Existence of 1-harmonic map flow

Transcription

1 Existence of 1-harmonic map flow Michał Łasica joint work with L. Giacomelli and S. Moll University of Warsaw, Sapienza University of Rome Banff, June 22, of 30

2 Setting Ω a bounded Lipschitz domain in R m 2 of 30

3 Setting Ω a bounded Lipschitz domain in R m (N, g) n-dim., complete Riemannian manifold embedded in R N 2 of 30

4 Setting Ω a bounded Lipschitz domain in R m (N, g) n-dim., complete Riemannian manifold embedded in R N given a subspace X(Ω, R N ) of L 1 loc (Ω, RN ) we denote X(Ω, N ) = {u X(Ω, R N ) s. t. u(x) N for a. e. x N } 2 of 30

5 Setting Ω a bounded Lipschitz domain in R m (N, g) n-dim., complete Riemannian manifold embedded in R N given a subspace X(Ω, R N ) of L 1 loc (Ω, RN ) we denote X(Ω, N ) = {u X(Ω, R N ) s. t. u(x) N for a. e. x N } 1-harmonic map flow L 2 -gradient flow of constrained total variation functional, given for u C 1 (Ω, N ) by T VΩ N (u) = u Ω 2 of 30

6 Setting manifold domain (M, γ) compact Riemannian manifold 3 of 30

7 Setting manifold domain (M, γ) compact Riemannian manifold given a subspace X(M, R N ) of L 1 loc (M, RN ) we denote X(M, N ) = {u X(M, R N ) s. t. u(x) N for a. e. x M} 3 of 30

8 Setting manifold domain (M, γ) compact Riemannian manifold given a subspace X(M, R N ) of L 1 loc (M, RN ) we denote X(M, N ) = {u X(M, R N ) s. t. u(x) N for a. e. x M} for u C 1 (M, N ) the total variation is now given by T VM(u) N = M u γ = M ( ) 1 γ αβ 2 u x αu x β 3 of 30

9 p-harmonic map flows for p > 1, p-harmonic flow L 2 -gradient flow of 1 p M u p γ 4 of 30

10 p-harmonic map flows for p > 1, p-harmonic flow L 2 -gradient flow of 1 p M u p γ formally given by equation (in normal coordinates on M) u t = π N (u)div( u p 2 u) 4 of 30

11 p-harmonic map flows for p > 1, p-harmonic flow L 2 -gradient flow of 1 p M u p γ formally given by equation (in normal coordinates on M) or equivalently u t = π N (u)div( u p 2 u) u t = div( u p 2 u) + A N (u)(u x i, u x i) (phmfe) 4 of 30

12 p-harmonic map flows for p > 1, p-harmonic flow L 2 -gradient flow of 1 p M u p γ formally given by equation (in normal coordinates on M) or equivalently u t = π N (u)div( u p 2 u) energy inequality u t = div( u p 2 u) + A N (u)(u x i, u x i) 1 u p γ + p M t 0 M u t 2 γ 1 p M u 0 p γ (phmfe) (EI) 4 of 30

13 Existence of harmonic map flows (p = 2) M, N compact, R N Riemann curvature tensor of N 5 of 30

14 Existence of harmonic map flows (p = 2) M, N compact, R N Riemann curvature tensor of N Theorem (Eells-Sampson, 1964) Let p = 2 and u 0 C (M, N ). If R N 0, unique smooth harmonic map flow u starting with u 0 exists for all t > 0. There exists a sequence (t i ) such that (u(t i )) converges uniformly to a harmonic map u. 5 of 30

15 Existence of harmonic map flows (p = 2) M, N compact, R N Riemann curvature tensor of N Theorem (Eells-Sampson, 1964) Let p = 2 and u 0 C (M, N ). If R N 0, unique smooth harmonic map flow u starting with u 0 exists for all t > 0. There exists a sequence (t i ) such that (u(t i )) converges uniformly to a harmonic map u. Theorem (Chen-Struwe, 1988) Let p = 2. For any u 0 C (M, N ) there exists a global weak solution to (phmfe) with initial datum u 0 satisfying (EI). There exists a sequence (t i ) such that (u(t i )) converges weakly in W 1,2 (M, N ) to a weakly harmonic map u. 5 of 30

16 p 2, part I: Hungerbühler Theorem (Hungerbühler, 1997) Let p = m, u 0 W 1,p (M, N ). There exists a global weak solution to (phmfe) with initial datum u 0 satisfying (EI). This solution is regular except finitely many time instances. There is at most one solution satisfying u L (]0, [ M). 6 of 30

17 p 2, part I: Hungerbühler Theorem (Hungerbühler, 1997) Let p = m, u 0 W 1,p (M, N ). There exists a global weak solution to (phmfe) with initial datum u 0 satisfying (EI). This solution is regular except finitely many time instances. There is at most one solution satisfying u L (]0, [ M). Theorem (Hungerbühler, 1996) Let N be a homogeneous space and let u 0 W 1,p (M, N ). There exists a global weak solution to (phmfe) with initial datum u 0 satisfying (EI). 6 of 30

18 p 2, part 2: Fardoun & Regbaoui Theorem (Fardoun-Regbaoui, ) Let u 0 C 2+α (M, N ). If R N 0 or M u p γ is small enough, there exists a regular global weak solution to (phmfe) with initial datum u 0. There exists a sequence (t i ) such that (u(t i )) converges uniformly to a p-harmonic map u. 7 of 30

19 Motivation for 1-harmonic map flows denoising of manifold-valued image/signal: 8 of 30

20 Motivation for 1-harmonic map flows denoising of manifold-valued image/signal: Ω a rectangle/interval/box 8 of 30

21 Motivation for 1-harmonic map flows denoising of manifold-valued image/signal: Ω a rectangle/interval/box examples of N : 8 of 30

22 Motivation for 1-harmonic map flows denoising of manifold-valued image/signal: Ω a rectangle/interval/box examples of N : S 2 color component of an image 8 of 30

23 Motivation for 1-harmonic map flows denoising of manifold-valued image/signal: Ω a rectangle/interval/box examples of N : S 2 color component of an image R 2 S 1 luminance-chromacity-hue space 8 of 30

24 Motivation for 1-harmonic map flows denoising of manifold-valued image/signal: Ω a rectangle/interval/box examples of N : S 2 color component of an image R 2 S 1 luminance-chromacity-hue space SO(3) or SE(3) orientations of objects (e. g. camera trajectories) 8 of 30

25 Motivation for 1-harmonic map flows denoising of manifold-valued image/signal: Ω a rectangle/interval/box examples of N : S 2 color component of an image R 2 S 1 luminance-chromacity-hue space SO(3) or SE(3) orientations of objects (e. g. camera trajectories) SP D(3) diffusion tensor space 8 of 30

26 The Euclidean case N = R n = R N T V Ω (u) = Ω { } u = sup u divϕ: ϕ Cc 1 (Ω), ϕ 1 Ω 9 of 30

27 The Euclidean case N = R n = R N T V Ω (u) = Ω { } u = sup u divϕ: ϕ Cc 1 (Ω), ϕ 1 Ω T V Ω convex, lower semicontinuous functional on L 2 (Ω) 9 of 30

28 The Euclidean case N = R n = R N T V Ω (u) = Ω { } u = sup u divϕ: ϕ Cc 1 (Ω), ϕ 1 Ω T V Ω convex, lower semicontinuous functional on L 2 (Ω) given u 0 BV (Ω) there exists a global in time L 2 -gradient flow (steepest descent curve) u satisfying for t > 0 u t T V (u) 9 of 30

29 The Euclidean case N = R n = R N T V Ω (u) = Ω { } u = sup u divϕ: ϕ Cc 1 (Ω), ϕ 1 Ω T V Ω convex, lower semicontinuous functional on L 2 (Ω) given u 0 BV (Ω) there exists a global in time L 2 -gradient flow (steepest descent curve) u satisfying for t > 0 u satisfies energy inequality u(t, ) + for t > 0 9 of 30 Ω u t T V (u) t 0 Ω u 2 t Ω u 0

30 The Euclidean case N = R n = R N Theorem (Andreu-Ballester-Caselles-Mazon, 2000) u H 1 (0, T ; L 2 (Ω)) L (0, T ; BV (Ω)) is a steepest descent curve of T V Ω iff there exists Z L (]0, T [ Ω) with divz L 2 (]0, T [ Ω) such that u t = divz a. e. in Ω, ( u, Z) = u as measures on Ω, for a. e. t ]0, T [ Z 1 Z ν Ω = 0 a. e. in Ω a. e. on Ω 10 of 30

31 Regular 1-harmonic map flow formally u t = π N (u)div u u ( ) + u u νω = 0 (1HMFE) 11 of 30

32 Regular 1-harmonic map flow formally u t = π N (u)div u u ( ) + u u νω = 0 (1HMFE) Definition We say that u W 1,2 (]0, T [ Ω, N ) with u L (]0, T [ Ω) is a regular solution to (1HMFE) if there exists Z L (]0, T [ Ω) such that u t = divz, Z T u N, Z 1, Z = u u if u 0 Z ν Ω = 0 a. e. on ]0, T [ Ω a. e. in ]0, T [ Ω, 11 of 30

33 Existence of regular 1-harmonic map flow Theorem (Giga-Kashima-Yamazaki, 2004) Suppose that N is compact, Ω = T m, u 0 C 2+α (Ω, N ) and u 0 L p (Ω) is small enough for some p > 1. There exists a local-in-time regular solution to (1HMFE) with initial datum u 0 satisfying (EI). 12 of 30

34 Existence of regular 1-harmonic map flow Theorem (Giga-Kashima-Yamazaki, 2004) Suppose that N is compact, Ω = T m, u 0 C 2+α (Ω, N ) and u 0 L p (Ω) is small enough for some p > 1. There exists a local-in-time regular solution to (1HMFE) with initial datum u 0 satisfying (EI). Theorem (Giacomelli-Ł-Moll, preprint 2017) Suppose that N is a closed submanifold in R N and Ω is a convex domain in R m. If u 0 W 1, (Ω, N ), there exists a unique local-in-time regular solution to (1HMFE) with initial datum u 0 satisfying (EI). 12 of 30

35 Existence of regular 1-harmonic map flow Theorem (Giga-Kashima-Yamazaki, 2004) Suppose that N is compact, Ω = T m, u 0 C 2+α (Ω, N ) and u 0 L p (Ω) is small enough for some p > 1. There exists a local-in-time regular solution to (1HMFE) with initial datum u 0 satisfying (EI). Theorem (Giacomelli-Ł-Moll, preprint 2017) Suppose that N is a closed submanifold in R N and Ω is a convex domain in R m. If u 0 W 1, (Ω, N ), there exists a unique local-in-time regular solution to (1HMFE) with initial datum u 0 satisfying (EI). If R N 0 or the image of the datum is small enough, the solution is global and becomes constant in finite time. 12 of 30

36 Sketch of proof approximation with gradient flow of (ε 2 + u 2 ) of 30

37 Sketch of proof approximation with gradient flow of (ε 2 + u 2 ) 1 2 totally geodesic metric h on R N for N : h restricted to N coincides with g h coincides with the Euclidean metric outside a neighbourhood of N there exists a neighbourhood U of N in R N and an involution i of U that is isometric with respect to h 13 of 30

38 Sketch of proof approximation with gradient flow of (ε 2 + u 2 ) 1 2 totally geodesic metric h on R N for N : h restricted to N coincides with g h coincides with the Euclidean metric outside a neighbourhood of N there exists a neighbourhood U of N in R N and an involution i of U that is isometric with respect to h local existence for approximation via a result by Acquistapace-Terreni 13 of 30

39 Sketch of proof approximation with gradient flow of (ε 2 + u 2 ) 1 2 totally geodesic metric h on R N for N : h restricted to N coincides with g h coincides with the Euclidean metric outside a neighbourhood of N there exists a neighbourhood U of N in R N and an involution i of U that is isometric with respect to h local existence for approximation via a result by Acquistapace-Terreni 13 of 30

40 Sketch of proof Bochner s formula 1 d 2 dt u 2 = div(u x i Z x i) (π N (u)u x i x j) Z i,x j + Z i R N (u)(u x i, u x j)u x j 14 of 30

41 Sketch of proof Bochner s formula 1 d 2 dt u 2 = div(u x i Z x i) (π N (u)u x i x j) Z i,x j + Z i R N (u)(u x i, u x j)u x j due to convexity of Ω, from Bochner s formula we get uniform estimate 1 d u p C(N ) u p+1, p dt Ω Ω = d dt u L (Ω) C(N ) u 2 L (Ω) 14 of 30

42 Sketch of proof Bochner s formula 1 d 2 dt u 2 = div(u x i Z x i) (π N (u)u x i x j) Z i,x j + Z i R N (u)(u x i, u x j)u x j due to convexity of Ω, from Bochner s formula we get uniform estimate 1 d u p C(N ) u p+1, p dt Ω Ω = d dt u L (Ω) C(N ) u 2 L (Ω) standard limit passage 14 of 30

43 Role of convexity suppose R N 0 (e. g. N = R N ) 15 of 30

44 Role of convexity suppose R N 0 (e. g. N = R N ) if Ω convex, then u(t, ) L p (Ω) u 0 L p (Ω) 15 of 30

45 Role of convexity suppose R N 0 (e. g. N = R N ) if Ω convex, then u(t, ) L p (Ω) u 0 L p (Ω) for Lipschitz (smooth) non-convex Ω, is it still true that u 0 W 1,p (Ω) = u(t, ) W 1,p (Ω)? 15 of 30

46 Role of convexity suppose R N 0 (e. g. N = R N ) if Ω convex, then u(t, ) L p (Ω) u 0 L p (Ω) for Lipschitz (smooth) non-convex Ω, is it still true that u 0 W 1,p (Ω) = u(t, ) W 1,p (Ω)? for the gradient flow of u x + u y there is a non-convex polygon Ω and u 0 W 1, (Ω) such that u(t, ) W 1,1 loc (Ω) for small t > 0 (Ł-Moll-Mucha, 2017) 15 of 30

47 Manifold domain Theorem (Giacomelli-Ł-Moll, preprint 2017) Suppose that N is a closed submanifold in R N and M is a compact, orientable Riemannian manifold. If u 0 W 1, (M, N ), there exists a unique local-in-time regular solution to (1HMFE) with initial datum u 0 satisfying (EI). 16 of 30

48 Manifold domain Theorem (Giacomelli-Ł-Moll, preprint 2017) Suppose that N is a closed submanifold in R N and M is a compact, orientable Riemannian manifold. If u 0 W 1, (M, N ), there exists a unique local-in-time regular solution to (1HMFE) with initial datum u 0 satisfying (EI). If R N 0, the solution is global. If furthermore Ric M 0, the solution converges uniformly to a 1-harmonic map. 16 of 30

49 BV solutions (N - hyperoctant) Theorem (Giacomelli-Mazon-Moll, ) Let N be a hyperoctant of S n and u 0 BV (Ω, N ). There exists a solution to u t = divz + u g u as measures on Ω, u t u = div(z u) a. e. on Ω, Z 1, Z T u N a. e. on Ω, Z ν Ω = 0 a. e. on Ω in a. e. t ]0, T [ for arbitrarily large T > of 30

50 BV solutions (N - hyperoctant) Theorem (Giacomelli-Mazon-Moll, ) Let N be a hyperoctant of S n and u 0 BV (Ω, N ). There exists a solution to u t = divz + u g u as measures on Ω, u t u = div(z u) a. e. on Ω, Z 1, Z T u N a. e. on Ω, Z ν Ω = 0 a. e. on Ω in a. e. t ]0, T [ for arbitrarily large T > 0. there holds ( u, Z) = u u as measures for a. e. t ]0, T [. 17 of 30

51 m = 1 localization of energy inequality Ω = I =]0, 1[ 18 of 30

52 m = 1 localization of energy inequality Ω = I =]0, 1[ Theorem (Giacomelli-Ł, preprint 2018) Let u H 1 (0, ; L 2 (I) n ) L (0, ; BV (I) n ) be the steepest descent curve of T V I emanating from u 0 BV (I) n. There holds as measures for t > 0. u x (t, ) u 0,x 18 of 30

53 Sketch of proof approximate with gradient flow of I (ε2 + u x 2 ) 1 2, mollify u 0 19 of 30

54 Sketch of proof approximate with gradient flow of I (ε2 + u x 2 ) 1 2, mollify u 0 take smooth cutoff function ϕ supported in B R (x 0 ) with ϕ = 1 in B r (x 0 ) 19 of 30

55 Sketch of proof approximate with gradient flow of I (ε2 + u x 2 ) 1 2, mollify u 0 take smooth cutoff function ϕ supported in B R (x 0 ) with ϕ = 1 in B r (x 0 ) for p > 1 calculate d dt ϕ 2 (ε 2 + u x 2 ) p 2 19 of 30

56 Sketch of proof approximate with gradient flow of I (ε2 + u x 2 ) 1 2, mollify u 0 take smooth cutoff function ϕ supported in B R (x 0 ) with ϕ = 1 in B r (x 0 ) for p > 1 calculate d dt ϕ 2 (ε 2 + u x 2 ) p 2 estimate 1 (ε 2 + u x (t, ) 2 ) p 1 2 p p B r(x 0 ) B R (x 0 ) (ε 2 + u 2 0,x) p ε p 1 t 2 + p 1 R r 19 of 30

57 Sketch of proof approximate with gradient flow of I (ε2 + u x 2 ) 1 2, mollify u 0 take smooth cutoff function ϕ supported in B R (x 0 ) with ϕ = 1 in B r (x 0 ) for p > 1 calculate d dt ϕ 2 (ε 2 + u x 2 ) p 2 estimate 1 (ε 2 + u x (t, ) 2 ) p 1 2 p p B r(x 0 ) B R (x 0 ) (ε 2 + u 2 0,x) p ε p 1 t 2 + p 1 R r pass to the limit ε 0 +, then p 1 +, R r +, relax initial datum 19 of 30

58 Completely local estimates Theorem (Bonforte-Figalli, 2012) Let u be a solution to the scalar total variaton flow with initial datum u 0 BV (I). Then u x ({x 0 }) u 0,x ({x 0 }) for any x 0 J u0 and osc A u osc A u 0 on any interval A I where u 0 is continuous. 20 of 30

59 Completely local estimates Theorem (Bonforte-Figalli, 2012) Let u be a solution to the scalar total variaton flow with initial datum u 0 BV (I). Then u x ({x 0 }) u 0,x ({x 0 }) for any x 0 J u0 and osc A u osc A u 0 on any interval A I where u 0 is continuous. Theorem (Briani-Chambolle-Novaga-Orlandi, 2012) Let Ω be an open domain in R n and let u 0 L 2 (Ω, R n ) be such that div u 0 is a Radon measure on Ω. The L 2 -gradient flow of functional Ω div u satisfies (div u(t, )) ± (div u 0 ) ±. 20 of 30

60 Completely local estimates Theorem (Bonforte-Figalli, 2012) Let u be a solution to the scalar total variaton flow with initial datum u 0 BV (I). Then u x ({x 0 }) u 0,x ({x 0 }) for any x 0 J u0 and osc A u osc A u 0 on any interval A I where u 0 is continuous. Theorem (Briani-Chambolle-Novaga-Orlandi, 2012) Let Ω be an open domain in R n and let u 0 L 2 (Ω, R n ) be such that div u 0 is a Radon measure on Ω. The L 2 -gradient flow of functional Ω div u satisfies (div u(t, )) ± (div u 0 ) ±. non-increase of jumps for the scalar total variation flow (Caselles-Jalalzai-Novaga, 2013) 20 of 30

61 Completely local estimates Theorem (Bonforte-Figalli, 2012) Let u be a solution to the scalar total variaton flow with initial datum u 0 BV (I). Then u x ({x 0 }) u 0,x ({x 0 }) for any x 0 J u0 and osc A u osc A u 0 on any interval A I where u 0 is continuous. Theorem (Briani-Chambolle-Novaga-Orlandi, 2012) Let Ω be an open domain in R n and let u 0 L 2 (Ω, R n ) be such that div u 0 is a Radon measure on Ω. The L 2 -gradient flow of functional Ω div u satisfies (div u(t, )) ± (div u 0 ) ±. non-increase of jumps for the scalar total variation flow (Caselles-Jalalzai-Novaga, 2013) for a solution u to the scalar TV flow with initial datum u 0 BV (Ω), does s u(t, ) s u 0? 20 of 30

62 A note about TV flow for m = 1 take u BV (I) n, Z W 1,1 (I) n with Z 1 a. e. 21 of 30

63 A note about TV flow for m = 1 take u BV (I) n, Z W 1,1 (I) n with Z 1 a. e. the condition (u x, Z) = u x is equivalent to (a measure derivative) Z = ux u x u x a. e. in I 21 of 30

64 1-harmonic map flow for m = 1 Definition Suppose that u W 1,2 (0, T ; L 2 (I, N )) L (0, T ; BV (I, N )) and dist g (u, u + ) < inj N on J u. We say that u is a solution to (1HMFE) if there exists Z L (]0, T [ I) n such that a. e. in ]0, T [ there holds 22 of 30 u t = π N (u)z x a. e. on I, Z T u N, Z 1 a. e. on I, Z = ux u x u x a. e. on I \ J u, Z = T (u ), Z + = T (u + ) on J u, Z ν Ω = 0 on I

65 1-harmonic flow for m = 1 Theorem (Giacomelli-Ł, in preparation) Let u 0 BV (I, N ) satisfy dist g (u 0, u+ 0 ) < R on J u, R = R (N ). For any T > 0 there exists a solution to (1HMFE) starting with u of 30

66 Relaxed T V for u BV (I, N ), define T V g (u) = { inf lim inf I u k x : (u k ) W 1, (I, N ), u k u in BV (I, N ) } 24 of 30

67 Relaxed T V for u BV (I, N ), define T V g (u) = { inf lim inf I u k x : (u k ) W 1, (I, N ), u k u in BV (I, N ) } there holds T V g (u) = I u x g, where u x g = u x I \ Ju + dist g(u, u+)h0 (Giaquinta-Mucci, 2006) Ju 24 of 30

68 Difficulties Z and u x not in complementary spaces 25 of 30

69 Difficulties Z and u x not in complementary spaces in the expanded form u t = Z x + A N (Z, u x ) the nonlinear term depends on Z (no sphere trick) 25 of 30

70 Difficulties Z and u x not in complementary spaces in the expanded form u t = Z x + A N (Z, u x ) the nonlinear term depends on Z (no sphere trick) lack of strong convergence of Z cannot pass to the limit timeslice-wise 25 of 30

71 Sketch of proof due to symmetry of R N, u x R N (u x, u x )u x = 0 26 of 30

72 Sketch of proof due to symmetry of R N, u x R N (u x, u x )u x = 0 for u 0 W 1, (I, N ), there exists a unique global regular solution to the 1-harmonic flow independently of N 26 of 30

73 Sketch of proof due to symmetry of R N, u x R N (u x, u x )u x = 0 for u 0 W 1, (I, N ), there exists a unique global regular solution to the 1-harmonic flow independently of N approximate with regular solutions 26 of 30

74 Sketch of proof due to symmetry of R N, u x R N (u x, u x )u x = 0 for u 0 W 1, (I, N ), there exists a unique global regular solution to the 1-harmonic flow independently of N approximate with regular solutions completely local estimate u x (t, ) g u 0,x g gives a good uniform bound and allows to calculate ux u 0,x, u x u 0,x outside of J u 0 26 of 30

75 Sketch of proof due to symmetry of R N, u x R N (u x, u x )u x = 0 for u 0 W 1, (I, N ), there exists a unique global regular solution to the 1-harmonic flow independently of N approximate with regular solutions completely local estimate u x (t, ) g u 0,x g gives a good uniform bound and allows to calculate ux u 0,x, u x u 0,x outside of J u 0 calculate ux u x by chain rule 26 of 30

76 Sketch of proof (jump part) Fermi coordinates in a neighborhood of the geodesic along the jump: on the geodesic g ij = δ ij, g ij,k = 0 27 of 30

77 Sketch of proof (jump part) Fermi coordinates in a neighborhood of the geodesic along the jump: on the geodesic g ij = δ ij, g ij,k = 0 in coordinates ũ i t = z i x + Γ i jk (u) zj ũ k x 27 of 30

78 Sketch of proof (jump part) Fermi coordinates in a neighborhood of the geodesic along the jump: on the geodesic g ij = δ ij, g ij,k = 0 in coordinates ũ i t = z i x + Γ i jk (u) zj ũ k x take ϕ 0 cutoff centered around the jump: I I u x ϕ = I g ij (u) ũ i t ũ j ϕ+ g ij (u) z i ũ j xϕ = g ij,k (u) ũ k x z i ũ j ϕ I g ij (u) Γ i jk (u) zj ũ k x ũ j ϕ g ij (u) z i ũ j ϕ x I I 27 of 30

79 Sketch of proof (jump part) slice-wise estimate dist(u(t, x), γ u(t,a),u(t,b) ) C b a u t (t, ) for x ]a, b[, t > 0, where γ u(t,a),u(t,b) is the minimal geodesic joining u(t, a) and u(t, b) 28 of 30

80 Sketch of proof (jump part) slice-wise estimate dist(u(t, x), γ u(t,a),u(t,b) ) C b a u t (t, ) for x ]a, b[, t > 0, where γ u(t,a),u(t,b) is the minimal geodesic joining u(t, a) and u(t, b) maximum principle in a convex ball 28 of 30

81 Sketch of proof (jump part) slice-wise estimate dist(u(t, x), γ u(t,a),u(t,b) ) C b a u t (t, ) for x ]a, b[, t > 0, where γ u(t,a),u(t,b) is the minimal geodesic joining u(t, a) and u(t, b) maximum principle in a convex ball relaxation estimate lim inf I u k x ϕ I u x g ϕ for the approximating sequence (u k ) W 1, (I, N ) converging to u weakly in BV (I, N ) 28 of 30

82 Other boundary conditions can be transferred to periodic or Dirichlet b. c. 29 of 30

83 Other boundary conditions can be transferred to periodic or Dirichlet b. c. no finite stopping time for Dirichlet 29 of 30

84 Other boundary conditions can be transferred to periodic or Dirichlet b. c. no finite stopping time for Dirichlet counterexample by Giga-Kuroda, 2015 for N = S 2 29 of 30

85 Other boundary conditions can be transferred to periodic or Dirichlet b. c. no finite stopping time for Dirichlet counterexample by Giga-Kuroda, 2015 for N = S 2 easy counterexample for N = R 2 29 of 30

86 30 of 30 Thank you for your attention!