The total variation flow in metric random walk spaces

Abstract

In this paper we study the (BV, L p )-decomposition, p = 1, 2, of functions in metric random walk spaces, developing a general theory that can be applied to weighted graphs and nonlocal models in Image Processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case p = 1 we also study the associated geometric problem and the thresholding parameters. m J x (A) := A J(x − y)dL N (y) for every Borel set A ⊂ R N , where J : R N → [0, +∞[ is a measurable, nonnegative and radially symmetric function verifying J =1. Furthermore, given a metric measure space (X, d, µ) we can obtain a metric random walk space Date: July 26, 2019.

Evolution problems of Leray–Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces

“…Hence, since by Theorem 3.5 and Theorem 2.8 we have that K L 2 (Ω,ν) = L 2 (Ω, ν), we conclude that, in fact, K = L 2 (Ω, ν). ] satisfies the Poincaré inequality (2.9) (see [21], note that slight modifications in the results given there are required to prove our statement). Let a p (x, y, r) = |r| p−2 r, which is positive homogeneous.…”

“…It is shown in [21] (see also [4] and [5]) that, under rather general conditions, there are metric random walk spaces satisfying this kind of inequality. Note that the proof of the existence of the Poincaré type inequality in [21] must be slightly modified in order to cover the inequality considered in (2.9).…”

“…Let Ω ⊂ V (G) be a finite set. It is easy to see (see [21]) that [Ω m G , d G , m G , ν G ] satisfies the Poincaré inequality (2.9). Therefore, if we consider the Theorem 3.9.…”

Least gradient functions in metric random walk spaces

“…We recall the following result proved in Theorem 2.22 of [18] (see also [12], Thm. 3.1), which links the total variation defined above and the nonlocal total variation of the type presented in Example 1.1(4): Theorem 1.12.…”

“…As in the local and nonlocal case, we have the following coarea formula, given in Theorem 2.7 of [18], relating the total variation of a function with the perimeter of its superlevel sets. While the result in [18] is stated for u ∈ L 1 (X, ν), this assumption is never used, and the coarea formula holds for all u ∈ BV m (X). Theorem 1.6 (Coarea formula).…”