω has a finite number of sides and f has a finite number of humps.…”