Symmetric Self-Shrinkers for the Fractional Mean Curvature Flow
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Symmetric Self-Shrinkers for the Fractional Mean Curvature Flow Annalisa Cesaroni1 · Matteo Novaga2 Received: 5 December 2018 © Mathematica Josephina, Inc. 2019
Abstract We show existence of homothetically shrinking solutions of the fractional mean curvature flow, whose boundary consists in a prescribed number of concentric spheres. We prove that all these solutions, except from the ball, are dynamically unstable. Keywords Fractional mean curvature flow · Self-similar solutions · Singularities Mathematics Subject Classification 53C44 · 35R11 · 49Q20
1 Introduction Let us introduce the geometric evolution which we consider in this paper. Given an initial set E ⊂ Rn , we define its evolution E t according to fractional mean curvature flow as follows: the velocity at a point x ∈ ∂ E t is given by
1 dy, |x − y|n+s (1.1) where s ∈ (0, 1) is a fixed parameter and ν is the outer normal at ∂ E t in x. The fractional mean curvature of a set has been introduced in [5] as the first variation of the fractional perimeter functional, and it has been proved in [1] that for sufficiently smooth sets E the rescaled fractional mean curvature (1 − s)Hs (x, E) converges as ∂t x · ν = −Hs (x, E t ) := − lim
ε→0 Rn \Bε (x)
B
χRn \E t (y) − χ E t (y)
Matteo Novaga [email protected] Annalisa Cesaroni [email protected]
1
Department of Statistical Sciences, University of Padova, Via Cesare Battisti 141, 35121 Padua, Italy
2
Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
123
A. Cesaroni, M. Novaga
s → 1 to the classical mean curvature of E at x. The evolution law (1.1) can be interpreted as the L 2 -gradient flow of the fractional perimeter. Existence and uniqueness of viscosity solutions to a level set formulation of (1.1) has been provided in [8,15], and qualitative properties of smooth solutions have been studied in [19]. However, we point out that the short-time existence of smooth solutions has not yet been proved. In [6], the convergence to the fractional mean curvature flow of a threshold dynamics scheme is proved; this result was adapted to the anisotropic case, even in presence of a driving force in [9], where it is also shown that the flow preserves convexity. It has also been observed that the geometric law (1.1) presents some different behavior with respect to the classical mean curvature flow: we refer for instance to the paper [10] about the formation of neck-pinch singularities, and to the paper [7] about fattening and non-fattening phenomena. In this paper, we are interested in the homothetically shrinking solutions for the flow (1.1). A homothetic solution to (1.1) is a self-similar solution to (1.1): substituting E t = λ(t)E in (1.1), it is easy to see, using scale invariance of the fractional mean 1 curvature, that this is equivalent to λ (t)x · ν = − λ(t) s Hs (x, E) for all x ∈ ∂ E. So homothetically shrinking solutions to (1.1) are given by the solutions to (1.1) with initial datum every set E ⊆ Rn of class C 1,1 which satisfies x · ν = c H
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