The Total Variation Flow in RN

Abstract

In this paper, we study the minimizing total variation flow u t ¼ divðDu=jDujÞ in R N for initial data u 0 in L

Anisotropic mean curvature on facets and relations with capillarity

“…Heuristically, if u is a solution of (7.1), and p(t) = (x, u(t, x)) is a point of graph(u(t)) ⊂ R 3 around which u(t) is sufficiently smooth with nonzero gradient, then the vertical velocity of p(t) equals the mean curvature of the level set of u(t) passing through x; strictly ϕc-calibrable flat regions F of graph(u(t)) evolve in vertical direction (8) with velocity equal to P(F)/|F|; vertical walls (provided u(t) is discontinuous) of graph(u(t)) do not move; finally, isolated points where the gradient of u(t) vanishes, such as local minima or local maxima, may develop instantaneously flat horizontal regions. See also [15], [16], [5], [37], and Section 7. Therefore, there are analogies between the total variation flow in R 2 and the anisotropic mean curvature flow of ϕc-calibrable facets; however the two motions differ immediately after the initial time.…”

“…(15) For m = 2, in [7] the authors study the problem for a more general notion of perimeter, and prove that the inner boundary of a solution of (3.1) is a Lipschitz curve out of a closed singular set of zero H 1 -measure. The result has been improved in [63,Theorem 4.5], with the following theorem.…”

“…Let v ∈ C 1 (Ω), and let E be the subgraph of v. Then the ϕc-perimeter (38) of E in Ω × R is Well-posedness of (7.1)-(7.2) follows using the theory in [34], see also [15]. In particular, it is possible to prove that there exists a unique u ∈ C([0, T]; L 2 (R 2 )) ∩ W 1,2 (0, T; L 2 (R 2 )) and there exists η ∈ L ∞ ((0, T) × R 2 ; R 2 ), with ||η||∞ ≤ 1, such that u t = divη in the sense of distributions, and (recalling also (5.3) and setting u(t) = u(t, ·)) ∫︁ for all w ∈ L 2 (R 2 ) ∩ BV(R 2 ); the initial datum is taken in the L 2 (R 2 ) sense.…”

“…In particular, it is possible to prove that there exists a unique u ∈ C([0, T]; L 2 (R 2 )) ∩ W 1,2 (0, T; L 2 (R 2 )) and there exists η ∈ L ∞ ((0, T) × R 2 ; R 2 ), with ||η||∞ ≤ 1, such that u t = divη in the sense of distributions, and (recalling also (5.3) and setting u(t) = u(t, ·)) ∫︁ for all w ∈ L 2 (R 2 ) ∩ BV(R 2 ); the initial datum is taken in the L 2 (R 2 ) sense. We also mention the existenceuniqueness result for the entropic solution, well-suited for more general initial data [15].…”

“…(39) In [8] (see also [19] for eventual regularity of solutions), solutions of (7.1)-(7.2) are shown to vanish in a finite time T(u 0 ), and the rescaled function u(t,·) T(u0)−t converges along subsequences, as t → (T(u 0 )) − , to a solution (40) The problem of detecting explicit solutions of (7.4) has an independent interest: first of all, given v solution of (7.4), the function u(t, ·) := (1 − t) + v(·) is the solution of (7.1) starting at v. Secondly, as shown in [15], any solution of (7.4) allows to construct an explicit solution of problem (7.3). (41) It is natural to look for special solutions of (7.4) of the form v := 1 Ω , for some bounded open set Ω ⊂ R 2 .…”