Abstract
<abstract><p>In this survey we present the state of the art about the asymptotic behavior and stability of the <italic>modified Mullins</italic>–<italic>Sekerka flow</italic> and the <italic>surface diffusion flow</italic> of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the <italic>strict stability</italic> property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $ W^{2, p} $–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently "close" to a smooth <italic>strictly stable critical</italic> set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $.</p></abstract>
Highlights
In this survey we present the state of the art about the asymptotic behavior and stability of the modified Mullins–Sekerka flow and the surface diffusion flow of smooth sets, mainly due to E
+ 4γvt uEt uEt dx on ∂Et in Ω \ ∂Et on ∂Et in Ω where γ is a nonnegative parameter, v, w : [0, T )×Ω → R are continuous functions such that, setting wt = w(t, ·) and vt = v(t, ·), the functions vt and wt are smooth in Ω \ ∂Et, for every t ∈ [0, T ); the functions νt, Ht are the “outer” normal and the relative mean curvature of ∂Et and uEt = 2χEt − 1; the velocity of the motion is given by [∂νtwt] which denotes ∂νtwt+ − ∂νtwt−, that, is the “jump” of the normal derivative of wt on ∂Et, where wt+ and wt− are the restrictions of wt to Ω \ Et and Et, respectively
Where ∆t is the Laplacian of the hypersurface ∂Et, for all t ∈ [0, T ). Such flow was first proposed by Mullins in [45] to study thermal grooving in material sciences, in particular, in the physically relevant case of three–dimensional space, it describes the evolution of interfaces between solid phases of a system, which are studied in a variety of physical settings including phase transitions, epitaxial deposition and grain growth
Summary
We start by introducing the nonlocal Area functional and its basic properties. In the following we denote by Tn the n–dimensional flat torus of unit volume which is defined as the Riemannian quotient of Rn with respect to the equivalence relation x ∼ y ⇐⇒ x − y ∈ Zn, with Zn the standard integer lattice of Rn. By Proposition 2.17, the second variation of the functional J under a volume constraint at a smooth critical set E is a quadratic form in the normal component on ∂E of the infinitesimal generator X ∈ C∞(Tn; Rn) of a volume–preserving variation, that is, on φ = X|νE This and the fact that the infinitesimal generators of the volume–preserving variations are “characterized” by having zero integral of such normal component on ∂E, by Lemma 2.8 and the discussion immediately before, motivate the following definition. ≤ C|ψ(y)|, the third follows by a straightforward computation (involving the map L defined by formula (2.53) and its Jacobian), as ∂E is a “normal graph” over ∂F with ψ as “height function”, the last one by the definition of the “distance” α, recalling that we possibly translated the “original” set F by a vector η ∈ Rn, at the beginning of this step.
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