Variational solutions to the total variation flow on metric measure spaces

We discuss a purely variational approach to the total variation flow on metric measure spaces with a doubling measure and a Poincar\'e inequality. We apply the concept of parabolic De Giorgi classes together with upper gradients, Newtonian spaces and functions of bounded variation to prove a necessary and sufficient condition for a variational solution to be continuous at a given point.


Introduction
The total variation flow (TVF) is the partial differential equation where Ω ⊂ R N is an open set and T > 0. There are several ways to define the concept of weak solution to the TVF. One possibility is to apply the so-called Anzellotti pairing [2]. This approach has been applied in existence and uniqueness results for the total variation flow in [3,4,5,6]. The variational inequality related to the TVF is for every ϕ ∈ C ∞ 0 (Ω T ), where the total variation Du(t) (Ω) is a Radon measure for almost every t ∈ (0, T ). A variational approach to existence and uniqueness questions has been discussed by Bögelein, Duzaar and Marcellini [10], see also Bögelein, Duzaar and Scheven [11], and for the corresponding obstacle problem in [12]. The natural function space for weak solutions to the TVF is bounded variation (BV). Functions of bounded variation on metric measure spaces have been studied in [1,39]. Instead of partial derivatives, this approach is based on the modulus of the gradient and using concepts such as minimal upper gradients and Newtonian spaces, see [7,23,24,25,32,41]. This is also the main advantage of the variational approach. A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as Riemannian manifolds, Heisenberg groups, graphs, etc. The regularity theory of nonlinear parabolic problems in the metric space context has been developed and studied in [26,27,31,35,36,37,38]. The comparison principle has been discussed in [28] and stability theory has been investigated in [18,19,20]. Existence for parabolic problems on metric spaces has been discussed in [13]. All of these results consider variational inequalities with p-growth for p > 1. For the case p = 1, which corresponds to the TVF, Buffa, Collins and Pacchiano [9] showed existence of a parabolic minimizer using the concept of global variational solution. Górny and Mazón [21] studied the existence and uniqueness of weak solutions of the Neumann and Dirichlet problems to the TVF in metric measure spaces. The main goal of the present paper is to extend the results of DiBenedetto, Gianazza and Klaus [15] to a metric measure space with a doubling measure and a Poincaré inequality. The main result gives a necessary and sufficient condition for a variational solution to be continuous at a given point, see Theorem 7.1. Our assumption on the time regularity of a variational solution is initially weaker than in [15] and thus our results may be interesting also in the Euclidean case. As fas as we know, this is the first time when regularity questions are discussed for parabolic problems with linear growth on metric measure spaces. The first step is to derive an energy estimate for variational solutions, in other words, to prove that variational solutions belong to a parabolic De Giorgi class, see Proposition 4.2. The regularity results are based only on this energy estimate and on the assumptions made on the underlying metric measure space, but there is a technical difficulty present when establishing energy estimates for variational solutions. It is not clear that the time regularity of a variational solution is a priori sufficient for placing it as the test function and performing the usual techniques used for obtaining an energy estimate. We resolve this issue by using a mollification technique. The idea of this technique is to prove the required energy estimate for mollified functions and finally to conclude the estimate at the limit. To establish the limiting estimate, we consider the upper gradient of a difference of functions, see Lemma 2.3. In the Euclidean case this poses no difficulties as we can use the linearity of the gradients, in the general metric setting the situation is not that simple, as taking an upper gradient is not a linear operation. This paper is organized as follows. In Section 2 we recall basic definitions and describe the general setup of our study. Several results related to the function spaces and Sobolev-Poincaré inequalities may be of independent interest. In Section 3 we concentrate on the definition and properties of a variational solution to the TVF. Section 4 explores the relationship between variational solutions to the TVF and the parabolic De Giorgi classes. In Section 5 and 6, respectively, we prove that functions in a parabolic De Giorgi class are locally bounded and give a time expansion of positivity result. Finally, in Section 7 we present the characterization of continuity, i.e. we prove necessary and sufficient conditions for a variational solution to the TVF to be continuous at a given point. The last three sections are extensions of the corresponding results on Euclidean spaces by DiBenedetto, Gianazza and Klaus in [15] to metric measure spaces.

Newtonian spaces
Let (X, d, µ) be a complete metric measure space endowed with a Borel measure µ. The measure µ is said to satisfy the doubling condition if there exists a constant C µ ≥ 1, called the doubling constant of µ, such that for every x ∈ X and r > 0. Here B r (x) = {y ∈ X : d(x, y) < r} is an open ball centered at x ∈ X with radius r > 0. We assume throughout that the measure µ is nontrivial in the sense that 0 < µ (B r (x)) < ∞ for every x ∈ X and r > 0. A complete metric metric measure space with a doubling measure is proper, that is, closed and bounded subsets are compact, see [7,Proposition 3.1]. The doubling condition implies that for any x ∈ X, we have for 0 < r < R with Q = log 2 C µ and C = C −2 µ , see [7,Lemma 3.3]. The exponent Q = log 2 C µ is sometimes called the homogeneous dimension of (X, d, µ).
A path γ is a continuous mapping from a compact subinterval of R to X. The p-modulus, with 1 ≤ p < ∞, of a path family Γ on X is where the infimum is taken over all nonnegative Borel functions ρ with γ ρds ≥ 1 for all γ ∈ Γ, see [7,Section 1.5]. We recall the definition of upper gradient introduced and studied by [24], [32] and [41]. General references for this theory are [7], [23] and [25].
whenever both u(x) and u(y) are finite, and γ g ds = ∞ otherwise. Here x and y are the endpoints of γ. Moreover, if a nonnegative µ-measurable function g satisfies (2.3) for p-almost every path, that is with the exception of a path family of zero p-modulus, then g is called a p-weak upper gradient of u.
For 1 ≤ p < ∞ and an open set Ω ⊂ X, let where the infimum is taken over all upper gradients g of u. Consider the collection of all functions u ∈ L p (Ω) with an upper gradient g ∈ L p (Ω) and let The Newtonian space is defined by The corresponding local Newtonian space is defined by u ∈ N 1,p loc (Ω) if u ∈ N 1,p (Ω ′ ) for all Ω ′ ⋐ Ω, see [7,Proposition 2.29]. Here Ω ′ ⋐ Ω means that Ω ′ is a compact subset of Ω. If u has an upper gradient g ∈ L p (Ω), there exists a unique minimal p-weak upper gradient g u ∈ L p (Ω) with g u ≤ g µ-almost everywhere for all p-weak upper gradients g ∈ L p (Ω) of u, see [7,Theorem 2.5]. Moreover, the minimal p-weak upper gradient is unique up to sets of measure zero. For u ∈ N 1,p (Ω) we have where g u is the minimal p-weak upper gradient of u. The main advantage is that p-weak upper gradients behave better under L p -convergence than upper gradients, see [7,Proposition 2.2]. However, the difference is relatively small, since every p-weak upper gradient can be approximated be a sequence of upper gradients in L p , see [7,Lemma 1.46]. This implies that that the N 1,p -norm above remains the same if the infimum is taken over upper gradients instead of p-weak upper gradients. We collect some calculus rules for upper gradients on metric measure spaces. Let u, v ∈ N 1,p loc (Ω) and let g u , g v ∈ L p loc (Ω) be the p-weak upper gradients of u and v, respectively. Then g u + g v and |u|g v + |v|g u are p-weak upper gradients for u + v and uv, respectively, see [7,Theorem 2.15]. Let η be Lipschitz continuous on Ω with 0 ≤ η ≤ 1 and consider w = u + η(u − v) = (1 − η)u + ηv. Then (1 − η)g u + ηg v + |v − u|g η is a p-weak upper gradient of w, see [7,Theorem 2.18]. Moreover, g u = g v , µ-almost everywhere on the set {x ∈ X : u(x) = v(x)}. In particular, if c ∈ R is a constant, then g u = 0 µ−almost everywhere on the set {x ∈ X : u(x) = c}, see [7,Corollary 2.21]. A metric measure space (X, d, µ) supports a weak Poincaré inequality, if there exist a constant C P and a dilation factor τ ≥ 1 such that for every ball B ρ (x 0 ) in X, for every u ∈ L 1 loc (X) and every upper gradient g of u, we have where the integral average is denoted by A space supporting a Poincaré inequality is connected, see [7,Proposition 4.2]. Throughout the work, we assume that the measure µ is doubling and that the metric measure space (X, d, µ) supports a weak Poincaré inequality. The weak Poincaré inequality and the doubling condition imply the Sobolev-Poincaré inequality for every u ∈ L 1 loc (X) and every 1-weak upper gradient g of u and for every ball B ρ (x 0 ) in X with C = C(C µ , C P ) and Q as in (2.2), see [7,Theorem 4.21]. Next we discuss parabolic Newtonian spaces.

Definition 2.2.
Let Ω ⊂ X be an open set, 0 < T < ∞ and 1 ≤ p < ∞. The parabolic Newtonian space L p (0, T ; N 1,p (Ω)) consists of strongly measurable functions u : (0, T ) → N 1,p (Ω) with the norm The integration over (0, T ) is taken with respect to the one-dimensional Lebesgue measure L 1 . We The strong measurability of u : (0, T ) → N 1,p (Ω) and the assumption u ∈ L p (0, T ; N 1,p (Ω)), imply that there exists a sequence (u k ) k∈N of simple functions u k : (0, T ) → N 1,p (Ω), is a L 1 -measurable pairwise disjoint partition of (0, T ) and v (k) i ∈ N 1,p (Ω), i = 1, . . . , n k , such that u k → u in L p (0, T ; N 1,p (Ω)) as k → ∞. In particular, we have u k (t) → u(t) in N 1,p (Ω) for L 1 -almost every t ∈ (0, T ). In other words, up to relabeling, we have with the sets E k and and simple functions u k as in (2.6). Next we consider upper gradients. Since u k (t) → u(t) in N 1,p (Ω) for L 1 -almost every t ∈ (0, T ) as k → ∞, we have u(t) ∈ N 1,p (Ω) for L 1 -almost every t ∈ (0, T ). Consider the minimal p-weak upper gradient g u(t) ∈ L p (Ω) of u(t) for L 1 -almost every t ∈ (0, T ). The parabolic p-weak upper gradient of u ∈ L p (0, T ; N 1,p (Ω)) is defined to be g u = g u(t) for L 1 -almost every t ∈ (0, T ). We note that the function g u is strongly measurable. For L 1 -almost every t ∈ (0, T ), the function u(t) is the limit of strongly measurable functions u k (t) defined in (2.6). By (2.7) and the locality of minimal p-weak upper gradients, we have for L 1 -almost every t ∈ (0, T ). Strong measurability follows, since u k ∈ N 1,p (Ω) and g u k ∈ L p (Ω) for every k ∈ N. In other words, g u(t) can be approximated in L p (Ω) by the functions g u k (t) ∈ L p (Ω), which we obtain from (2.6) by arguing as in (2.8). Since u k → u in L p (0, T ; N 1,p (Ω)), we have u k → u in L p (0, T ; L p (Ω)) and g u k → g u in L p (0, T ; L p (Ω)) as k → ∞. The product measure in the space X × (0, T ), T > 0, is denoted by µ ⊗ L 1 . For T > 0, we denote the space-time cylinder over an open subset Ω ⊂ X as Ω T = Ω × (0, T ). For u ∈ L p (0, T ; L p (Ω)), there exists a (µ ⊗ L 1 )-measurable representative u : Ω T → [−∞, ∞] such that u(t) = u(·, t) for L 1 -almost every t ∈ (0, T ) and See [33,Theorem 23.21] and [40, Section 2.
With these observations we may consider the parabolic Newtonian space Let u ∈ L p loc (0, T ; N 1,p loc (Ω)), 1 ≤ p < ∞, and consider the time mollification The following approximation result was proved in more generality in [8]. We include a slightly modified version together with its full proof for reader's convenience. We say that where Ω ′ ⋐ Ω and 0 < t 1 < t 2 < T .

Lemma 2.3. Let Ω ⊂ X be an open set and assume that
Proof. Since for L 1 -almost every t ∈ (0, T ), by the definition of the time mollification we have for L 1 -almost every t ∈ (0, T ). By a standard mollifier argument we conclude that u ε → u in L p loc (Ω T ) as ε → 0. By properties of minimal p-weak upper gradients and standard mollifications, we obtain for every s > 0.
Let Ω ′ × (t 1 , t 2 ) ⋐ Ω T . By Fubini's theorem and Minkowski's inequality, we have Since 1 E k ∈ L p (0, T ), k ∈ N, the expression above vanishes as s → 0 by the continuity of translations on L p functions.

BV functions
Next we recall the definition and basic properties of functions of bounded variation on metric spaces, see [39]. The total variation of u ∈ L 1 loc (X) is defined as where the infimum is taken over all sequences (u i ) i∈N with u i ∈ Lip loc (X) for every i ∈ N and Here g ui is a 1-weak upper gradient of u i and Lip loc (X) denotes the class of functions that are Lipschitz continuous on compact subsets of X. We say that a function u ∈ L 1 (X) is of bounded variation, and denote u ∈ BV (X), if Du (X) < ∞. By replacing X with an open set Ω ⊂ X in the definition of the total variation, we can define Du (Ω). A function u ∈ BV loc (Ω) if u ∈ BV (Ω ′ ) for all open sets Ω ′ ⋐ Ω. For an arbitrary set A ⊂ X, we set where g ui is the minimal 1-weak upper gradient of u i in Ω.
If the space supports the Poincaré inequality in (2.4), by an approximation argument, for every u ∈ BV loc (X) and every ball where the constant C and the dilation factor τ are the same as in the Sobolev-Poincaré inequality in (2.5). Next we state a Sobolev type inequality for BV functions which vanish on a large set, see [30] where Q is as in (2.2).
Hölder's inequality and the fact that where we used the fact that The following isoperimetric inequality in [ is a ball in X and u ∈ BV loc (X), then for k < l real numbers we get Notice that By the Sobolev-Poincaré inequality for BV in (2.9), we have On the other hand where, by Bernoulli's inequality, we have This implies This proves the claim.
We also apply parabolic BV functions.

Total variation flow
We discuss a definition of a variational solution to the total variation flow.

Definition 3.1.
Let Ω ⊂ X be an open set and 0 < T < ∞. A function u ∈ L 1 loc (0, T ; BV loc (Ω)) is a variational solution to the total variation flow in Ω T , if for every ϕ ∈ Lip(Ω T ) with supp ϕ ⋐ Ω T .
Boundary terms appear for test functions that do not necessarily vanish on the initial and the last moment of time.

Proposition 3.2.
Let u ∈ L 1 loc (0, T ; BV loc (Ω)) be a variational solution to the total variation flow in Ω T and let Ω ′ × (t 1 , t 2 ) ⋐ Ω T , where 0 < t 1 < t 2 < T are such that the boundary terms below are defined. Then For small enough h > 0, ϕ h = ϕζ h ∈ Lip(Ω T ) with supp ϕ h ⋐ Ω × (0, T ) and therefore admissible as a test function in the definition of variational solution. Thus Notice that For the first term on the left-hand side of (3.2), we find By the dominated convergence theorem, we get on the other hand, by the Lebesgue differentiation theorem, we obtain for L 1 -almost every t 1 , t 2 ∈ (0, T ). This implies that for L 1 -almost every t 1 , t 2 ∈ (0, T ). For the second term on the left-hand side of (3.2), we find Here we used the fact that supp ϕ ⋐ Ω ′ × (0, T ). Substituting these in (3.2), we obtain Eliminating the repeated elements gives

Parabolic De Giorgi class
Next we define the class of functions for which we prove the regularity results. For (x 0 , t 0 ) ∈ X × R and ρ, θ > 0, we denote Q − ρ,θ (x 0 , t 0 ) = B ρ (x 0 ) × (t 0 − θρ, t 0 ]. The positive and negative parts of u are denoted by u ± = max{±u, 0}, respectively. for every Q − ρ,θ (x 0 , t 0 ) ⋐ Ω T , k ∈ R and ϕ ∈ Lip(Ω T ) with supp ϕ ⋐ B ρ (x 0 ) × (0, T ) and 0 ≤ ϕ ≤ 1. The parabolic De Giorgi class DG(Ω T ; γ) is defined as The proof of the necessary and sufficient conditions for continuity of a variational solution to the total variation flow, Theorem 7.1, will only use the local integral inequalities in (4.1). We show that a variational solution to the total variation flow belongs to the parabolic De Giorgi class.

Proposition 4.2. Let u be variational solution to the total variation flow in Ω T . Then u ∈ DG(Ω T ; 8).
Proof. Let φ ∈ Lip(Ω T ) with supp φ ⋐ Ω T . There exists h 0 > 0 such that for every 0 < h < h 0 , we have φ h ∈ Lip(Ω T ) with supp φ h ⋐ Ω T and thus we may apply it as test function in (3.1). Here φ h denotes the time mollification of φ. For a small enough s the translated function v(t) = u(t − s) fulfills (3.1). For 0 < t 2 < t 1 < T , to be specified later, Proposition 3.2 implies where v i (t) = u i (t − s) for every i ∈ N. Let ǫ > 0. There exists i ǫ ∈ N such that, for every i ≥ i ǫ , we have t1 t2 This implies We multiply both sides of the inequality above by a standard mollifier η ε = η ε (s) with support [−ε, ε] for small enough ε > 0. By integrating the resulting expression in the variable s we obtain Applying integration by parts and Fubini's theorem, we have Lemma 2.3 implies that the last term on the right-hand side converges to zero as h → 0. By passing to the limit h → 0, we have g vi+φ (t)η ε (s) dµ dt ds + ǫ.
We apply the test function φ = −ϕζ h ((u i ) ε − k) + in (4.2). For the right-hand side of (4.2) we have For the last integral in (4.3), we have t1 t2 Bρ(x0) The Leibniz rule for upper gradients implies t1 t2 For the integral on the right-hand side of (4.2), we have By letting h → 0, we obtain where Here we used the following observation By Fubini's Theorem, we get This implies that for L 1 -almost every k. Let k ∈ R be such that (4.4) holds. Applying Fatou's lemma one more time, we get It follows that lim for L 1 -almost every t ∈ (0, T ). Therefore, we conclude χ Aε(t) → χ A(t) in L 1 (B ρ (x 0 )) as ε → 0, for L 1 -almost every t ∈ (0, T ) and for L 1 -almost every k ∈ R. For the first term on the left-hand side of (4.2) we find For the second term on the left-hand side of (4.2), by Lemma 2.3, we have Thus, by (4.2) we get g ui (t) dµ dt, and consequently by absorbing terms, this implies By the Leibniz rule we obtain t2 t1 Bρ(x0) Thus we have Letting i → ∞, we obtain (4.5) Choosing t 2 ∈ (0, T ), with t 2 < t 0 − θρ then, for all t 0 − θρ ≤ t 1 ≤ t 0 , by (4.5), we have We conclude that and this holds for any t 0 − θρ ≤ t 1 ≤ t 0 . Therefore ess sup On the other hand, by choosing t 2 = t 0 − θρ and t 1 = t 0 in (4.5), we have (4.7) Adding (4.6) and (4.7) we conclude that ess sup This implies u ∈ DG + (Ω T ; 8). A similar argument shows that u ∈ DG − (Ω T ; 8) and thus u ∈ DG(Ω T ; 8)

De Giorgi lemma
This short section is devoted to prove that functions in a parabolic De Giorgi class are bounded from below. We apply the following standard iteration lemma in the proof, see [16,Lemma 5.1]. where C, b > 1 and α > 0 are given numbers.
In {u ≤ k n } we have It follows that This implies ess sup tn<t<t0 Bn where By Proposition 2.8 there exists a constant C = C(C µ , C P ) such that, for κ = Q+2 Q , we have where γ 2 = Cγ On the other hand, we have for n ∈ N. By (5.3) and (5.2) we obtain By the doubling property we have for every n ∈ N. By (5.4) we conclude By Lemma 5.1, we have Y n → 0 as n → ∞ provided = ν − = ν − (γ, C µ , C P , ω, ξ, a, θ, θ).
The proof of (ii) is almost identical. One starts from inequalities (4.1) for the truncated functions (u − k n ) + with k n = µ + − ξ n ω for the same choice of ξ n . The parameter θ will be determined by the proof. Let ξ ∈ (0, 1) be a fixed parameter.

Characterization of continuity
Finally we are ready to prove the main result of this paper. By repeating the same argument starting from the cylinder Q − ρ1,1 (x 0 , t 0 ) and proceeding recursively, we generate a decreasing sequence of radii ρ n → 0 such that ω 0 ≤ ess osc Q − ρn ,1 (x0,t0) u ≤ η n ω, for every n ∈ N. This is a contradiction with the assumption u is not continuous at (x 0 , t 0 ).