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On length-preserving and area-preserving nonlocal flow of convex closed plane curves. (English) Zbl 1339.53066

Let \(X_0\) be a smooth convex and closed curve in \(\mathbb{R}^2\) and \(X~: \mathbb{S}^1 \times [0,T) \to \mathbb{R}^2\) be a family of curves starting from \(X_0\) and evolving according to a nonlocal curvature flow \[ \partial_t X(\theta,t) = \left[ F(k(\theta,t)) - \lambda(t) \right] N_{\text{in}}(\theta,t)\;, \qquad (\theta,t)\in \mathbb{S}^1 \times (0,T) \;, \] where \(N_{in}(\cdot,t)\) is the inward normal vector field to \(X(\cdot, t)\) and \(k(\cdot,t)\) its curvature. Denote the length and area of the curve \(X(\cdot,t)\) by \(L(t)\) and \(A(t)\), respectively. If \[ F(k)=k^\alpha\;, \qquad \lambda(t) = \frac{1}{2\pi} \int_{X(\cdot,t)} k^{\alpha+1}(\cdot,t)\;ds \] for some \(\alpha>0\), it is shown that \(X\) is well-defined for all times (\(T=\infty\)) and \(X(\cdot,t)\) is a smooth convex and closed curve for each \(t\geq 0\) which satisfies \(L(t)=L(0)\) and converges in \(C^\infty(\mathbb{S}^1)\) to a circle of radius \(L(0)/2\pi\) as \(t\to\infty\). A similar result is obtained when \[ F(k)=k^\alpha\;, \qquad \lambda(t) = \frac{1}{L(t)} \int_{X(\cdot,t)} k^{\alpha}(\cdot,t)\;ds\;, \] but it is now the area which is time-independent (\(A(t)=A(0)\)) and the radius of the limit circle is \(\sqrt{A(0)/\pi}\).
The analysis relies on the fact that the curvature \(k\) solves the nonlocal parabolic equation \[ \partial_t k(\theta,t) = k^2(\theta,t) \left[ \partial_\theta^2 k^\alpha(\theta,t) + k^\alpha(\theta,t) - \lambda(t) \right]\;, \qquad (\theta,t)\in \mathbb{S}^1 \times (0,\infty)\;. \] In particular, combining geometric inequalitites with comparison arguments provides positive upper and lower bounds on \(k\) which are time-independent.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
53A04 Curves in Euclidean and related spaces
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References:

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