Evolution of Plane Curves Driven by a Nonlinear Function of Curvature and Anisotropy

We study the intrinsic heat equation governing the motion of plane curves. The normal velocity v of the motion is assumed to be a nonlinear function of the curvature and tangential angle of a plane curve Γ. By contrast to the usual approach, the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional α. Short time existence of a regular family of evolving curves is shown in the case when v = γ(ν)|k|m−1k, 0 < m ≤ 2 and the governing system of equations includes a nontrivial tangential velocity functional. We study the evolution of a closed smooth plane curve Γ : S → R with the normal velocity speed v depending on the curvature k and the tangential angle ν, i.e. v = β(k, ν). As a motivation one can consider e.g. the multiphase thermomechanics where the plane curve evolution satisfying v = β(k, ν) is an appropriate model for describing the motion of phase interfaces (see [AG]). Another application arises from the image processing where the affine invariant scale with v = k has special conceptual and practical importance (see [ST], [AST]). In our approach a family of evolving curves Γ = Image(x(., t)), t ∈ [0, T ], is represented by the position vector x : QT = S 1 × (0, T )→ R satisfying the intrinsic heat equation ∂tx = θ −1 1 ∂s ( θ 2 ∂sx ) , x(., 0) = x(.) (1) where s is the arc-length parameter and θ1, θ2 are geometric quantities, i.e. functions whose definition is independent of particular parameterization of Γ. By using Frenet’s formulae, the intrinsic heat equation can be rewritten as ∂tx = β ~ N + α~ T , where θ1θ2 = k/β(k, ν) and α = θ −1 1 ∂s ( θ 2 ) . (2) Given a function β, the only constraint imposed on θ1, θ2 is the condition θ1θ2 = k/β. This gives raise to various choices of θ1, θ2 and subsequently to various tangential velocities α. It is well known (cf. [AST]) that the tangential velocity functional α does not change the shape of evolving curves. On the other hand, the presence of a suitable tangential velocity is very important in order to suggest a powerful numerical scheme for solving the geometric equation v = β(k, ν) (cf. [MS1], [MS2]). The choice of a trivial α = 0 may lead to computational instabilities caused by merging of numerical grid points representing a discrete curve or by formation of the so-called swallow tails. If we denote g = |∂ux| then the intrinsic heat equation can be rewritten in terms k, ν and g as follows ∂tk = g ∂u ( g∂uβ(k, ν) ) + αg∂uk + k β(k, ν) ∂tν = β ′ k(k, ν)g ∂u ( g∂uν ) + k(α + β ν(k, ν)) ∂tg = −gkβ(k, ν) + ∂uα (u, t) ∈ QT (3) (cf. [MS2]). A solution of (3) is subject to the initial conditions k(., 0) = k, ν(., 0) = ν, g(., 0) = g corresponding to the initial curve Γ = Image(x). Notice that ∂uν 0 = gk. In this paper we propose a special choice of the tangential velocity functional α such that that the ratio of the local length element g = |∂ux| to the the total length |Γ| is constant with respect to time, i.e. ∂t(g/|Γ|) = 0. Combining the third equation in (3) with the equation for the total length d dt |Γ|+ ∫ Γt kβ(k, ν)ds = 0 it turns out that ∂t(g/|Γ|) = 0 iff α is a solution of the nonlocal equation