Integrability of the derivative of solutions to a singular one-dimensional parabolic problem

Abstract

We study integrability of the derivative of solutions to a singular one-dimensional parabolic equation with initial data in W 1,1 . In order to avoid additional difficulties we consider only the periodic boundary conditions. The problem we study is a gradient flow of a convex, linear growth variational functional. We also prove a similar result for the elliptic companion problem, i.e. the time semidiscretization.

A local estimate for vectorial total variation minimization in one dimension

“…We note that preservation of C 0,α (Ω) class is also known for the scalar total variation flow on a convex domain Ω ⊂ R m [9,19]. On the other hand, preservation of W 1,1 (I) regularity has recently been obtained for gradient flows of more general functionals of linear growth at infinity on an interval [21]. In [7], the authors consider the gradient flow of a functional u → Ω |div u|, where Ω is a bounded domain in R m .…”

“…Property (21) is satisfied for every x 0 ∈ U and almost every r > 0 such that B r (x 0 ) ⊂ U . Hence, by [12, 1.5.2., Corollary 1], up to a set of zero |u ′ | measure we can fill any open set V ⊂ U with a countable family of disjoint closed balls B r j (x j ) contained in V and satisfying (21): hence…”

Existence of $W^{1,1}$ solutions to a class of variational problems with linear growth on convex domains

“…Let us recall a result in linear algebra, which permits us to generalize the results of [23] to higher dimensions. Proof.…”

“…In particular, if f ∈ W 1,∞ (Ω), then u ∈ W 1,∞ (Ω) ⊂ W 1,1 (Ω). On the other hand, in [23], the case m = n = 1 is considered (with Ω = T). In this setting it is proved for any convex Φ with linear growth that if f ∈ W 1,1 (Ω), then u ∈ W 1,1 (Ω) as well.…”

Local estimates for vectorial Rudin–Osher–Fatemi type problems in one dimension

“…4). However, it has been proved in [11] for general convex F of linear growth in m = n = 1 that if h ∈ W 1,1 (Ω), then u ∈ W 1,1 (Ω). Estimate (1.7) is also known to fail for m > 1, see explicit examples in [12], Section 4.…”