Elastic flow of networks: short-time existence result
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Journal of Evolution Equations
Elastic flow of networks: short-time existence result Anna Dall’Acqua, Chun-Chi Lin
and Paola Pozzi
Abstract. In this paper we study the L 2 -gradient flow of the penalized elastic energy on networks of qcurves in Rn for q ≥ 3. Each curve is fixed at one end-point and at the other is joint to the other curves at a movable q-junction. For this geometric evolution problem with natural boundary condition we show the existence of smooth solutions for a (possibly) short interval of time. Since the geometric problem is not well-posed, due to the freedom in reparametrization of curves, we consider a fourth-order non-degenerate parabolic quasilinear system, called the analytic problem, and show first a short-time existence result for this parabolic system. The proof relies on applying Solonnikov’s theory on linear parabolic systems and Banach fixed point theorem in proper Hölder spaces. Then the original geometric problem is solved by establishing the relation between the analytical solutions and the solutions to the geometrical problem.
1. Introduction The elastic energy of a smooth regular curve immersed in Rn , f : I¯ → Rn , n ≥ 2, I = (0, 1), is given by 1 | κ |2 ds, (1.1) E( f ) = 2 I where ds = |∂x f |d x is the arc-length element and κ is the curvature vector of the curve. The latter is given by κ = ∂s2 f where ∂s = |∂x f |−1 ∂x denotes the differentiation with respect to the arclength parameter. The elastic energy in (1.1) is also called bending energy of curves. It was proposed by Jacob Bernoulli in 1691 for studying the equilibrium shape of curves, called elasticae or elastic curves, [21]. Besides being used as a simple model in mechanics, the elastic energy has also been used for defining and studying the so-called nonlinear splines in computer graphics, see e.g., [15] and the references therein. Since the elastic energy of a curve can be made arbitrarily small by enlarging the curve, in minimization problems one usually penalizes the length or consider curves with fixed length. In the first case one is led to consider the energy Eλ ( f ) = E( f ) + λL( f ) , λ > 0, Mathematics Subject Classification: Primary 35K52; Secondary 53C44, 35K61, 35K41 Keywords: Geometric evolution, Elastic networks, Junctions, Short-time existence.
(1.2)
Anna Dall’Acqua et al.
J. Evol. Equ.
where L( f ) =
ds I
is the length of the curve. The term λL( f ), when λ > 0, in (1.2) is a natural term to be considered, since it could be viewed as the energy naively responsible for the stretching of curves in elasticity. In both cases, critical points of the energy satisfy the equation 1 2 κ | κ − λ κ = 0, ∇ L 2 Eλ ( f ) = ∇s2 κ + | 2 where in the case of fixed length λ is a Lagrange multiplier (see for istance [9], [18]). Here ∇s is an operator that on a smooth vector field φ acts as follows ∇s φ = ∂s φ − ∂s φ, ∂s f ∂s f , i.e., it is the normal projection of ∂s φ. It may also be understood as a covariant differentiation. The attempt to associate the elastic energy to networks appears in
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