Existence of parabolic minimizers to the total variation flow on metric measure spaces

Abstract

We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ ( X , d , μ ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum $$u_0$$ u 0 on the parabolic boundary of a space-time-cylinder $$\Omega \times (0, T)$$ Ω × ( 0 , T ) with $$\Omega \subset {\mathcal {X}}$$ Ω ⊂ X an open set and $$T > 0$$ T > 0 , we prove existence in the weak parabolic function space $$L^1_w(0, T; \mathrm {BV}(\Omega ))$$ L w 1 ( 0 , T ; BV ( Ω ) ) . In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $$\mathrm {BV}$$ BV -valued parabolic function spaces. We argue completely on a variational level.

The Cheeger problem in abstract measure spaces

“…Quite often, the ambient space is a (metric) measure spaces. For example, such a general point of view has been adopted for the variational mean curvature of a set [19], for shape optimization problems [37], for Anzellotti-Gauss-Green formulas [78], for the total variation flow [34,35], and, very recently, for the existence of isoperimetric clusters [111].…”

The Cheeger problem in abstract measure spaces

“…Quite often, the ambient space is a (metric) measure spaces. For example, such a general point of view has been adopted for the variational mean curvature of a set [20], for shape optimization problems [37], for Anzellotti-Gauss-Green formulas [74], for the total variation flow [34,35] and, very recently, for the existence of isoperimetric clusters [107].…”

Time-smoothing for parabolic variational problems in metric measure spaces

“…Our study originates from the work in progress [5] where, inspired by the Euclidean results of [8], the issue of the regularity for the minimizers of the Total Variation Flow (tvf) is treated in the context of a metric measure space (X, d, µ) equipped with a doubling measure µ and satisfying a weak (1, 1)-Poincaré inequality.…”

“…In [5], the strategy chosen to prove that the minimizers f in the class (1.3) satisfy the De Giorgi estimate (1.4) is reminiscent of [23], where the authors establish a similar condition for the (quasi) minimizers of the variational problem related to the parabolic p-Laplace equation, p > 2, in the setting of a doubling metric measure space (X, d, µ) supporting a weak (1, p)-Poincaré inequality. In their case, the functional space under consideration is the parabolic Newton-Sobolev space L p (0, τ ; N 1,p (Ω)) (or its local version).…”