Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

We prove that, in the flat torus and in any dimension, the volume-preserving mean curvature flow and the surface diffusion flow, starting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,1}$$\end{document}C1,1-close to a strictly stable critical set of the perimeter E, exist for all times and converge to a translate of E exponentially fast as time goes to infinity.


Introduction
In this paper we establish global in time existence and convergence towards equilibrium of two physically relevant volume-preserving geometric motions, namely the volumepreserving mean curvature flow and the surface diffusion flow.
On the one hand, the volume-preserving mean curvature flow is the volume-preserving counterpart of the well-known mean curvature flow, and it is defined as a smooth evolution of sets E t governed by the law (0.1) where V t and H Et are the outer normal velocity and the mean curvature of ∂E t , respectively, while HEt = ffl ∂Et H Et .The mean curvature flow is a well-known evolution model, with far-reaching geometric and physical applications, which has a rich history dating back to its use in material science.One notable application is in physical systems involving multiple phases, such as the motion of grain boundaries in materials science, as first discussed by Mullins [34].
On the other hand, the surface diffusion flow is a smooth flow of sets E t evolving according to the law (0.2) where ∆ Et denotes the Laplace-Beltrami operator on ∂E t .Similar to the mean curvature flow, the surface diffusion flow has important applications in material science, especially in physical systems with multiple phases.It has been proposed in the physical literature by Mullins [33] to model surface dynamics for phase interfaces when the evolution is governed by mass diffusion in the interface.
The volume preserving mean curvature flow can be seen as a simplified, second-order version of the surface diffusion flow as both flows share several common properties.Indeed, from the evolution laws (0.1) and (0.2) it follows that the volume of the evolving sets is preserved along the two flows, as can be easily seen from the following computation d dt the perimeter is decreasing, since the evolution (0.1) satisfies d dt and an integration by parts shows for (0.2) that d dt Moreover, these two evolutions can be regarded (at least formally) as gradient flows of the perimeter according to suitable metrics.In particular, the mean curvature flow can be considered as (a volume preserving modification of) the L 2 -gradient flow of the perimeter, while the surface diffusion can be interpreted as its H −1 -gradient flow.
In both cases, singularities may appear in a finite time even for initial smooth sets (see [31]), therefore in general only short-time existence results are available, see for instance [15,23] for the mean curvature flow and [14] for the surface diffusion flow.Therefore, it is an important question is to identify sufficient conditions for global existence.In this paper, we will focus on the flat torus T N , which is particularly interesting due to the great variety of possible limit points for the flows, namely periodic constant mean curvature hypersurfaces.In the Euclidean space only unions of balls have constant mean curvature, whereas the flat torus admits a much broader range of such surfaces.However, a full characterization of constant mean curvature hypersurfaces in T N is not available in any dimension.In dimension N = 2, the only sets with constant mean curvature are discs and stripes (also called lamellae), while for N ≥ 3 there exist many nontrivial examples, as stripes, cylinders and triply periodic surfaces known as gyroids.Given the (formal) gradient flow structure of the two evolutions, it is natural to expect that the flow starting close to "stable" sets for the perimeter exists for all times and asymptotically converges to those "stable" sets.We refer to this property as dynamical stability.We will properly define the concepts of stability and criticality in Definition 1.1, however we can summarize them as follows: critical sets are those with boundary with constant mean curvature, while stable sets are critical sets with positive definite second variation of the perimeter (i.e., they are "stable" for the perimeter functional).
It is a classical approach in the study of this type of flows to restrict the analysis to small enough neighborhoods of strictly stable sets, in order to show global existence and convergence of the flows.This method was employed in [2,17,18] (see also [11] for a complete survey), where the authors considered the surface diffusion and the Mullins-Sekerka flows in the 2, 3-dimensional flat torus.In these works, it was shown that strictly stable sets are dinamically stable for the aforementioned flows.To be more precise, they proved that for initial data that are small deformations of a strictly stable set (in a suitably high Sobolev norm), the flows exist globally and they converge exponentially fast to a translate of the stable set.It should be noted that the flows considered in these works include nonlocal terms accounting for the presence of elastic potentials, but their results also apply to the evolution driven solely by the perimeter energy as in our case.
Regarding the volume preserving mean curvature flow, recent progresses have been made in proving the dynamical stability of strictly stable sets in the flat torus of dimension 3 [35], while older results mainly concern convex sets, balls, or the 2-dimensional setting.The dynamical stability of balls has been proven in the Euclidean setting under various hypoteses on the dimension or on the initial set in [15,19,23,28] (see also the approach based on weak solutions of [24] in R 2 and in [6] in the anisotropic and crystalline setting, for convex initial data).For the surface diffusion flow, most famous results deal with the stability of balls [14,36], infinite cylinders [27], and two-dimensional triple junctions [20], as well double bubbles [1].
Building upon recent developments in the study of geometric flows, we are able to extend to all dimensions the aforementioned results on the dynamical stability of strictly stable sets in the flat torus, both for the surface diffusion flow and the volume preserving mean curvature flow.Specifically, we will employ a quantitative Alexandrov-type estimate recently established in [9], based on prior results in the Euclidean setting [32] (see also [10,7] for similar result in the nonlocal setting).Regarding our result, simply assuming the initial set to be close in the C 1,1 -topology to a strictly stable set, we obtain global existence and asymptotic convergence of both the flows to (a translated of) the underlying stable set.This is surprising for the surface diffusion flow, which is a fourth-order flow, but is made possible by applying estimates from [22], in the spirit of [26], which provide a fourth-order counterpart to the results for mean curvature flows with rough initial data [26].Our main result is the following.Theorem 0.1.Let E ⊂ T N be a strictly stable set and let i) the volume-preserving mean curvature flow (E t ) t starting from E 0 (defined in (1.10)) exists smooth for all times t ≥ 0, and E t → E + τ as t → ∞, for some τ ∈ T N , in C k for every k ∈ N exponentially fast; (ii) the surface diffusion flow (E t ) t starting from E 0 (defined in (1.17)) exists smooth for all times t ≥ 0, and Where with exponentially fast we mean that the sets E t can be written as normal deformations of E + τ induced by functions u( We conclude this introduction by outlining the strategy of our proof, which is based on the gradient flow structure of the evolution.As we said above, the novelty is the application of the Alexandrov-type inequality [9,Theorem 1.3], which allows to bound the velocity in terms of the displacement.By iterating this procedure for the whole time of existence and using higher order estimates, we can extend the flow for all times.In order to do so, we need to show that the short-time existence and regularity results depend only on the bounds of the initial datum.This is not a priori clear from previous existence results [14,15], and so we rely on Schauder estimates on the linearized problem solved by the flows.This is a quasilinear perturbation of the heat equation for the mean curvature flow, and a quasilinear perturbation of the biharmonic heat equation for the surface diffusion flow.While Schauder-type estimates for general quasilinear parabolic PDEs of the second order are well known (see for istance [16]), we couldn't find a precise reference for the fourth-order equation.Thus, we used some recent reformulation provided in [22], where time-weighted Hölder norms are employed.Although an approach by scaling (in the spirit of [26]) could be feasible by working in local coordinates, we preferred to rely on the estimates provided in [22].After establishing the global existence of both flows, we obtain the exponential convergence up to translations via a Gronwall-type inequality.Notably, the optimality of the exponent in the Alexandrov theorem [9,Theorem 1.3] yields the exponential rate of convergence.Finally, we prove the convergence of these translations by exploiting the decay of geometric quantities along the flow, in the spirit of [2].
We also note that we expect the same arguments to be applicable in treating the Mullins-Sekerka flow.However, further work will be necessary to address the regularity of the linearized problem, which in this case is a quasilinear perturbation of a fractional heat equation.We plan to tackle this problem in future research.
We conclude by highlighting that a similar stability result for the surface diffusion flow has been proved by the second author and collaborators in [13] using different techniques (that are shown in details in dimension N = 4 and listed for any general N ).In particular, they consider initial sets E 0 close to the strictly stable set in C 1 and with associated energy, that is sufficiently small.Even if the final result is the same, we consider appropriate to stress the differences in the hypotheses on the initial datum.

Preliminary results
In this section we collect some preliminary results and we fix the notations.We denote by T N the N -dimensional flat torus, which is the quotient of R N by Z N .The function spaces C k (T N ) and W k,p (T N ), for k ∈ N and p ∈ [1, ∞], are defined as the restriction of C k (R N ) and W k,p loc (R N ), respectively, to the functions that are one-periodic.With B r (x) we denote the ball in R N of center x and radius r, while B r will be a shorthand notation for B r (0).Given x ∈ R N , we will write x = (x , x N ) where x ∈ R N −1 and x N ∈ R. Similarly, we denote by B r (x ) ⊂ R N −1 the ball in R N −1 with radius r > 0 and center x ∈ R N −1 .
Moreover, we denote by c, C some constants, which could be changing from line to line and always depend on the dimension N , and by ∂ ∂t (or equivalently ∂ t ) the partial derivative with respect to the variable t.If F ⊂ T N we denote with dist F (•) the distance from the set F .
Given a smooth closed (N − 1)-manifold Σ ⊂ T N we denote by ν Σ : Σ → S N the outer normal to Σ, by B Σ the second fundamental form of Σ, and by H Σ its mean curvature, that is the trace of B Σ .For every vector field X : Σ → R N we let X τ to be the tangential part of X, that is X τ (x) = X(x) − X(x) • ν Σ (x)ν Σ (x), and for every function the mean of f over Σ.
Let E ⊂ T N be a open set with smooth boundary and let X : T N → R N be a vector field of class C 2 .We consider the associated flow Φ : where I : T N → T N denotes the identity, and we say that E t = Φ(E, t) is the variation of E associated to Φ (or to X).If in addition it holds |E t | = |E| for every t ∈ (−1, 1), we say that E t is a volume-preserving variation of E.
We now recall some results on sets of finite perimeter, referring to [29] for the basic definitions and proofs.We say that a measurable set E ⊂ T N is a set of finite perimeter if Moreover, by De Giorgi's structure theorem, we have The first and second variation of the perimeter at E with respect to the flow Φ are defined as follows It is well known that, for any set of finite perimeter E, we have where div τ (X) is the tangential divergence of X on E and it is given by Finally, the second variation formula for perimeter on open sets of class C 2 (see for instance [3,Section 3]) is given by where D τ f (x) = (Df ) τ (x) denotes the tangential derivative of E. Since the expression above only depends on the normal component of the velocity field X, we also denote by δP (E)[ϕ] and δ 2 P (E)[ϕ], respectively, the first and the second variation of the perimeter at E, where ϕ = X • ν E .
Let E be a critical point of the perimeter.It should be noted that the translation invariance of the perimeter implies that the second variation becomes degenerate along flows of the form Φ(x, t) = x + tη, where η ∈ R N .Because of that, we denote by and by T (∂E) the subspace generated by the functions ν i : ∂E → R for i = 1, . . ., N .We then set T ⊥ (∂E) to be the orthogonal subspace of T (∂E) in the L 2 −sense, that is After defining all the spaces, we can finally give the notion of stability.
Definition 1.1.We say that a set E ⊂ T N of class C 2 is a strictly stable set if it is a critical set, that is δP (E)[ϕ] = 0 for all ϕ ∈ H1 (∂E), and the second variation of the perimeter is positive definite, in the sense that We now recall some technical results that will be useful in the following.We start by recalling the definition of normal deformation of a set and a result which ensures that any W 2,p -small normal deformation of a smooth set can be translated in a way so the projection on the subspace T ⊥ (E) becomes small.
is sufficiently small, we define the normal deformation of E induced by f the set E f having as boundary and We now recall the definition of inner and outer ball condition.
Definition 1.4.We say that a open set E ⊂ T N satisfies a uniform inner (respectively outer) ball condition with radius r if there exists r > 0 such that for every x ∈ ∂E there exists a ball B r (y) ⊂ E (resp.B r (y) ⊂ E c ) with x ∈ ∂B r (y).
Note that all sets E ⊂ T N of class C 1,1 satisfy a uniform inner and outer ball condition (see e.g.[8]).Arguing as in the proof of [3,Lemma 3.8], we can prove the following result. where Proof.Being the set E smooth, it satisfy the uniform inner and outer ball condition, hence there exists a positive radius r > 0 such that the signed distance sd E from the set E, defined by is a function of class C k+1 (from the regularity of ∂E) in the r-tubular neighborhood (∂E) r , that is (∂E) r := {x : dist ∂E (x) < r} (for further properties of the distance function see [21, section 14.6]).Moreover, since u has C k -norm bounded, by interpolation we get u C 1,1 (∂E) ≤ C.Then, there exists a radius ρ = ρ(C, E) such that ∂E u satisfies a uniform inner and outer ball condition of radius ρ.We can assume without loss of generality that ρ < r.
We now let η ≤ ρ/2 to be chosen later, take any |σ| < η and set F = E u + σ.Clearly, F still satisfies a uniform inner and outer ball condition of radius ρ.Then, for every y ∈ ∂F there exists x ∈ ∂E u such that y = x + σ, hence we have and in particular ∂F ⊂ (∂E) 2η ⊂ (∂E) r .We now define the map T u : ∂E → ∂E as where π E is the projection map on ∂E and y Moreover, using again (1.4) and the invertibility of the map x → x + u(x)ν E (x) + σ, we obtain Using the fact that T u is a diffeomorphism and (1.4), we can find a function v : ∂E → R such that F is the normal deformation of E induced by v, more precisely for every x ∈ ∂E ).Finally, using the above expression and the bounds in (1.5) and (1.6), we conclude that Let E, F ⊂ T N be measurable sets.We define a L 1 −distance between E, F modulo translations (also known as the Fraenkel asymmetry of the set E related to F ) as The following quantitative isoperimetric inequality has been proved in [3].As a consequence of this result, strictly stable sets are of class C ∞ (see [29]).
We now recall the quantitative version of Alexandrov's theorem proved in [9, Theorem 1.3], which can be also seen as a Lojasiewicz-Simon inequality with sharp exponents.It will be the main tool to prove the exponential stability of the geometric flows considered.We slightly rephrase the conclusion as it will be more useful in the following.
Theorem 1.7 ([9, Theorem 1.3]).Let E ⊂ T N be a strictly stable critical set.There exist δ * ∈ (0, 1/2) and C = C(E) > 0 with the following property: for any f ∈ C 1 (∂E)∩H 2 (∂E) such that f C 1 (∂E) ≤ δ * and satisfying setting Remark 1.8.Note that equation (1.8) in particular implies that, under the hypotheses of Theorem 1.7, for any λ ∈ R it holds We conclude this section by recalling the Poincaré and Gagliardo-Nieremberg inequalities on smooth hypersurfaces (see [5] for instance).Lemma 1.9.Let Σ ⊂ T N be a smooth closed hypersurface and f ∈ H 1 (Σ).There exists Theorem 1.10.Let Σ ⊂ T N be a smooth closed hypersurface.Let l, m, k ∈ N be such that 1 ≤ l < m, and let 1 ≤ r ≤ ∞.There exists a constant C, depending on these constants and on Σ, with the following property: for every u ∈ W l,p (∂Σ) we have 1 q for all θ ∈ [l/m, 1) for which p is nonnegative.
1.1.Short-time existence for the mean curvature flow.Given T > 0 and E 0 ⊂ T N an open smooth set, the volume-preserving mean curvature flow in [0, T ) starting from E 0 is the family of sets (E t ) 0≤t<T whose outer normal velocity is given by We remark that this equation should be intended as follows: there exist a smooth open set E ⊂ T N and a 1-parameter family of smooth diffeomorphism Φ t : ∂E → T N given by Φ Assuming that the flow starting from E 0 exists, following classical computation (see for instance [30]) one can deduce that the evolution equation satisfied by u is where ∆ E is the Laplace-Beltrami operator on ∂E, A is a smooth tensor such that A(•, 0, 0) = 0, and J is a smooth function.
In order to prove the stability of such flow, we need the following short-time existence result.
Theorem 1.11.Let ε > 0, let β ∈ (0, 1) and let E ⊂ T N be a smooth open set.There exists δ = δ(ε, E, β) > 0 with the following property: if E 0 is the normal deformation of E induced by u 0 ∈ C 1,1 (∂E), u 0 C 1,1 (∂E) ≤ δ, and |E 0 | = |E|, then there exists T > 0, which only depends on E, β and the bound on u 0 C 1,1 (∂E) , such that the volume preserving mean curvature flow E t starting from E 0 exists in [0, T ), the sets E t are normal deformation of E induced by u(•, t) ∈ C ∞ (∂E) for all t ∈ (0, T ), and Moreover, for every k ∈ N, there exist two constants We remark that the proof of this result is classical and can be derived from the Schauder estimates for quasi-linear parabolic equations, as u solves a lower-order, nonlinear perturbation of the heat equation.In the following section we will provide a brief outline of the proof for an analogous short-time existence result for the surface diffusion flow (see Theorem 1.21).Similar and simplified arguments would prove the previous result for the mean curvature flow, which is a second order flow.
For the sake of completeness, we provide here an alternative proof of Theorem 1.11 which follows from some results found in the literature.Even if these results are shown in the ambient space R N , the same arguments can be repeated in the flat torus.The first part of the Theorem is the short-time existence result of [15].Theorem 1.12 ([15, Main Theorem]).Let E ⊂ T N be a smooth open set and β ∈ (0, 1).There exists δ = δ(E, β) > 0 with the following property: if E 0 is the normal deformation of E induced by u 0 ∈ C 1,1 (∂E), u 0 C 1,1 (∂E) ≤ δ, and |E 0 | = |E|, then there exists T > 0, only depending on E, β and the bound on u 0 C 1,1 (∂E) , such that the volume-preserving mean curvature flow E t starting from E 0 exists in [0, T ), and the sets E t are normal deformations induced by u(•, t) ∈ C ∞ (∂E) for all t ∈ (0, T ).Furthermore, the mapping We remark that the local smooth semiflow property in particular implies that u(•) C 1,β depends continuously on u 0 C 1,β (see for instance [4, pg. 66]).In particular, for every ε > 0 there exists δ(E, ε, β) > 0 and In order to obtain the higher-order regularity inequalities, we apply some curvature estimates obtained recently in [25].
an open bounded set satisfying a uniform inner and outer ball condition with radius r.Then, there exists a time T = T (r, N ) > 0 such that the volume preserving mean curvature flow E t starting from E 0 exists in [0, T ) and it satisfies a uniform inner and outer ball condition of radius r/2.Moreover, it is smooth in (0, T ) and satisfies for every k ∈ N (1.14) sup where C k depends on k, |E 0 |, r.
Before proving the short time existence result, we remark a classical result concerning the uniform ball condition.
Remark 1.14.Let E be a smooth set satisfying a uniform ball condition of radius r E .Then every small C 1,1 -normal deformations of E satisfy a uniform ball condition of radius r ≈ r E .Indeed, it is easy to see that if E f is the normal deformation of E induced by f ∈ C 1,1 (∂E), then the Hausdorff distance between E and E f is bounded by f C 0 (∂E) .Furthermore, since ∇sd E f = ν E f and ν E f can be written as where the family {v i } i=1,...,N −1 denotes an orthonormal frame of the tangent space on ∂E (see [9, eq.(3.
3)]), by differentiating (1.15) one can see that [8,Theorem 2.6] and [8, Remark 2.7] one infers that the radius r of the uniform ball condition of the set E f depends continuously on f C 1,1 when it is small enough.In particular, for every ε > 0 there exists δ(r Proof of Theorem 1.11.By Theorem 1.12 there exist a time T > 0 and a family of evolving functions u(•, t), which are smooth in (0, T ) and satisfy the inequality (1.11).The second bound follows from classic elliptic regularity arguments that we now sketch.Fix t ∈ (0, T ), from the bound on sup t∈(0,T ) u C 1,β (∂E) and (up to rotations) for any given point x = (x , x N ) ∈ ∂E we can parametrize in a cylinder C = B r (x) × (−L, L) both ∂E and ∂E t as graphs of smooth functions g, g t .From Theorem 1.13 there exists a time T (depending on E, δ by Remark(1.14))such that the evolving sets E t satisfy a uniform inner and outer ball condition of radius r/2 for any t ∈ (0, T ).Let us set T = min{T , T }.From estimate (1.14) we get that Then, we conclude by means of Sobolev embeddings and by a covering argument.
1.2.Short-time existence for the surface diffusion flow.We now consider the evolution called surface diffusion flow, defined by (1.17) As for the mean curvature flow, the equation above means that there exist a smooth open set E ⊂ T N and a 1-parameter family of smooth diffeomorphism Φ t : E → T N such that Φ t (x) = x + u(x, t)ν E (x), Φ t (∂E) = ∂E t and Assuming that the diffeomorphisms above exist, arguing as in [30, pag. 21], one can deduce that the evolution equation satisfied by u is where P is a smooth function (assuming that u and ∇u are small), the function J can be written as and B1 , B2 , B3 and b4 are tensor-valued, respectively scalar-valued functions depending on (x, u, ∇u) and smooth if their arguments are small enough.Here ∇ denote the covariant derivative on ∂E.
On the other hand, linearizing the Laplace-Beltrami operator yields the evolution equation (compare with [18, Section 3.1]) where A is a smooth 4th-order tensor, vanishing when both h and ∇h vanish, and J is given by where B i , i = 1, . . . 5 and b 6 are smooth tensor-valued, respectively scalar-valued functions depending on (x, u, ∇u).
In this subsection we want to prove a short-time existence result for the surface diffusion flow, in particular we will obtain a priori estimates that will be used to prove the stability of the flow.We will follow the classical approach of linearization and fixed point to solve the nonlinear evolution problem, and then employ Shauder-type estimates to show higher order regularity of the flow.We will follow closely what has been done in [18], combining it with the results of [22].
To start we recall some classical results concerning the Cauchy problem for the biharmonic heat equation on a smooth Riemannian manifold Σ with metric g, which is the solution to the following problem once the functions f, u 0 are assigned.
We now collect some results, which are shown in [22], about the solution of (1.21).The following Schauder-type estimates on the solution of the homogeneous problem (1.24) can then be proved, see [22,Theorem 3.8].In particular, we modify slightly the formulation of the result, to fit our purposes.One can inspect the proof of [22, Theorem 3.8] (see pag. 7487,7489 in particular) to check the result.Theorem 1.16.Suppose u 0 ∈ C 1,1 (Σ) and fix T > 0. Then there exists C 1 (Σ, T ) > 0 such that (1.25) sup Furthermore, for any l, k ∈ N, we have for some constants C l,k > 0 depending on l, k, Σ and T .
The proof of the completeness of the spaces Y T and X T is standard, indeed one can prove directly that all Cauchy sequence converge to a function in the space and the candidate limit is obtained using a diagonal argument.
Remark 1.18.Since the norm 4 k=0 ∇ k u C 0 is equivalent to the norm u C 0 + ∇ 4 u C 0 for C 4 (Σ), we have that the norm • X T defined in (1.30) is equivalent to the following norm .
and there exists a constant C > 0 depending on Σ, T such that We now turn our attention to the evolution equation (1.19), and use the results above for the particular choice Σ = ∂E with the Riemaniann metric induced by the Euclidean one.We consider the map (1.34) f [u](x) := A(x, u, ∇u), ∇ 4 u + J(x, u, ∇u, ∇ 2 u, ∇ 3 u), where A, J are the operators defined in (1.19).We now provide the fundamentals estimates on f [u], which represents the nonlinear error generated linearizing (1.19).
Lemma 1.20.For any ε, m > 0 there exist T, δ > 0 depending on E, ε with the following properties.For every u 0 ∈ C 1,1 (Σ) and ψ ∈ X T satisfying ψ X T ≤ m it holds Proof.Let T < 1 to be chosen later and fic ε, m > 0. We prove only equation (1.36), giving a sketch of the proof for (1.37) and (1.35) as they are analogous; we also drop the dependence on the set E in the norms.For clarity of exposition, we prove the results for the simplified error term where B is a (constant) tensor of the same dimension of ∇ 3 u ⊗ ∇ 2 u with B < 1.The general case is explained in the appendix, but follows by analogous computations.We will also write A(x, t) and assume implicitly the dependence on u, ∇u.
Firstly, we prove (1.36).In what follows we use the short-hand notation u = Su 0 .From the definition of f (1.39)Then, we multiply by t 1 2 the first equation in (1.39) to get By (1.26), with the choice of l = 0, k = 0, 1, 2, we have that all the terms t (times a constant that depends on E which we can suppose equal to one for simplicity).We now fix δ > 0 sufficiently small, depending on ε and E, so that A C 0 is bounded by ε, which can be done since A is a smooth tensor and A(•, 0, 0) = 0. Finally, taking T small enough, depending on ε and E, we conclude sup Therefore, taking into account the full expression for the error term f [u] given by (1.34), one can show that sup t∈(0,T ) where the last constant comes from the term b 6 .Concerning the Hölder seminorm in space, we first remark that sup where ∂ 2 A and ∂ 3 A denote the derivative of A(x, y, z) with respect to the second and third components.Therefore, employing again the bounds in (1.25) and (1.26) we can bound where we took δ > 0 sufficiently small, depending on ε and E, such that which is possible since A is smooth and A(•, 0, 0) = 0. Thus, multiplying by t 4 the second equation in (1.39) we obtain Then, all the terms in (1.41) with the norms of u can be bounded employing (1.25) and (1.26), thus we can make the right-hand side above as small as needed taking T, δ small enough.Analogous calculations show a similar inequality for the complete error term f [u].
Finally, we show how to bound the Hölder seminorm in time appearing in f [u] Y T .We fix t ∈ (0, T ), h ∈ (0, T − t).To ease notation, we omit to write the evaluation at x in the following.We have by the very definition of f [u](t) that

Now by the triangular inequality we obtain
Therefore from formulas (1.42) and (1.43), we obtain Applying again (1.25), (1.26), and using the smallness of A C 0 , we conclude (1.36) by taking T, δ small enough.
Following the computations above one can easily prove that if u 0 ∈ C 1,1 (Σ) and The only difference is that, in addition to (1.25), (1.26) one can directly exploit the definition of • X T to obtain the required bounds.Also the proof for (1.37) is essentially the same, only much more tedious to write.We show the computations only for the term sup t∈(0,T ) t 1/2 • C 0 appearing in the norm of Y T and for the simplified error term (1.38).For u i := ψ i + Su 0 we can write Multiplying the inequality above by t 1 2 we have Again, by definition of • X T and by (1.25),(1.26)we conclude taking T, δ small enough.
We are now able to prove a short-time existence result for the surface diffusion evolution.Thanks to the previous lemmas, we provide also higher order regularity estimates depending on the C 1,1 −bound on the initial datum only.The proof follows closely the corresponding one in [22,18].Theorem 1.21.Let ε > 0 and let E ⊂ T N be a smooth open set.There exist δ = δ(ε, E), T = T (ε, E) > 0 with the following property: if E 0 is the normal deformations of E induced by u 0 ∈ C 1,1 (∂E), u 0 C 1,1 (∂E) ≤ δ, and |E 0 | = |E|, then the surface diffusion flow E t starting from E 0 exists in [0, T ), the sets E t are normal deformation of E induced by u(•, t) ∈ C ∞ (∂E) for all t ∈ (0, T ), and Moreover, for every k ∈ N \ {0}, there exist constants Proof.In this proof we denote by C > 0 a constant that depends on N and E and may change from line to line.Fix ε > 0.
Step 1: We show existence for (1.19) via a fixed point argument.Let T < 1, δ < 1 to be chosen later, and let u 1 ∈ C ∞ ((0, T ); C ∞ (∂E)) be the solution of The solution exists and it is given by (1.23), that is u 1 = 0 + Su 0 =: ψ 1 + Su 0 .Moreover (1.44) and (1.45) are satisfied by u 1 thanks to Theorem 1.16, for δ small enough depending on ε.Let now u 2 be the solution of where f [u] is defined as in (1.34).By (1.23) and (1.32), the unique solution is given by for m sufficiently large.We are then led to define an iterative scheme.We set u 1 , u 2 as above and for n ≥ 3 we let u n be the solution to and we split it as u n = Su 0 + V f [u n−1 ] =: ψ n + Su 0 .We will show that the sequence ψ n is converging in X T .To do so, assume that ψ j ∈ X T for j = 1, . . ., n − 1 with Then, by Theorem 1.19 and Lemma 1.20 we get ψ n ∈ X T and therefore ψ n is a Cauchy sequence and admits a limit point ψ satisfying We thus showed the existence of a fixed point u = ψ + Su 0 for the problem (1.46).Finally, by (1.25) and (1.48) it holds Step 2: By (1.49) we get straightforwardly that (1.45) holds for k = 0, 1, 2. In order to prove (1.45) for k ≥ 3, we consider x ∈ ∂E and we work under local coordinate, E is uniformly elliptic in U .In the following we identify B r and U ⊂ ∂E.We also set g t as the metric on ∂E t (see [30, pag. 20] for details).Observe that u restricted to B r × [ T 2 , T ) is of class C ∞ by the previous step.Recalling that u = ψ + Su 0 , we have that the function ψ satisfies (1.50) (1.50) shows that the function ∇ g ψ satisfies the equation where the error term F contains the derivative of ψ up to order four.To estimate we first observe that, by (1.26), it follows ).Secondly, we remark that the other terms of F can be bounded analogously, recalling that they contain derivatives of ψ up to order four and using (1.48), to show that gt is a uniformly parabolic operator, since the coefficients of ∆ 2 gt are close to the ones of ∆ 2 E depending on u(•, t) C 1,1 (∂E) as g ij Eu − g ij E = B(x, u, ∇u) and B is a smooth function with B(x, 0, 0) = 0, see again [30, pag. 20].Since ∇ g ψ solves (1.51), by the standard interior Schauder estimates and the bound (1.52) we get where we noted that ψ C 1 ((B r ×[ T 2 ,T ))) ≤ ψ X T and employed again (1.48).Finally, we conclude sup .
By induction, one can prove (1.45) for every k ∈ N.

Stability
2.1.Stability of the volume preserving mean curvature flow.In this subsection, we study the evolution by mean curvature (1.10) of normal deformations of a strictly stable set, as defined in Definition 1.1.Suppose that E is a strictly stable set and that E 0 = E u 0 is a smooth normal deformation of E. By Theorem 1.11, the volume preserving mean curvature flow starting from E 0 exists in a short time interval, and the evolving sets E t can be parametrized as normal deformations of the set E induced by functions u(•, t) satisfying where p = x+u(x, t)ν E (x) and HEt = ffl ∂Et H Et .The scalar product above (see for instance [9, eq.(3.4)]) can be written as , where κ j (x) and τ j (x) are, respectively, the principal curvatures and the principal directions of E at x.In particular, we remark that ν Et (p) • ν E (x) = 1 + O( u(•, t) H 1 ).We can then prove the first part of the main result, that is Theorem 0.1, concerning the long time behaviour of the volume preserving mean curvature flow.
Proof of (i) Theorem 0.1.Let ε, δ(ε) ∈ (0, 1) to be chosen later.In the following, if not otherwise stated, the constants depends on N, E and may change from line to line.Fix for instance β = 1/2 and suppose that δ is smaller than the constant given by Theorem 1.11.We also use the short-hand notation Step 1.We start by proving that P (E t ) − P (E) ≤ Ce −ct as long as the flow exists.
Let u 0 ∈ C 1,1 (∂E) with u 0 C 1,1 ≤ δ < 1.By Theorem 1.11 there exist a time T > 0, which depends on E and the bound on u 0 C 1,1 < 1, and a smooth flow E t starting from E 0 for t ∈ [0, T ).Moreover, E t = E u(•,t) and u(•, t) satisfies (1.11) and (1.12).Without loss of generality we can assume T < ∞.We notice that, considering ε, δ smaller, the value of T does not change.
We recall the following well-known identities, holding along the smooth flow Let δ * be the constant given by Theorem 1.
Furthermore, Lemma 1.5 (taking δ smaller if needed) implies that ũ(•, t) C 1 (∂E) ≤ δ * .We then apply Theorem 1.7 to the set E t + σ t to obtain for any λ ∈ R, where we recall H Et+σt (x) = H Et (x + ũ(x)ν E (x)).From the previous equation, first by the change of variable y = x + ũ(x, t)ν E (x) (estimating the Jacobian with the bounds on ũ and Lemma 1.5), and then by translation invariance, we arrive at We now claim that (2.4) which is a classical result but we provide a proof for the sake of completeness.Let us define, for every x ∈ ∂E, the function where τ 1 (x), . . ., τ N −1 (x) and κ 1 (x), . . ., κ N −1 (x) are, respectively, the principal directions and curvatures of ∂E at x. Then by [9, Lemma 3.1] we have where we have used that H ).Hence, we prove the claim in (2.4).We now define the Lyapunov functional E (t) = P (E t ) − P (E), which is non increasing by (2.1).Moreover, by translation invariance, from (2.3), (2.4) and for any λ ∈ R we have Since for any t ∈ (0, T ) equation (2.5) for the particular choice of λ = HEt implies Step 2. We now show that the flow exists for every t ≥ 0 and it converges exponentially fast to E up to translations.Up to taking δ smaller, we can use the quantitative isoperimetric inequality in Theorem 1.6 to find the existence of translations τ t such that Furthermore, since all the evolving sets {E t } t∈[T /2,T ) satisfy a uniform inner and outer ball condition by Remark 1.14, by classical convergence results (see e.g.[8, Theorem 3.2]) we have that E t + τ t is C 1 −close to E. In particular, there exist smooth (by the implicit map theorem) functions v(•, t) up to taking δ smaller.Therefore, recalling (2.6), we have For every t ∈ [T /2, T ), by combining the previous estimate with (1.12), (2.7) and interpolation inequalities, for any l ∈ N there exist k(l) ∈ N, θ(l) ∈ (0, 1) and Choosing E (0) = P (E 0 ) − P (E) small (hence choosing δ small) we can then apply again Theorem 1.11 with the new initial set E v(•,T /2) = E T /2 + τ T /2 to get existence of the translated flow up to the time 3T /2.We remark that, by uniqueness, the flow above is well defined since it coincides in [T /2, T ) with the flow E t translated by τ t and estimate (2.6) now holds for all t ∈ [T /2, 3T /2).By induction, choosing at every step the times t = nT /2, we can iterate the procedure above to prove that the flow exists for all times t ∈ [0, ∞).Moreover, for every t ∈ (0, ∞) there exists a translation τ t such that E t + τ t = E v(•,t) with v satisfying (2.8).In particular, we have that v → 0 exponentially in C k for any k, as t → ∞ and thus E t + τ t → E in C k for every k.This also implies (reasoning as in (2.3)) that H Et − HEt L 2 (∂E) → 0 exponentially fast.
Step 3. We conclude by showing the convergence of the whole flow to a translate of E.
Let us prove the convergence of the translations {τ t } t≥0 .By compactness we can find a sequence t n → ∞ such that τ tn → τ .Defining (2.9) where we recall that V t is the velocity of the flow in the normal direction (see (1.10)).Clearly, condition (2.10) implies that D(E t , E − τ ) admits a limit as t → +∞.By the previous step and since τ tn → τ , we deduce that Assume now that σ ∈ T N is the limit of τ sn along a subsequence s n → ∞ as n → +∞.By the previous step, which implies σ = τ by definition (2.9).This concludes the proof as the exponential convergence follows from Step 2.
2.2.Stability of the surface diffusion flow.We now focus on surface diffusion flow, which we defined in (1.17).As in the previous subsection, we consider E a strictly stable set and E 0 = E u 0 a smooth normal deformation of E. By Theorem 1.21, the surface diffusion flow starting from E 0 exists smooth in an interval [0, T ), moreover the evolving sets E t can be written as normal deformations of E induced by functions u(•, t) satisfying where p = x + u(x, t)ν E (x).Now, we aim to show the stability result (ii) of Theorem 0.1 for the surface diffusion flow.Due to the similarity of the arguments needed with those employed to prove item (i) of Theorem 0.1, we will only highlight the main differencies between the two.
Proof of (ii) Theorem 0.1.Firstly, Theorem 1.21 ensures the existence of a smooth flow E t for t ∈ (0, T ) of normal deformations of E induced by functions u(•, t) ∈ C ∞ (∂E) and satisfying (1.44) and (1.45).We recall the following identities, holding along the flow E t as long as it exists smooth, Since u(•, t) C 1,1 (∂E) ≤ c for every t ∈ (0, T ), the Poincaré constants C Et are uniformly bounded in the same time interval and the bound depends on E, u C 1,1 (∂E) (see e.g. the results in [12]).Thus, we obtain the estimate d dt P (E t ) ≤ −C H Et − HEt L 2 (∂Et) uniformly in (0, T ).We then conclude by following the same arguments of part (i).
3. Appendix: sketch of a general proof of the Lemma 1.20 In this appendix we complete the proof of Lemma 1.20 in the general case, i.e. considering the full nonlinear error term given by (1.20).
Since the estimates for the first term of f [u] have been presented in the proof of Lemma 1.20, we focus on bounding the terms of J(x, u, ∇u, ∇ 2 u, ∇ 3 u) with respect to the norm • C 0 .Considering the term B 1 , ∇ 3 u ⊗ ∇ 2 u , we have as long as u C 1 is small.Hence, with the same arguments presented for the functional B, ∇ 3 u ⊗ ∇ 2 u we obtain sup t∈(0,T ) by choosing T = T (ε) small enough.We analogously treat the other terms, so we have In the end we have that b 6 C 0 ≤ C = C(E).Therefore, taking T small we obtain We now focus on the Hölder seminorm in space.We present the calculations only for B 1 , ∇ 3 u ⊗ ∇ 2 u , being the other analogous.A straightforward computation shows (using the triangular inequality) that | B 1 (x + h, u(x + h), ∇u(x + h)), ∇ 3 u(x + h) ⊗ ∇ 2 u(x + h) − B 1 (x, u(x), ∇u(x)), ∇ 3 u(x) ⊗ ∇ 2 u(x) Therefore, as in the case J(x, u, ∇u, ∇ 2 u, ∇ 3 u) = B, ∇ 3 u × ∇ 2 u , using formula (1.25) and (1.26) we obtain the thesis.
Finally, we show how to bound the Hölder seminorm in time appearing in f [u] Y T .We fix t ∈ (0, T ), τ ∈ (0, T − t) and, for simplicity, we omit the dependence on x.For the first term, we have Then, for the second, third and fourth terms we get, respectively, Therefore, we can conclude with the same arguments used for B, ∇ 3 u ⊗ ∇ 2 u .
[12, constant which depends on |E 0 |, T, r.Then, by uniform geometric Calderon-Zygmund inequality (see[12, Section 3]or [3, Lemma 7.2]) we deduce that, for some ρ < r, in the ball B ρ (x ) the function g t is bounded in H 2 (B ρ (x )) by a constant, depending only on the L 2 -bound on H Et , the norm of the coefficients of the elliptic operator, which are in turn bounded by u 0 C 1,1 thanks to the previous step.Iterating this procedure, we bound the higher norms H k (B ρ (x )) of g t , for every k ∈ N.