Convergence of Nonlocal Threshold Dynamics Approximations to Front Propagation

In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order α ∈ (0, 2) converge to moving fronts. When α ≧ 1 the resulting interface moves by weighted mean curvature, while for α < 1 the normal velocity is nonlocal of “fractional-type.” The results easily extend to general nonlocal anisotropic threshold dynamics schemes. Introduction We study here the convergence of a class of threshold dynamics-type approximations to moving fronts. Although the arguments extend easily to general anisotropic kernels to keep the presentation simple, here we concentrate on a particular isotropic case, namely, fractional Laplacian of order α ∈ (0, 2). The resulting interfaces move either by weighted mean curvature, if α ∈ [1, 2), or by a nonlocal fractional-type normal velocity, if α ∈ (0, 1). Threshold dynamics is a general term used to describe approximations to motion of boundaries of open sets in RN by “measuring” interactions with the environment. The general scheme we consider here is described as follows: Let Ω0 be an open subset of R N with boundary Γ0. The goal is to come up with an explicit approximation evolution scheme with time step h > 0, so that, as nh → t, the “approximate” front Γnh, which is the boundary of an open set Ωnh identified as the level set of a sign-function, converges, in a suitable sense, to a moving front Γt, the boundary of an open set Ωt, and to identify the limiting velocity. For each n ∈ N, let Γnh = ∂{x ∈ R N : uh(·, nh) = 1} , where (0.1) uh(·, 0) = 1| Ω0 −1| Ω̄c 0 in R , and, for n ≧ 1, (0.2) uh(·, (n + 1)h) = sign (Jh ∗ uh(·, nh)) . Here sign(t) = 1 if t > 0 and −1 otherwise, 1| A denotes the characteristic function of A ⊂ R N , and, for α ∈ (0, 2), (0.3) Jh(x) = pα(x, σα(h)) , Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257 USA. email: [email protected]; [email protected] . Both authors partially supported by the National Science Foundation. 1 2 Caffarelli and Souganidis where pα is the fundamental solution of the fractional Laplacian pde (0.4) Wt − L W = 0 in R × (0,∞) , with (0.5) LW (x) = ∫ |y − x|(W (y)−W (x)) dy . Hence, at each time step, we solve the equation (0.4) with initial datum W (·, 0) = uh(·, nh) in R N , and for time σα(h). Then we define uh(·, (n + 1)h) by uh(x, (n + 1)h) = { 1 if W (x, σα(h)) > 0, −1 otherwise. The algorithm generates functions uh(·, nh) and open sets Ω h nh defined by Ωnh = {x ∈ R N : Jh ∗ uh(·, (n − 1)h)(x) > 0} and uh(·, nh) = 1| Ωnh −1| (Ωnh) in R . We prove that, when h → 0, the discrete evolution Γ0 → Γ h nh = ∂Ω h nh converges, in a suitable sense, to the motion Γ0 → Γt with nonlocal fractional normal velocity, if α ∈ (0, 1), and normal velocity equal to a multiple of the mean curvature, if α ∈ [1, 2). If the kernel J is a Gaussian, it is a classical result that the algorithm generates movement by mean curvature — see below for an extensive discussion and references. To state the main result we recall that the geometric evolution of a front Γt = ∂Ωt with normal velocity v(Dn,n,Ωt), starting at Γ0 = ∂Ω0, is best described by “the level set” partial differential equation (0.6)    ut + F (D u,Du, {u(·, t) ≧ u(x, t)}, {u(·, t) ≦ u(x, t)} = 0 in R × (0,∞) , u = g on RN × {0} , with g such that (0.7) Ω0 = {x ∈ R N : g(x) > 0} and Γ0 = {x ∈ R N : g(x) = 0} , and F (Du,Du, {u(·, t) ≧ u(x, t)}, {u(·, t) ≦ u(x, t)}) = |Du|v(−D( Du |Du| ),− Du |Du| , {u(·, t) ≧ u(x, t)}, {u(·, t) < u(x, t)}) . The basic fact of the level set approach is that the sets Ωt = {x ∈ R N : u(x, t) > 0} and Γt = {x ∈ R N : u(x, t) = 0} are independent of the choice of the initial datum g provided the latter is positive in Ω0 and zero in Γ0. The weighted mean curvature motion corresponds to the level set pde (0.8) ut − Cα tr(I − D̂u⊗ D̂u)D u = 0 in R × (0,∞) , Convergence of nonlocal threshold dynamics approximations to front propagation 3 where, for p ∈ RN \ {0}, p̂ = p/|p|, while the equation corresponding to the nonlocal motion is (0.9) ut − Cα|Du| ∫ (1| (u(x+ y, t)− u(x, t))− 1| (u(x+ y, t)− u(x, t)))|y| dy = 0 , where 1| + and 1| − denote respectively the characteristic functions of [0,∞) and (−∞, 0) and, in both cases, Cα is an explicit constant specified later in the paper. Although the heuristic meaning of (0.8) is well known, some discussion about (0.9) is in order It is implicit in (0.9) that there are sufficient cancellations in the term 1| +(u(x+ y, t)− u(x, t)) − 1| −(u(x+ y, t)− u(x, t)) to compensate for the lack of integrability of the kernel y 7→ |y|−N−α at the origin. Indeed if we write the integral in polar coordinates, ∫ ∞