Abstract
We consider the flat flow solution, obtained via a discrete minimizing movement scheme, to the volume preserving mean curvature flow starting from \(C^{1,1}\)-regular set. We prove the consistency principle, which states that (any) flat flow solution agrees with the classical solution as long as the latter exists. In particular the flat flow solution is unique and smooth up to the first singular time. We obtain the result by proving the full regularity for the discrete time approximation of the flat flow such that the regularity estimates are stable with respect to the time discretization. Our method can also be applied in the case of the mean curvature flow and thus it provides an alternative proof, not relying on comparison principle, for the consistency between the flat flow solution and the classical solution for \(C^{1,1}\)-regular initial sets.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
1.1 Statement of the Main Theorem
In this paper we consider the flat flow solution to the volume preserving mean curvature flow, which is a weak notion of solution obtained via discrete minimizing movement scheme. Our main goal is to prove the full regularity of the flat flow up to the first singular time when the initial set is \(C^{1,1}\)-regular. As a corollary we obtain the consistency principle between the flat flow and the classical solution.
Let us begin by recalling that a smooth family of sets \((E_t)_{t\in [0,T)}\subset \mathbb {R}^{n+1}\), for some \(T>0\), is a solution to the volume preserving mean curvature flow if it satisfies
where \(V_t\) denotes the normal velocity, \(H_{E_t}\) the mean curvature and 
the integral average of the mean curvature of the evolving boundary \(\partial E_t\). An important feature is that (1.1) can be seen as a \(L^2\)-gradient flow of the surface area. Since it also preserves the volume, it can be regarded as the evolutionary counterpart to the isoperimetric problem.
If the initial set \(E_0\) is regular enough, e.g. it satisfies interior and exterior ball conditions, the equation (1.1) has a unique smooth solution for a short interval of time [19]. The classical result by Huisken [28] states that for convex initial sets the classical solution exists for all times and converges exponentially fast to a sphere. Similarly, it follows from [19, 44] that if the initial set is close to a local minimum of the isoperimetric problem, the equation (1.1) does not develop singularities and convergences exponentially fast. However, for generic initial sets the equation (1.1) may develop singularities in finite time [40, 41]. In fact, unlike the standard mean curvature flow, (1.1) may develop singularities even in the plane and the boundary may also collapse such that the curvature of the evolving boundary stays uniformly bounded up to the singular time. It is therefore natural to find a proper notion of weak solution for (1.1) which is defined for all times even if the flow develops singularities. The crucial difference between (1.1) and the mean curvature flow is that the former is nonlocal and does not satisfy the comparison priciple. Therefore we cannot directly use the notion of viscosity solution to define the level-set solution via the methods introduced by Chen-Giga-Goto [15] and Evans-Spruck [20], although in [33] Kim-Kwon are able to find a viscosity solution for (1.1) for star-shaped sets. Instead, we may use the gradient flow structure to obtain a weak solution called flat flow via discrete minimizing movement scheme as first introduced by Almgren-Taylor-Wang [3] and Luckhaus-Stürzenhecker [36] for the mean curvature flow, and then implemented to the volume preserving setting (1.1) by Mugnai-Seis-Spadaro [43]. We give the precise definition in Section 3. The existence of the flat flow solution of (1.1) is proven in [43] and the recent results [16, 23, 31, 32, 42] indicate that it has the expected asymptotic behavior. Indeed, it is proven in [31] that in the plane any flat flow solution of (1.1), starting from any set of finite perimeter, converges exponentially fast to a union of equisize disks.
One of the main issues with the flat flow solution is that it has a priori very low regularity. The second issue is that it is not clear if the procedure provides a solution to the equation (1.1) in some weak sense. The first issue is related to the regularity and the second one is the problem of consistency, and it is rather clear that these are closely related to each other. Indeed, the flat flow is obtained as a limit of a discrete minimizing scheme, in the spirit of the Euler implicit method, where the time disretization is led to zero. If the flow remains smooth enough, as the time discretization goes to zero, then one can show that the limiting flat flow provides a solution to the equation (1.1). However, the only case when this seems to be known is the case when the initial set is convex. In this case the construction in [8], which however is slightly different than [43], provides a flow of sets which remains convex and thus gives a solution to (1.1). One may also define a distributional solution to (1.1) (see [43]) and in a recent work Laux [34] proves that this notion of solution, and in fact any gradient-flow calibration, agrees with the classical solution as long as the latter exists (see also [26]).
The issue with regularity and consistency is better understood in the case of the standard mean curvature flow. It is proven in [3] that the flat flow for the mean curvature equation agrees with the classical solution as long as the latter exists. If we are in a situation where the level-set solution is unique, i.e., it does not develop fattening, then due to the result by Chambolle [12] we know that the flat flow coincides with the level-set solution, see also [13, 14]. We may then use the result in [21] to conclude that the flat flow is a ’subsolution’ to the mean curvature flow in the sense of Brakke and has the partial regularity proven in [9]. Thus we have the consistency and partial regularity for the mean curvature flow when the flow does not develop fattening. In addition, due to the recent result by DePhilippis-Laux [17] together with the classical result in [36], we know that the flat flow is a distributional solution to the mean curvature flow equation when the initial set is mean convex.
As we mentioned above, here we study the regularity of the flat flow solution of (1.1) when the initial set is \(C^{1,1}\)-regular, which is the same as to say that the set satisfies interior and exterior ball conditions. Throughout the paper we will say that an open set \(E \subset \mathbb {R}^{n+1}\) satisfies uniform ball condition (which we refer as UBC) with radius \(r>0\) if it satisfies interior and exterior ball condition with radius \(r>0\). If we do not want to emphasize the radius r, we simply say that E satisfies UBC. Our main theorem reads as follows:
Theorem 1.1
Assume that \(E_0 \subset \mathbb {R}^{n+1}\) is an open and bounded set which satisfies UBC with radius \(r_0\). There is time \(T_0>0\), which depends on \(r_0\) and n, such that any flat flow solution \((E_t)_{t \ge 0}\) of (1.1) starting from \(E_0\) satisfies UBC with radius \(r_0/2\) for all \(t \le T_0\). This condition is open in the sense that if \((E_t)_{t \ge 0}\) satisfies UBC with radius r for all \(t \le T\), then there is \(\delta >0\) such that it satisfies UBC with radius r/2 for all \(t < T+\delta \).
Moreover, the flat flow \((E_t)_{t \ge 0}\) becomes instantaneously smooth and remains smooth as long as it satisfies UBC. To be more precise, if \((E_t)_{t \ge 0}\) satisfies UBC with radius r for all \(t \le T\), then for every \(k \in \mathbb {N}\) it holds that
where \(C_k\) depends on T, n, k, r and \(|E_0|\).
In fact, we obtain even stronger result since we prove UBC and the estimate (1.2) directly for the discrete approximative flat flow \((E_t^h)_{t \ge 0}\) such that the estimates hold for all \(h \le h_0\) for constants independent of h. However, we choose to state the regularity result only for the limiting flow since the precise statement, which can be found in Theorem 4.7 and Theorem 5.2, is rather technical. The first part of the theorem is related to the result by Swartz-Yip [47], where the authors prove curvature bounds for the Merriman-Bence-Osher thresholding algorithm for the mean curvature flow.
It is well-known that we have uniqueness among smooth solutions of (1.1). Therefore an important consequence of Theorem 1.1 is the consistency between the notion of flat flow solution and the classical solution of (1.1) when the initial set is \(C^{1,1}\)-regular.
Corollary 1.2
Assume that \(E_0 \subset \mathbb {R}^{n+1}\) is an open and bounded set which satisfies UBC. Let \(({\hat{E}}_t)_{t\in [0,T)}\subset \mathbb {R}^{n+1}\) be the classical solution of (1.1) starting from \(E_0\), where \(T>0\) is the maximal time of existence, and let \((E_t)_{t\ge 0}\subset \mathbb {R}^{n+1}\) be a flat flow solution of (1.1) starting from \(E_0\). Then
Let us next briefly comment on the regularity estimate (1.2). The first part of Theorem 1.1 (see Theorem 4.7 in Section 4) provides a bound on UBC for a short time \([0,T_0]\) and the proof of Theorem 4.7 also provides an estimate how the curvature grows in time for the approximative flat flow \((E_t^h)_{t \ge 0}\). However, without higher order regularity bounds we are not able to pass these growth-estimates to the limit as \(h \rightarrow 0\). Therefore the results of Section 4 only imply the consistency for a short time interval \([0,T_0]\) (see the discussion at the end of Sect. 5). Our main motivation to prove (1.2) is to pass the previously mentioned curvature estimates to the limit as \(h \rightarrow 0\) by Ascoli-Arzela theorem, and deduce that UBC is, in fact, an open condition and therefore the flat flow agrees with the classical solution over the whole maximal time of existence. Of course, in addition to that, (1.2) quantifies the smoothing effect of the equation in a sharp way.
1.2 An Overview of the Proof
The proof of Theorem 1.1 is divided in three sections and therefore we give here a short overview. We recall that in the minimizing movements scheme, for a fixed time discretization step \(h>0\), we obtain a sequence of sets \(E_{k}^h\) such that \(E_0^h = E_0\) is the initial set and \(E_{k+1}^h\) is defined inductively as a minimizer of the functional
where \(d_{E_{k}^h}\) denotes the signed distance function and \(m_0=|E_0|\). A flat flow is then defined as any cluster point of the discrete flow as \(h \rightarrow 0\). We first prove in Proposition 3.1 via energy comparison argument, that if \(E_{k}^h\) satisfies UBC with radius \(r_0\) then the subsequent set \(E_{k+1}^h\) satisfies the distance estimate
The above estimate is crucial as it implies that the speed of the discrete flow is sublinear. It also implies a bound for the mean curvature and the regularity of \(E_{k+1}^h\) by applying the Allard’s regularity theory [2]. The most crucial part of the proof of the main theorem is then to show that the subsequent set \( E_{k+1}^h\) also satisfies UBC with a quantified radius.
We solve this problem by adopting the two-point function method due to Huisken [27] to the discrete setting (see also the works by Andrews [4] and Brendle [10] for an overview of the topic). The idea is to double the variables and to study the maximum and minimum values of the function
for \(x\ne y \in \partial E_{k}^h\). The point is that the extremal values of \(S_{E_{k}^h}\) are related to the maximal UBC radius of the set \(E_{k}^h\) (see Lemma 4.1). We use the maximum principle to prove the following familiar inequality (see Lemma 4.6):
By iterating the above estimate, we obtain that the sets \(E_{k}^h\) satisfy UBC for all \(k \le T_0 h^{-1}\), where the constant \(T_0\) is related to the UBC of the initial set. This implies the first part of Theorem 1.1 (see Theorem 4.7). An important technical part in this argument is the discrete version of the formula for \(\frac{d}{dt} \nu _{E_t}\) which we derive in Lemma 4.4.
The formula in Lemma 4.4 is, in fact, so simple that we are able to differentiate it multiple times and obtain in Proposition 5.1 a discrete analog for the formula
where \(\Delta \) denotes the Laplace-Beltrami operator (see e.g. [38]). The lower order terms are due to the nonlinearity of the equation (1.1) and we need the notation and tools from differential geometry in order to control them. We stress that this is the only part in the paper where we need to introduce higher order covariant derivatives. After we have obtained the discrete version of the formula (1.3) and bounded the lower order error terms, we may adopt the argument from [22] to the discrete setting and obtain the full regularity of the flow. Finally, we point out that the argument can be adopted to the case of the mean curvature flow essentially without any modifications.
2 Notation and Preliminary Results
Throughout this paper, \(C_n \in \mathbb {R}_+\) stands for a generic dimensional constant which may change from line to line. We denote the open ball with radius r centered at x by \(B_r(x) \subset \mathbb {R}^{n+1}\) and by \(B_r\) if it is centered at the origin. We denote by \({\textbf {C}}(x,r,R) \subset \mathbb {R}^{n+1}\) the open cylinder
where \(B_r^n \subset \mathbb {R}^n\) denotes the n-dimensional ball and \(x = (x',x_{n+1}) \in \mathbb {R}^n \times \mathbb {R}\). For a given set \(E \subset \mathbb {R}^{n+1}\) and a radius \(r \in \mathbb {R}_+\) we set its r-enlargement \({\mathcal {N}}_r(E) = \{ x \in \mathbb {R}^{n+1}: \textrm{dist}(x,E) < r\}\). Note that we may alternatively write this as the Minkowski sum \(E + B_r\). The notation \(\nabla ^k F\) stands for k:th order differential of a vector field \(F: \mathbb {R}^{n+1} \rightarrow \mathbb {R}^m\). For a matrix \({\mathcal {A}} \in \mathbb {R}^k \otimes \mathbb {R}^k\) we denote by \(|{\mathcal {A}}|\) its Frobenius norm \(\sqrt{\text {Tr}({\mathcal {A}}^\text {T} {\mathcal {A}})}\) and by \(|\mathcal A|_{\textrm{op}}\) its operator norm \(\max \{|{\mathcal {A}} \, \xi |: \xi \in \mathbb {R}^k, |\xi |=1\}\).
If a set \(S \subset \mathbb {R}^{k}\) is Lebesgue-measurable, we denote its k-dimensional Lebesgue measure (or volume) by |S|. Given a non-empty set \(E \subset \mathbb {R}^{n+1}\) we denote the distance function by \(\text {dist}_E(x):= \inf _{y \in E}|x-y|\) and the signed distance function by \(d_E:\mathbb {R}^{n+1} \rightarrow \mathbb {R}\), which is defined as
Then clearly it holds that \(\text {dist}_{\partial E} = |d_E|\). If for a given point \(x \in \mathbb {R}^{n+1}\) there is a unique distance minimizer \(y_x\) on \(\partial E\) (that is \(|x-y_x|=\text {dist}_{\partial E}(x)\)), we denote \(y_x\) by \(\pi _{\partial E}(x)\) and call it the projection of x onto \(\partial E\). For a set of finite perimeter \(E \subset \mathbb {R}^{n+1}\) we denote its reduced boundary by \(\partial ^* E\). Then \(P(E;F)= \mathcal {H}^n(\partial ^* E \cap F)\) for every Borel set \(F \subset \mathbb {R}^{n+1}\) and \(P(E) = \mathcal {H}^n(\partial ^* E)\).
2.1 Regular Sets and Tangential Differentiation
We will mostly deal with regular and bounded sets \(E \subset \mathbb {R}^{n+1}\). As usual, a bounded set \(E \subset \mathbb {R}^{n+1}\) is said to be \(C^{k,\alpha }\)-regular, with \(k \ge 1\) and \(0\le \alpha \le 1\), if for every \(x \in \partial E\) we find a cylinder \({\textbf {C}}(x,r,R)\) and a function \(f \in C^{k,\alpha }(B^n_r(x'))\) with \(|f-x_{n+1}|<R\) such that, up to rotating the coordinates, we may write
In particular, \(\partial E\) is a compact and embedded \(C^{k,\alpha }\)-hypersurface. Again, if \(\alpha = 0\), we say that E is \(C^k\)-regular and if \(k=\infty \), we say that E is smooth. If r and R are independent of the choice of x and the \(C^{k,\alpha }\)-norm of g has a bound, also independent of x, then we say that E is uniformly \(C^{k,\alpha }\)-regular. We denote the outer unit normal by \(\nu _E\), or simply \(\nu \) if the meaning is clear from the context. Note that \(\nu _E \in C^{k-1,\alpha }(\partial E; \partial B_1)\). We always assume that the orientation of \(\partial E\) is induced by \(\nu _E\). We define the matrix field \(P_{\partial E}: \partial E \rightarrow \mathbb {R}^{n+1} \otimes \mathbb {R}^{n+1}\) by setting \(P_{\partial E} = I - \nu _E\otimes \nu _E\). For a given point \(x \in \partial E\) the map \(P_{\partial E}(x)\) is the orthogonal projection onto the geometric tangent plane \(G_x \partial E:= \langle \nu _E (x) \rangle ^\perp \).
For given a vector field \(F \in C^l(\mathbb {R}^{n+1};\mathbb {R}^m)\) with \(1 \le l \le k\) we define its tangential differential along \(\Sigma = \partial E\) as a matrix field \(\nabla _{\tau _E} F: \partial E \rightarrow \mathbb {R}^m \otimes \mathbb {R}^{n+1}\) by setting
When the meaning is clear from the context, we abbreviate E from the notation and write simply \(\nabla _\tau F\). In the case \(m=n+1\), the tangential divergence of F is defined as \({\text {div}}_\tau F = \text {Tr}(\nabla _\tau F)\) and the tangential Jacobian \(J_\tau F\) of F is defined on \(\partial E\) as
where \(\iota _\tau (x)\) at \(x \in \partial E\) is the inclusion \(G_x \partial E \hookrightarrow \mathbb {R}^{n+1}\). In the case \(m=1\), the notation \(\nabla _\tau F\) also stands for the tangential gradient \(P_{\partial E} \nabla F\). Note that \(\nabla _\tau F\) is \(C^{l-1}\)-regular and independent of how F is extended beyond \(\partial E\). On the other hand, every \( G \in C^l(\partial E,\mathbb {R}^m)\), with \(1 \le l \le k\), admits a \(C^l\)-extension \( F: \mathbb {R}^n \rightarrow \mathbb {R}^m\) so we may extend the concept of tangential differential to concern G simply by setting \(\nabla _\tau G = \nabla _\tau F\) and further define the other introduced concepts in a similar manner.
If E is \(C^k\)-regular for \(k \ge 2\), we may define its second fundamental form, with respect to the orientation \(\nu _E\), as a matrix field \(B_E: \partial E\rightarrow \mathbb {R}^{n+1}\otimes \mathbb {R}^{n+1}\) given by
where the (unit) principal directions \(\kappa _1 (x), \ldots , \kappa _n (x) \in \langle \nu _E(x) \rangle ^\perp \) and the principal curvatures \(\lambda _1 (x), \ldots , \lambda _n (x)\) at \(x \in \partial E\) are given by the orientation \(\nu _E\). The corresponding (scalar) mean curvature field \(H_E\) is then given pointwise as the sum of the principal curvatures, i.e., \(H_E = \text {Tr}(B_E)\). Note that we may simply write
Finally, we define the tangential Hessian for given \(u \in C^2(\partial E)\) as \(\nabla _\tau ^2 u = \nabla _\tau (\nabla _\tau u)\) and further the tangential Laplacian or the Laplace-Beltrami of u as
The tangential Laplacian \(\Delta _\tau F\) for \(F \in C^2(\partial E;\mathbb {R}^{n+1})\) is defined as \(\sum _i \Delta _\tau (F \cdot e_i) e_i\). We will need the following identities on \(\partial E\):
The importance of the mean curvature \(H_E\) lies in the surface divergence theorem which states that for every \(G \in C^1(\partial E;\mathbb {R}^{n+1})\) it holds that
The concept of mean curvature can be generalized to the setting of bounded sets of finite perimeter in the varifold sense. Indeed, for a set of finite perimeter \(E \subset \mathbb {R}^{n+1}\), we may define the tangential divergence \({\text {div}}_\tau F\) of \(F \in C^1(\mathbb {R}^{n+1};\mathbb {R}^{n+1})\) along \(\partial ^*E\) in the same way as in the regular case by replacing the outer unit normal field with the measure theoretic normal field \(\partial ^* E \rightarrow \partial B_1\) which we also denote by \(\nu _E\). Then, if E is a bounded set of finite perimeter and there is \(g \in L^1(\partial ^*E,\mathcal {H}^n|_{\partial ^* E})\) such that
for every \(F \in C^1(\mathbb {R}^{n+1};\mathbb {R}^{n+1})\), we say that g is a generalized mean curvature of E and denote it by \(H_E\). As mentioned, this is a concept from the context of varifold theory for which we refer to [46] as a standard introduction. Since \(\partial ^* E\) is \(\mathcal {H}^n\)-rectifiable set, one may treat the pair \((\partial ^* E, \mathcal {H}^n|_{\partial ^*E})\) as an rectifiable integral varifold of multiplicity one.
2.2 Riemannian Geometry
We need the notation related to Riemannian geometry and as an introduction to the topic we refer to [35]. Let us assume that \(E \subset \mathbb {R}^{n+1}\) is a smooth and bounded set and denote \(\Sigma = \partial E\). Since \(\Sigma \) is embedded in \(\mathbb {R}^{n+1}\) it has natural metric g induced by the Euclidean metric. Then \((\Sigma , g)\) is a Riemannian manifold and we denote the inner product on each tangent space \(X, Y \in T_x \Sigma \) by \(\langle X, Y \rangle \), which we may write in local coordinates as
We extend the inner product in a natural way for tensors. Note that \(x \cdot y\) denotes the inner product of two vectors in \(\mathbb {R}^{n+1}\). We denote smooth vector fields on \(\Sigma \) by \(\mathscr {T}(\Sigma )\) and by a slight abuse of notation we denote smooth k:th order tensor fields on \(\Sigma \) by \(\mathscr {T}^k(\Sigma )\). We write \(X^i\) for vectors and \(Z_i\) for covectors in local coordinates. We denote the Riemannian connection on \(\Sigma \) by \(\tilde{\nabla }\) and recall that for a function \(u \in C^\infty (\Sigma )\) the covariant derivative \(\tilde{\nabla }u \) is a 1-tensor field defined for \(X \in \mathscr {T}(\Sigma )\) as
i.e., the derivative of u in the direction of X. The covariant derivative of a smooth k-tensor field \(F \in \mathscr {T}^k( \Sigma )\), denoted by \(\tilde{\nabla }F\), is a \((k+1)\)-tensor field and for \( Y_1, \dots , Y_k, X \in \mathscr {T}( \Sigma )\) we have the recursive formula
where
Here \(\tilde{\nabla }_X Y\) is the covariant derivative of Y in the direction of X (see [35]) and since \(\tilde{\nabla }\) is the Riemannian connection it holds that \(\tilde{\nabla }_X Y = \tilde{\nabla }_Y X + [X,Y]\) for every \(X, Y \in \mathscr {T}( \Sigma )\). We denote the k:th order covariant derivative of a function u on \(\Sigma \) by \(\tilde{\nabla }^k u \in \mathscr {T}^k( \Sigma )\) and the Laplace-Beltrami operator by \(\Delta \). Note that for functions it holds that \(\Delta u = \Delta _\tau u\). The notation \(\tilde{\nabla }_{i_k} \cdots \tilde{\nabla }_{i_1} u\) means a coefficient of \(\tilde{\nabla }^k u\) in local coordinates. We may raise the index of \(\tilde{\nabla }_i u\) by using the inverse of the metric tensor \(g^{ij}\) as \(\tilde{\nabla }^i u = g^{ij}\tilde{\nabla }_j u\). We note that the tangential gradient of \(u: \Sigma \rightarrow \mathbb {R}\) is equivalent to its covariant derivative in the sense that for every vector field \(X \in \mathscr {T}(\Sigma )\) we find a unique vector field \(\tilde{X}: \Sigma \rightarrow \mathbb {R}^{n+1}\) which satisfies \(\tilde{X}\cdot \nu _E = 0\) and
Similarly it holds that \(\tilde{\nabla }^2 u(X,Y) = \nabla _\tau ^2 u \tilde{X} \cdot {\tilde{Y}}\). Finally we recall that the notation \(\nabla ^k\) always stands for the standard Euclidean k:th order differential for an ambient function.
We define the Riemann curvature tensor \(R \in \mathscr {T}^4(\Sigma )\) [35, 39] via interchange of covariant derivatives of a vector field \(Y^i\) and a covector field \(Z_i\) as
where we have used the Einstein summation convention. We may write the Riemann tensor in local coordinates by using the second fundamental form B, which in the Riemannian setting is understood to be 2-form, as
We will also need Simon’s identity, which reads as
Let us next fix our notation for the function spaces. We define the Sobolev space \(W^{l,p}(\Sigma )\) in a standard way for \(p \in [1,\infty ]\), see e.g. [6], denote the Hilbert space \(H^l(\Sigma ) = W^{l,2}(\Sigma )\) and define the associated norm for \(u \in W^{l,p}(\Sigma )\) as
and, for \(p = \infty \),
The above definition extends naturally for tensor fields. We adopt the convention that \(\Vert u\Vert _{H^0(\Sigma ) } = \Vert u\Vert _{L^2(\Sigma )}\) and denote \(\Vert u\Vert _{C^{m}(\Sigma )} = \Vert u\Vert _{W^{m,\infty }(\Sigma )}\). We remark that we may define the k:th order covariant derivative of a function \(u \in C^k(\Sigma )\) and the space \(W^{k,p}(\Sigma )\) for \(k \ge 2\) as above assuming only that \(\Sigma \) (i.e. the set E for which \(\Sigma = \partial E\)) is \(C^k\)-regular.
Finally we adopt the notation \(S \star T\) from [25, 38] to denote a tensor formed by contracting some indexes of tensors S and T using the coefficients of the metric tensor \(g_{ij}\). This notation is useful as it implies
where the constant C depends on the ’structure’ of \(S \star T\).
2.3 Functional and Geometric Inequalities
We will need standard interpolation inequalities on smooth hypersurfaces. Since we will apply them on the moving boundary given by the flow, we need to control the constants in the inequalities. We begin with a simple interpolation on Hölder norms.
Lemma 2.1
Let \(\Omega \subset \mathbb {R}^{k}\) be an open set and let \(u \in C^1(\Omega )\), then for every \(\alpha \in (0,1)\)
Proof
The inequality follows from
\(\square \)
We continue to introduce functional and geometric inequalities that we need in order to prove the higher order regularity estimates stated at the end of Theorem 1.1. As we already mentioned we do not need any deep results from differential geometry in order to prove the estimate for UBC stated in the beginning of Theorem 1.1. It is only when we deal with higher order derivatives, i.e., higher than two, we need the notation of covariant derivatives. Recall that we always assume that \(\Sigma = \partial E\) for a bounded set \(E \subset \mathbb {R}^{n+1}\).
Let us first recall the interpolation inequality with Sobolev-norms on embedded surfaces. We use the result from [38, Proposition 6.5] which states that under curvature bound the standard interpolation inequality holds for a uniform constant.
Proposition 2.2
Assume \(\Vert B_{\Sigma }\Vert _{L^{\infty }}, \mathcal {H}^n(\Sigma )\le C_0\) and \(\Sigma \) is \(C^{m}\)-regular for \(m \ge 2\). Then for integers \(0\le k \le l \le m\) and numbers \(p,q,r \in [1,\infty )\), there is \(\theta \in [k/l,1]\) such that for every \(C^l\)-regular covariant tensor field T on \(\Sigma \) it holds
for a constant \(C=C(k,l,n,p,q,r,\theta ,C_0) \in \mathbb {R}_+\) provided that the following compatibility condition is satisfied
We denote an index vector by \(\alpha \in \mathbb {N}^k\), i.e., \(\alpha = (\alpha _1, \dots , \alpha _k)\) where \(\alpha _i \in \mathbb {N}\), and define its norm by
The following inequality is well-known but we prove it for the reader’s convenience:
Proposition 2.3
Assume \(\Vert B_{\Sigma }\Vert _{L^{\infty }}, \mathcal {H}^n(\Sigma ) \le C\) and \(\Sigma \) is \(C^{m}\)-regular for \(m \ge 2\). Assume \(u_1, \dots , u_l\) are \(C^m\)-regular functions such that \(\Vert u_i\Vert _{L^\infty } \le C\). Then for an index vector \(\alpha \in \mathbb {N}^l\) with \(|\alpha | \le k \le m\) and \(p \in (1,\infty )\) it holds that
Proof
Without loss of generality we may assume that \(|\alpha | = k\). We first use Hölder’s inequality to get that
By the interpolation inequality in Proposition 2.2 and by \(\Vert u_i\Vert _{L^\infty } \le C\) it holds that
Hence we have
Since \(\alpha _1 + \dots + \alpha _l = |\alpha | = k\), the claim follows, from Young’s inequality. \(\quad \square \)
If \(u: \mathbb {R}^{n+1} \rightarrow \mathbb {R}\) is a regular function then its restriction on \(\Sigma \) is also regular. In the next lemma we bound the covariant derivatives of u on \(\Sigma \) with the Euclidean ones. The statement of the lemma is not optimal but it is sharp enough for our purpose. In the proof we will repeatedly use the fact that the k:th order derivative of the composition \(f \circ h\) and the product \(f \cdot g\) of functions \(f,g: \mathbb {R}^m \rightarrow \mathbb {R}^k\) and \(h: \mathbb {R}^n \rightarrow \mathbb {R}^m\) can be written as
Lemma 2.4
Assume \(\Sigma \) is \(C^{k+2}\)-regular and \(u \in C^{k+1}(\mathbb {R}^{n+1})\). Then it holds for all \(x \in \Sigma \) that
Recall that \(\tilde{\nabla }^k \) denotes the k:th order covariant derivative on \(\Sigma \) while \(\nabla ^k \) is the k:th order Euclidean derivative.
Proof
The proof follows from basic theory of differential geometry and we merely sketch it. Let us fix \(x \in \Sigma \) and choose the coordinates such that \(x = 0\) and \(\nu _E(0) = e_{n+1}\). Since \(\Sigma \) is \(C^{k+2}\)-regular hypersurface we may write it locally as a graph of \(f \in C^{k+2}(\mathbb {R}^n)\), i.e., \(\Sigma \cap B_r(0) \subset \{ (x,f(x)): x \in \mathbb {R}^n\}\). Note that since \(\nu _E(0) = e_{n+1}\) then \(\nabla _{\mathbb {R}^n} f(0)= 0 \).
We consider the graph coordinates \(\Phi ^{-1}: B_r^n \rightarrow \Phi ^{-1}(B_r^n) \subset \Sigma \), \(\Phi ^{-1}(x) = (x,f(x))\). We denote the points on \(\mathbb {R}^n\) by x, the points on \(\Sigma \) by p, \(\Phi (p) = \big (x^1(p), \dots , x^n(p)\big )\) and \(U = \Phi ^{-1}(B_r^n) \). Then the chart \(\big (U,(x^i) \big )\) determines coordinate vector fields which we denote by \(\frac{\partial }{\partial x^i}\Big |_{p}\) and recall that they act on smooth functions \(v: U \rightarrow \mathbb {R}\) at \(p = \Phi (x)\) as
where \(\partial _i\) denotes the standard partial derivative in \(\mathbb {R}^n\). It holds for the metric tensor and for the Christoffel symbol \(\Gamma _{jk}^i \) (see [35]) for \(x \in B_r^n\) that
Moreover by the recursive formula (2.8) we may write the \((k+1)\):th order covariant derivative of u iteratively (see [35, Lemma 4.8]) as
Recall that \(\tilde{\nabla }u \left( \frac{\partial }{\partial x^{i}}\right) (p) = \frac{\partial }{\partial x^i}\Big |_{p} u \).
Using (2.12) we have
We use (2.13) and (2.12), and obtain after long but straightforward calculation that
Note that \(\nu _{E} \circ \Phi ^{-1} = \frac{(-\nabla _{\mathbb {R}^n} f,1)}{\sqrt{1 + |\nabla _{\mathbb {R}^n}f|^2}}\). We thus obtain by (2.12) that
and the claim follows. \(\quad \square \)
Next we turn our focus on geometric inequalities on compact hypersufaces. Recall that by classical results e.g. from [6] it holds that \(\Vert u\Vert _{H^2(\Sigma )}\le C(\Vert \Delta u\Vert _{L^2(\Sigma )} + \Vert u\Vert _{L^2(\Sigma )})\) and e.g. in [22] it is proven that \(\Vert u\Vert _{H^{2k}(\Sigma )}\le C(\Vert \Delta u\Vert _{H^k(\Sigma )} + \Vert u\Vert _{L^2(\Sigma )})\). We need these results with a quantitative control on the constant.
Lemma 2.5
Assume \(\Sigma \) is \(C^{2k+2}\)-regular and \(\Vert B_{\Sigma }\Vert _{L^{\infty }}, \mathcal {H}^n(\Sigma )\le C\). Then for all \(u \in C^{2k+1}(\Sigma )\) it holds
and
Proof
We only prove the first inequality since the second follows from the same argument. The proof is similar to [30, Proposition 2.11] but we sketch it for the reader’s convenience. Denote \(l = 2k\). We begin by noticing that we may interchange the derivatives of the \((l+1)\):th order covariant derivative of u by using (2.9), (2.10), (2.13) and the curvature bound \(\Vert B_{\Sigma }\Vert _{L^{\infty }}\le C\) (see also [38, Proof of Lemma 7.3])
We leave the details for the reader. This holds pointwise on \(\Sigma \) and we use it without further mentioning. Let us denote \(F = \tilde{\nabla }^{2k-2} u\) and denote its components simply by \(F_{\beta }\), where \(\beta = (i_1, \dots , i_{2k-2})\). Then it holds by divergence theorem, by interchanging the derivatives and by Proposition 2.3
By interchanging the derivatives and arguing as above we obtain
By repeating the argument by replacing u with \(\Delta ^j u\), for \(j = 1, \dots , k-1\), we deduce that
The claim follows from interpolation inequality (Proposition 2.2) as for \(\theta \in (0,1)\) it holds that
where the last inequality follows from Young’s inequality. \(\quad \square \)
Lemma 2.5 together with Simon’s identity (2.11), imply
Proposition 2.6
Assume that \(\Sigma \) is \(C^{2k+3}\)-regular and that \(\Vert B_{\Sigma }\Vert _{L^{\infty }}, \mathcal {H}^n(\Sigma )\le C\). Then it holds that
and
2.4 Uniform Ball Condition and Signed Distance Function
In this subsection, we recall some properties related to sets which satisfy UBC as well as properties of signed distance function defined in (2.1). Most of them can be found e.g. in [5, 7] while others are more difficult to find. We recall that a set \(E \subset \mathbb {R}^{n+1}\) satisfies UBC with given a radius \(r \in \mathbb {R}_+\), if it simultaneously satisfies the exterior and interior ball condition with radius r at every boundary point. That is, for every \(x \in \partial E\) there are balls \(B_r(x_+)\) and \(B_r(x_-)\) such that
It is well known, for the experts at least, that UBC for a set implies its boundary being a uniformly \(C^{1,1}\)-regular hypersurface. We need this property in a quantitative form which states that if \(E \subset \mathbb {R}^{n+1}\) satisfies UBC with radius r, then it can be written locally in a cylinder of width r/2 as a graph of a \(C^{1,1}\)-function. Since this result is not easy to find in the literature, we state it and provide a proof here.
Proposition 2.7
Assume \(E \subset \mathbb {R}^{n+1}\) satisfies UBC with radius \(r>0\). Then for every point \(x \in \partial E\) we may, by rotating the coordinates if necessary, write the interior of the set locally as a subgraph of a function \(g: B_{r/2}^n(x') \rightarrow \mathbb {R}\), i.e.,
The function g is \(C^{1,1}\)-regular and it holds for all \(y' \in B_{r/2}^n(x')\) and \(s \in (0,r/2]\)
Moreover, the outer unit normal \(\nu _E\) on \(\partial E\) is 1/r-Lipschitz continuous in Euclidean metric.
Remark 2.8
We remark, that the converse of Proposition 2.7 also holds true. That is, if \(E \subset \mathbb {R}^{n+1}\) is a set such that for every \(x \in \partial E\), we may write its boundary locally, by rotating the coordinates, as \(\partial E \cap \textbf{C}(x,r,2r) \subset \{(y',g(y')): y' \in B_r^n(x') \}\) with \(\Vert g\Vert _{C^{1,1}(B_r^n(x'))} \le C/r\), then E satisfies UBC with radius \(c \, r\), for a constant \(c>0\) which depends on n and C. This is fairly straightforward to show and we leave it to the reader.
Proof of Proposition 2.7
We remark that UBC with r implies for every \(x \in \partial E\) an existence of a unique unit vector \(\nu _E(x)\) such that \(B_r(x-r\nu _E(x)) \subset E\) and \(B_r(x+r\nu _E(x)) \subset \mathbb {R}^{n+1} {\setminus } E\). Therefore, we have a vector field \(\nu _E: \partial E \rightarrow \partial B_1\) which later turns out to be the outer unit normal field of E. We first show that \(\nu _E\) is 1/r-Lipschitz continuous with respect to Euclidean distance. To this end, fix \(x,y \in \partial E\). By the previous observation \(B_r(x + r\nu _E(x)) \subset \mathbb {R}^{n+1} {\setminus } E\) and \(B_r(y-r\nu _E(x)) \subset E\) so the balls are disjoint. Similarly, the balls \(B_r(x - r\nu _E(x))\) and \(B_r(y + r\nu _E(y))\) are disjoint. Hence the distances between the corresponding centerpoints are at least 2r and we obtain the inequalities
By summing the above inequalities gives us \( 8r^2 \le 2|x-y|^2 + 4r^2\left( 1+ \nu _E(x) \cdot \nu _E(y)\right) \) and, again, by subtracting and dividing terms we further obtain
In particular, \(\nu _E\) is 1/r-Lipschitz.
For given a point \(x \in \partial E\), we show the existence of g as claimed. Without loss of generality we may assume \(x = 0\) and \(\nu _E(0)= e_{n+1}\). Then it holds \(B_r(-re_{n+1}) \subset E\) and \(B_r(re_{n+1}) \subset \mathbb {R}^{n+1} \setminus E\). Thus, for every \(y' \in B_{r/2}^n\) there is a number \(t_{y'}\) such that \((y',t_{y' }) \in \partial E\) and
In particular, \(|t_{y'}|<|y'|\). Combining (2.15) and (2.16) yields
Let us show that such a number \(t_{y'}\) is unique.
We suppose by contradiction there is \(s_{y'} \in (-r,r) {\setminus } \{t_{y'}\}\) such that \((y',s_{y'}) \in \partial E\). Without loss of generality, we may assume \(s_{y'} > t_{y'}\). Since \(B_r\big ( (y',t_{y'}) + r \nu _E(y',t_{y'}) \big )\subset \mathbb {R}^{n+1} {\setminus } E\) and \((y',s_{y'}) \in \partial E\), then the point \((y',s_{y'}) \) is not in the ball \(B_r\big ( (y',t_{y'}) + r \nu _E(y',t_{y'}) \big )\). Hence, we obtain
We first subtract \(r^2\), then divide by \(s_{y'} -t_{y'}\) and finally use the estimates (2.16), (2.17) as well as \(|y'|<r/2\) to deduce
This implies together with \(s_{y'}<r\) that \((y',s_{y'}) \in B_r(re_{n+1}) \subset \mathbb {R}^{n+1} {\setminus } E\) which, in turn, contradicts \((y',s_{y'}) \in \partial E\) and, hence, \(t_{y'}\) is a unique value in \((-r,r)\) satisfying \((y',t_{y'}) \in \partial E\).
Thus, the function \(g: B_{r/2}^n \rightarrow \mathbb {R}\), given by the relation \(g(y')=t_{y'}\), satisfies
Again, (2.16) gives us the bound on \(|g(y')|\) as claimed. The condition (2.17) implies that for every \(y' \in B_{r/2}^n\) there are open sets \(y' \in V \subset B^n_{r/2}\), \((y',g(y')) \in U \subset {\textbf {C}}(0,r/2, r/2)\) and functions \(\psi _+,\psi _- \in C^\infty (V)\) such that \(\partial B_r((y',g(y' )) \pm r \nu _E(y',g(y')) \cap U\) are the graphs of \(\psi _\pm \) respectively. Then \(\psi _- \le g \le \psi _+\) in V and \(\psi _-(w)=g(w)=\psi _+(w)\) implying the differentiability of g at \(y'\) with \(\nabla g (y') = \nabla \psi _\pm (y')\). Moreover, we deduce that \(\nu _E(y',g(y'))\) is the outer unit normal of \(\{ (z',z_{n+1}) \in V \times \mathbb {R}: z_{n+1} > \psi _+(z')\}\) at \((y',g(y'))\) and thus
Since now g and \(\nu _E\) are continuous, (2.19) implies that \(\nabla g\) is continuous too. Thus, E is \(C^1\)-regular and \(\nu _E\) is the actual outer unit normal of E. We combine (2.17) and (2.19) to observe
To conclude the Lipschitz estimate, if \(y'_1,y'_2 \in B_s^n\) for given \(s \in (0,r/2]\), then the uniform ball condition implies that \((y'_1,g(y'_1)) \notin B_r((y'_2,g(y_2'))\pm r\nu _E(y'_2,g(y'_2))\) and \((y'_2,g(y'_2)) \notin B_r((y'_1,g(y'_1))\pm r\nu _E(y'_1,g(y'_1))\). Hence, using (2.19), we obtain the estimates
By summing these inequalities and simplifying, we have
Thus, by recalling, (2.20) we further estimate that
The desired estimate then follows from (2.21) via a standard mollification argument. \(\quad \square \)
We recall that a signed distance function \(d_E\) of a non-empty set \(E \subset \mathbb {R}^{n+1}\) is always 1-Lipschitz and it is differentiable at \(x \in \mathbb {R}^n \setminus \partial E\) exactly when the projection \(\pi _{\partial E} (x)\) exists on \(\partial E\). Again, UBC for E means the differentiability of \(d_E\) in a tubular neighborhood. Indeed, one may show that for a non-empty open set \(E \subset \mathbb {R}^{n+1}\) and \(r \in \mathbb {R}_+\) the conditions
-
(i)
\(d_E\) is differentiable in \({\mathcal {N}}_r(\partial E)\) and
-
(ii)
E satisfies UBC with radius r
are equivalent. In such a case, the projection \(\pi _{\partial E}\) onto \(\partial E\) is defined in \({\mathcal {N}}_r(\partial E)\) as a continuous map and the following fundamental identities hold in \({\mathcal {N}}_r(\partial E)\):
In particular, \(d_E \in C^1({\mathcal {N}}_r(\partial E))\). Further, it is fairly simple to conclude that for every \(t \in (-r,r)\) the sublevel set \(E_t = \{ x \in \mathbb {R}^{n+1}: d_E(x) <t\}\) has the level set \(\{x \in \mathbb {R}^{n+1}: d_E(x)=t\}\) as the boundary and satisfies UBC with radius \(r-|t|\). Moreover, it holds that
We may then improve the regularity by showing \(\nabla d_E\) and \(\pi _{\partial E}\) are locally Lipschitz continuous in \({\mathcal {N}}_r(\partial E)\) and obtain quantitative estimates for the Lipschitz constants in smaller tubes.
Lemma 2.9
Assume \(E \subset \mathbb {R}^{n+1}\) satisfies UBC with radius \(r>0\). Then for every \(0<\rho <r\) and \(x,y \in \overline{{\mathcal {N}}_\rho (\partial E)}\) it holds that
Proof
It is enough to prove the first estimate, since the second estimate follows from the first via Proposition 2.7 and the second identity of (2.22). We first show that the estimates hold locally, i.e., for every \(x \in {\mathcal {N}}_r(\partial E)\),
To this end, we show that, for every \(x \in \partial E\) and \(y \in B_{r/4}(x)\), it holds that
We may assume that \(x=0\), \(\nu _E(0)=e_{n+1}\) and \(y \notin E\). Let \(g: B^n_{r/2} \rightarrow \mathbb {R}\) be as in Proposition 2.7. Since \(|y|<r/4\), then \(y \in \textbf{C}(r/2,r/2,0)\) implying \(|d_E(y)| \le |y_{n+1}-g_n(y')|\) and, hence, we make a technical observation
Thus, using Proposition 2.7, (2.22), (2.26) and Young’s inequality, we estimate that
and (2.25) follows. Suppose next \(y_1,y_2 \in B_\rho (x)\) for given \(x \in \partial E\) and \(0<\rho <r/9\). The sublevel set \(E_t\), for \(t=d_E(y_2)\), satisfies UBC with radius \(r-\rho \) and \(y_2 \in \partial E_t\). Since \(|y_1 - y_2| < 2 \rho \le (r-\rho )/4\), then by applying (2.25) for \(\partial E_t\) we have
On the other hand, first recalling the second identity in (2.23) and then applying Proposition 2.7 gives us
so, by combining the estimate above with (2.27) yields \(\textrm{Lip}(x, \pi _{\partial E}) = 1\). Hence, we deduce that
for every \(t \in (-r,r)\) and \(x \in \partial E_t\). By using (2.23) and Proposition 2.7 similarly as previous, we infer (2.24) from (2.28).
Finally, for the first estimate of the claim, we may assume \(x,y \in {\mathcal {N}}_\rho (\partial E)\). Let \(J_{yx}:=\{tx+(1-t)y: t \in [0,1]\}\) be the line segment between them. If \(J_{yx} \subset \mathcal N_\rho (\partial E)\), then the first estimate of the claim follows from (2.24). Otherwise, there are \(0<t_1\le t_2 < 1\) such that \(tx+(1-t)y \in {\mathcal {N}}_\rho (\partial E)\) for every \(t \in [0,t_1) \cup (t_2,1]\) and \(z_i = t_i x +(1-t_i)y \in \partial \mathcal N_{\rho }(\partial E)\) for \(i=1,2\). Since \(d_E(z_1)=\rho =d_E(z_2)\), then Proposition 2.7 and (2.22) imply
On the other hand, due to (2.24) we have
and we conclude the proof. \(\quad \square \)
If E is \(C^{k,\alpha }\)-regular, with \(k \ge 2\) and \(0 \le \alpha \le 1\), then \(d_E \in C^{k,\alpha }({\mathcal {N}}_r(\partial E))\) and \(\pi _{\partial E} \in C^{k-1,\alpha }({\mathcal {N}}_r(\partial E);\mathbb {R}^{n+1})\). In particular, (2.22) holds everywhere in \({\mathcal {N}}_r(\partial E)\). Then it holds
In particular, we deduce from Lemma 2.9 and (2.29) that
Differentiating \(\nabla d_E \cdot \nabla d_E = 1\) yields \(\nabla ^2 d_E \nabla d_E = 0\) in \({\mathcal {N}}_r(\partial E)\). Again, by differentiating the first identity in (2.22) we obtain
The second identity in (2.22) says that \(\nabla d_E = \nabla d_E \circ \pi _{\partial E}\) in \({\mathcal {N}}_r(\partial E)\). Thus, by differentiating this and by using the properties of the distance function mentioned before we have
We write this as
It follows from (2.30) that the matrix field \( I + d_{E} ( B_E \circ \pi _{\partial E})\) is invertible in \(\mathcal {N}_r(\partial E)\). Therefore, we have
By combining (2.22), (2.31), (2.29) and (2.33), we may decompose \(\nabla \pi _{\partial E}\) as
By using a fairly standard calibration argument (see e.g. [1, Lemma 4.1]) we conclude that UBC implies so called \(\Lambda \)-minimizer condition.
Lemma 2.10
Assume that \(E \subset \mathbb {R}^{n+1}\) is an open and bounded set which satisfies UBC with radius \(r>0\). Then for every set of finite perimeter F it holds that
In particular, \(P(E) \le \frac{n+1}{r} |E|\).
Proof
The argument is a quantitative version of [1, Lemma 4.1]. We will prove that for every set of finite perimeter F it holds that
Then the two inequalities in the statement follow by using (2.35) with \(E \cup F\) and \(E \cap F\) in place of F and using the fact [37, Lemma 12.22] that
The third inequality follows by using (2.35) with \(F = \emptyset \).
By a standard approximation argument for the sets of finite perimeter [37, Thm 13.8 ] we may assume that F is smooth. In turn, we may approximate also E by a sequence of smooth sets \(E_k\) in the \(C^1\)-sense such that \( E_k\) satisfies UBC with radius \(r_k\) such that \(r_k \rightarrow r\). Therefore, by simplicity we assume that also E is smooth.
For each \(k \in \mathbb {N}\) we construct a vector-field \(X_k \in C_0^{0,1}(\mathbb {R}^{n+1}; \mathbb {R}^{n+1})\) such that
-
(i)
\(X_k = \nu _E\) on \(\partial E\),
-
(ii)
\(|X_k| \le 1\) in \(\mathbb {R}^{n+1}\) and
-
(iii)
\(\Vert {\text {div}}X_k\Vert _{L^\infty (\mathbb {R}^{n+1})} \le (n+1 + k^{-1})/r\).
To this aim, we first define \(\eta _k: \mathbb {R}\rightarrow \mathbb {R}\) by setting \(\eta _k(t) = \max \left\{ 0, 1 - (1+1/k)|t|/r\right\} \) and then set \(X_k = (\eta _k \circ d_E) \nabla d_E\). Clearly \(X_k\) is a Lipschitz continuous vector field supported in \(\overline{\mathcal N_{r/(1+k^{-1})}(\partial E)}\) and satisfies the properties (i) and (ii). We further compute that
Hence, it follows from Lemma 2.9, \(\nabla ^2 d_E \nabla d_E = 0\) in \({\mathcal {N}}_r(\partial E)\) as well as the definition of \(\eta _k\) that \(|{\text {div}}X_k| \le (n+1+k^{-1})/r\) in \(\mathcal N_{r/(1+k^{-1})}(\partial E) {\setminus } \partial E\). Since the sets \(\overline{{\mathcal {N}}_{r/(1+k^{-1})}(\partial E)}\) and \(\mathcal N_{r/(1+k^{-1})}(\partial E) {\setminus } \partial E\) agree in the \(L^1\)-sense we infer that \(X_k\) satisfies (iii). By using the properties (i) – (iii) for \(X_k\) as well as the divergence theorem we estimate
and thus, by letting \(k \rightarrow \infty \), the above yields (2.35). \(\quad \square \)
Suppose that \(E'\) is a connected component of a set E which satisfies UBC with r. If \(|E'|<\infty \), then \(E'\) is bounded and we may control its diameter in terms of r and \(|E'|\). Indeed, by the above approximation we may assume that E is smooth. Then by (2.29) we have \(|H_{E}| \le n /r\) on \(\partial E\). Thus, combining Lemma 2.10 and Topping’s generalization [48] of Simon’s diameter control [45] gives us the estimate
for a dimensional constant \(C_n \in \mathbb {R}_+\). Finally, we need the following interpolation result:
Lemma 2.11
Assume \(E \subset \mathbb {R}^{n+1}\) is an open and bounded set which satisfies UBC with radius \(r >0\). If U is an open set containing \(\partial E\) and \(u \in C^2(U)\), then
Proof
By the above approximation argument we may assume that E is smooth.
We first observe that for a bounded function \(f \in C^2(\mathbb {R})\) it holds
Indeed, let us fix a \(t \in \mathbb {R}_+\). We may assume that \(f'(t)> 0\), since otherwise we consider the function \(-f\) instead of f. Let I be a maximal open interval containing t such that \(f'>0\) in I so f is strictly increasing there. Then there is a decreasing sequence \(({\tilde{t}}_i)_i \in (\inf I, t)\) converging to \(\inf I\) such that \(f'({\tilde{t}}_i) \rightarrow 0\) as \(i \rightarrow \infty \). Since f is strictly increasing in I, it is invertible there. Hence, we may compute, for every \(i \in \mathbb {N}\),
and thus, by letting \(i \rightarrow \infty \), we obtain \(|f'(t)|^2 \le 4 \Vert f\Vert _{L^\infty (\mathbb {R})}\Vert f''\Vert _{L^\infty (\mathbb {R})}\), and (2.37) follows.
Since \(\partial E\) is compact we find \(x \in \partial E\) such that \(|\nabla _\tau u (x)| = \Vert \nabla _\tau u \Vert _{L^\infty (\partial E)}\). We may assume that \(|\nabla _\tau u (x)| > 0\). The connected component of \(\partial E\) containing x is geodesically complete and, hence, we find a smooth unit speed geodesic curve \(\gamma : \mathbb {R}\rightarrow \partial E\) satisfying \(\gamma (0)=x\) and \(\gamma '(0) = \nabla _\tau u (x)/|\nabla _\tau u (x)|\). Then we define a \(C^2\)-regular function \(f = u \circ \gamma \). Note that \(f'(0)= \Vert \nabla _\tau u\Vert _{L^\infty (\partial E)}\) and that
By differentiating the identity \(0 = d_E \circ \gamma \) twice and recalling the identities (2.22) and (2.29) we obtain \(0 = \gamma ' \cdot (B_E \circ \gamma ) \gamma ' + \gamma '' \cdot (\nu _E \circ \gamma )\). Since \(\gamma \) is a geodesic curve, then \(|\gamma '' \cdot (\nu _E \circ \gamma )|= |\gamma ''|\) and hence we infer from the previous that \(|\gamma ''| \le |B_E \circ \gamma |_\textrm{op}\). By combing this with (2.38) and using (2.30) gives us
Thus, by observing \(\Vert f\Vert _{L^\infty (\mathbb {R})} \le \Vert u\Vert _{L^\infty (\partial E)}\), the claim follows from (2.37). \(\quad \square \)
3 Definition of the Flat Flow and the First Regularity Estimates
Let us begin by recalling the definition of the minimizing movements scheme and the flat flow solution of (1.1) from [43]. Assume that \(E_0 \subset \mathbb {R}^{n+1}\) is a bounded set of finite perimeter. For given a time step \(h \in \mathbb {R}_+\) we construct a parametrized family \((E_t^h)_{t\ge 0}^\infty \) of sets of finite perimeter by an iterative minimizing procedure called minimizing movements, where
Here for a generic bounded set of finite perimeter \(E \subset \mathbb {R}^{n+1}\) the functional \(\mathcal {F}_h( \ \cdot \, E)\), in the class of the bounded set of finite perimeter, is defined as
for \(m_0 = |E_0|\). We call the family \((E_t^h)_{t\ge 0}^\infty \), defined in (3.1), an approximative flat flow solution of (1.1) starting from \(E_0\). We note that there is always a minimizer for (3.2) but it might not be unique. By [43] we know that there is a subsequence of approximative flat flows \((E^{h_l}_t)_{t \ge 0}\) which converges to a parametrized family \((E_t)_{t \ge 0}\) for a.e. t in the \(L^1\)-sense, where for every \(t >0\) the set \(E_t\) is a set of finite perimeter with \(|E_t| = |E_0|\). Any such limit is called a flat flow solution of (1.1) starting from \(E_0\).
Let us turn our focus back on a generic minimizer of (3.2), where we assume that \(|E|=m_0\). We then simply denote any minimizer for \(\mathcal {F}_h( \ \cdot \, E)\) by \(E^h_{\min }\). One has to be careful in the definition of the functional in (3.2), since the sets of finite perimeter are only defined up to measure zero. We avoid this issue by modifying a set of finite perimeter in a \(L^1(\mathbb {R}^{n+1})\) -negligible set and choose as in [37, Rmk 15.3] a representative which topological boundary agrees with the closure of its measure theoretical boundary. Thus, we always use the convention \(\partial F = \overline{\partial ^* F}\) for the initial set and the minimizers. We also remark that if E is empty, then we use the convention \(d_E = \infty \) everywhere to ensure that \(E^h_{\min }\) is empty too.
Next, we recall some basic properties regarding the minimizers. First, it is easy to conclude \(P(E^h_{\min }) \le P(E)\). Moreover, \(E^h_{\min }\) satisfies the distance property
for a dimensional constant \(\gamma _n \in \mathbb {R}_+\), see [43, Prop 3.2]. Second, \(E^h_{\min }\) has a generalized mean curvature satisfying the Euler-Lagrange equation
in the distributional sense (2.7) on \(\partial ^* E^h_{\min }\), where the Lagrange multiplier satisfies \(|\lambda ^h| = 1 / \sqrt{h}\) in the case \(|E^h_{\min }| \ne m_0\), see [43, Lemma 3.7]. Third, it is easy to see that \(E^h_{\min }\) is always a so called \((\Lambda ,r)\) -minimizer with suitable \(\Lambda , r \in \mathbb {R}_+\) satisfying \(\Lambda r \le 1\) (see [37] for the definition). Thus, by the standard regularity theory [37, Thm 26.5 and Thm 28.1] the reduced boundary \(\partial ^* E^h_{\min }\) is relatively open in \(\partial E^h_{\min }\) and an embedded \(C^{1,\alpha }\)-regular hypersurface with any \(0<\alpha <1/2\), and the Hausdorff dimension of the singular part \(\partial E^h_{\min } \setminus \partial ^* E^h_{\min }\) is at most \(n-7\). Thus, by standard Schauder estimates one may show that \(\partial ^*E^h_{\min }\) is in fact \(C^{2,\alpha }\)-regular and (3.4) holds in the classical sense on \(\partial ^* E^h_{\min }\). Consequently, we may always consider \(E^h_{\min }\) as an open set.
We may improve the distance estimate (3.3) as well as regularity properties of \(E^h_{\min }\), if we impose more regularity on E. We divide our approach into two steps. The first result states that if E is bounded and satisfies UBC with radius \(r_0>0\) and h is sufficiently small, then the left hand side of (3.3) is bounded linearly in h, the Lagrange multiplier \(\lambda ^h\) is bounded, the generalized mean curvature \(H_{E^h_{\min }}\) is bounded in the \(L^\infty \)-sense and \(E^h_{\min }\) has the volume \(m_0\).
Proposition 3.1
Assume \(E \subset \mathbb {R}^{n+1}\) is an open and bounded set of volume \(m_0\) which satisfies UBC with radius \(r_0\). There are positive numbers \(h_0 = h_0(n,m_0,r_0)\) and \(C_0=C_0(n,m_0,r_0)\) and a dimensional constant \(C_n \in \mathbb {R}_+\) such that if \( h \le h_0\), then
Proof
We prove first part of the claim, i.e., the distance estimate. If \(|E^h_{\min } \Delta E| = 0\), then it follows from the openness of \(E^h_{\min }\) and E as well as the property \(\partial E^h_{\min } = \overline{\partial ^* E^h_{\min }}\) and \(\partial E= \overline{\partial ^* E}\) that \(E^h_{\min } \Delta E = \emptyset \) and there is nothing to prove. Thus, we may assume that \(|E^h_{\min } \Delta E|>0\) and further set that
To conclude the first part of the claim, we show, under the assumption \(|E^h_{\min } \Delta E|>0\), the validity of the implication
Thus, (3.5) and our earlier observation gives us the implication
To prove (3.5), we assume by contradiction that \(d_-\ge 0\) which implies \(E \subset E^h_{\min }\) due to the openness of E and, hence, \(|E^h_{\min }{\setminus } E| = |E^h_{\min } \Delta E|>0\). Using (2.35) with \(r=r_0\), the previous observation, \(|E|=m_0\), and the assumption on h yields
contradicting the minimality of \(E^h_{\min }\) and, hence, \(d_- < 0\). Similarly we obtain \(d_+> 0\).
On the other hand, \(\sqrt{h} \le r_0 / (8 \gamma _n)\) implies via (3.3) that \(E^h_{\min } \Delta E \subset \subset \mathcal N_{r_0/4}(\partial E)\). In particular, \(-r_0/2< d_-<0<d_+<r_0/2\) and for every \(t \in (d_-,d_+)\) the sublevel set \(E_t = \{ x: d_E(x) < t\}\) satisfies UBC with \(r_0/2\) and \(|E^h_{\min } {\setminus } E_t|, |E_t {\setminus } E^h_{\min }|>0\). By using a suitable continuity argument, we infer from the previous that for every \(t < d_+\), sufficiently close to \(d_+\), there is \({\tilde{t}} \in (d_-, r_+)\) such that \(|E^h_{\min } {\setminus } E_t|=|E_{{\tilde{t}}} {\setminus } E^h_{\min }| > 0\) and \({\tilde{t}} \rightarrow d_-\) as \(t \rightarrow d_+\). For such a pair \((t,{\tilde{t}})\) we set
Clearly, F is a bounded set of finite perimeter and \( |F| = |E^h_{\min }|. \) Thus, using F as a competitor against \(E^h_{\min }\) with respect to \({\mathcal {F}}_h( \ \cdot \, E)\) we obtain
In turn, applying Lemma 2.10 to \(E_t\) and \(E_{{\tilde{t}}}\) gives us
We combine (3.7) and (3.8) and recall \(|E_{{\tilde{t}}} {\setminus } E^h_{\min }|>0\) to observe that
Thus, by letting \(t \rightarrow d_+\), we obtain the second estimate in (3.5).
To prove the second part of the claim, we denote by C a generic positive constant which may change its value from the line to line but depends only on \(n,m_0\) and \(r_0\). We fix any connected component \(E^i\) of E. By Lemma 2.10 and (2.36) we have \(\textrm{diam}(E^i) \le C\) and \(P(E) \le C\). If \(E^j\) is a connected component of E distinct to \(E^i\), then UBC with \(r_0\) guarantees \(\textrm{dist}(E^i,E^j) \ge r_0\). Assuming that \(\sqrt{h} \le r_0/ \max \{n+1,8\gamma _n\}\) we have, by (3.3), (3.6), openness of \(E^h_{\min }\) and \(\overline{\partial ^* E^h_{\min }} = \partial E^h_{\min }\), that \(E^h_{\min } \Delta E \subset \subset {\mathcal {N}}_{r_0/4}(\partial E)\) and \(|d_E/h|\le 4(n+1)/r_0\) on \(\partial ^* E^h_{\min }\). Again, we infer from the previous observations that for the intersection \({\tilde{E}}^i = E^h_{\min } \cap (E^i + B_{r_0/4})\) it holds \(\partial ^* {\tilde{E}}^i = \partial ^* E^h_{\min } \cap (E^i + B_{r_0/4})\), \(H_{{\tilde{E}}^i} = H_{E^h_{\min }}|_{\partial ^* {\tilde{E}}^i}\), \(\textrm{diam}({\tilde{E}}^i) \le C + r_0/2 \le C\) and \(|{\tilde{E}}^i|\ge |B_{r_0/2}|\). Using the divergence theorems and the Euler-Lagrange equation (3.4), which holds in the sense of (2.7) on \(\partial ^* \tilde{E}^i\), we compute that
By translating the coordinates, we may assume \(0 \in {\tilde{E}}^i\) so \(|\textrm{id}| \le \textrm{diam}({\tilde{E}}^i) \le C\) on \(\partial ^* E^i\). Since we also have \(P({\tilde{E}}^i) \le P(E^h_{\min }) \le P(E) \le C\), \(|\tilde{E}^i|\ge |B_{r_0/2}|\) and \(|d_E/h|\le 4(n+1)/r_0\) on \(\partial ^* \tilde{E}^i\), we infer from the previous computation \(|\lambda ^h| \le C_0\) for \(C_0=C_0(n,m_0,r_0) \in \mathbb {R}_+\). Therefore, using the Euler-Lagrange equation (3.4) and the first estimate again we have, by possibly increasing \(C_0\), that \(\Vert H_{E_{\min }^h}\Vert _{L^\infty \left( \partial ^*E_{\min }^h\right) } + |\lambda ^h| \le C_0\). Finally, if \(|E^h_{\min }| \ne m_0\), then \(|\lambda ^h| = 1 /\sqrt{h}\). Thus, assuming \(h \le (2 C_0)^{-2}\) excludes this possibility and hence it must hold \(|E^h_{\min }| = m_0\). \(\quad \square \)
Proposition 3.1, allows us to deduce, via Allard’s regularity theorem, that the singular set of minimizer is in fact empty. Further, standard Schauder estimates gives us a quantitative, albeit non-sharp, UBC for a minimizer.
Lemma 3.2
Assume \(E \subset \mathbb {R}^{n+1}\) is an open and bounded set of volume \(m_0\) which satisfies UBC with radius \(r_0\). There are positive numbers \(h_0 = h_0(n,m_0,r_0)\) and \(c_0=c_0(n,m_0,r_0)\) such that if \( h \le h_0\), then \(\partial E {\setminus } \partial E^* = \varnothing \), \(E^h_{\min }\) is \(C^{3,\alpha }\)-regular with any \(0<\alpha < 1\) and \(E^h_{\min }\) satisfies UBC with radius \(c_0 h^{1/3}\). In particular, (3.4) is satisfied in the classical sense on \(\partial E^h_{\min }\). Moreover, if E is \(C^k\)-regular, with \(k \ge 2\), then \(E^h_{\min }\) is \(C^{k+2}\)-regular.
Proof
We divide the proof into two steps. Recall that we may assume \(E^h_{\min }\) to be open. In the proof, C denotes a generic positive constant which may change its value from line to line but it depends only on \(n,m_0\) and \(r_0\).
Step 1: By using Allard’s regularity theorem we show that the topological boundary \(\partial E^h_{\min }\) agrees with the reduced boundary \(\partial ^*E^h_{\min }\) when h is sufficiently small. To be more precise, we show that there exist positive numbers \(\rho =\rho (n,m_0,r_0)\) and \(h_1 = h_1(n,m_0,r_0, \rho )\) such that if \(h \le h_1\) and \(x \in \partial E^h_{\min }\), then, by possibly rotating the coordinates, there is a function \(f \in C^{1,1/3} \left( B^n_{\rho }(x')\right) \) such that
and f satisfies the estimates
In particular, (3.9) implies that \(\partial ^*E = \partial E\) and hence, by our earlier discussion, we conclude that \(E^h_{\min }\) is \(C^{2,\alpha }\)-regular with any \(0<\alpha <1/2\). We may assume that \(h_1\) is chosen so small that via Proposition 3.1 the boundary \(\partial E^h_{\min }\) is contained in \({\mathcal {N}}_{r_0/2}(\partial E)\). Since \(d_E \in C^{1,1}(\mathcal N_{r_0/2}(\partial E))\), then recalling the Euler-Lagrange equation (3.4) we may write the generalized mean curvature of \(E^h_{\min }\) as a restriction of a \(C^{1,1}\)-function to \(\partial E^h_{\min }\). Therefore, by using standard Schauder estimates, one may show that \(E^h_{\min }\) is actually \(C^{3,\alpha }\)-regular with any \(0<\alpha <1\). Also, the same method gives us \(C^{k+2,\alpha }\)-regularity for any \(k \ge 2\), if E is already known to be \(C^{k,\alpha }\)-regular. This is well-known procedure and we leave it to the reader.
The claim of Step 1 follows essentially from [46, Thm 2.5.2], if we prove that for every \(x \in \partial E^h_{\min }\) and \(\varepsilon \in \mathbb {R}_+\) there are positive numbers \(\rho =\rho (n,m_0,r_0,\varepsilon )\) and \({\tilde{h}} = {\tilde{h}}(n,m_0,r_0,\rho ,\varepsilon )\) such that if \(h \le {\tilde{h}}\), then
We fix \(\varepsilon >0\) and initially assume \(h \le h_0\), where \(h_0\) is from Proposition 3.1. It follows from Proposition 3.1 and the fact \(\partial E^h_{\min }=\overline{\partial ^* E^h_{\min }}\) that
Thus, we may assume that \((\overline{E^h_{\min }} \cup {\overline{E}}) {\setminus } (E^h_{\min } \cap E) \subset {\mathcal {N}}_{r_0/2}(\partial E)\) where the projection \(\pi _{\partial E}\) is well-defined. Proposition 3.1 also gives us \(|E^h_{\min }|=m_0\). Next, we fix \(x \in \partial E^h_{\min }\). Without loss of generality, we may assume \(\pi _{\partial E}(x)=0\) and \(\nu _E(0)=e_{n+1}\). Then it follows from Proposition 2.7 that there is \(g \in C^{1,1}(B^n_{r_0/2})\) such that \(|g(y')| < |y'|^2/r_0\), \(|\nabla g(y')|< 2|y'|/r_0\) for every \(y' \in B^n_{r_0/2}\) and
We have, for every \(0<\rho <r_0/4\), a density bound
Suppose that \(y \in \textbf{C}(0,\rho ,r_0/2) \cap ((\overline{E^h_{\min }} \cup {\overline{E}}) {\setminus } (E^h_{\min } \cap E))\) for \(0<\rho <r_0/4\). Recalling (3.13), we may assume that \(\pi _{\partial E} (y) \in \textbf{C}(0, r_0/2,r_0/2)\) and since \(|\nabla g| \le C\) in \(B^n_{r_0/2}\) we estimate that
It follows then from Fubini’s theorem that
for \(0<\rho <r_0/4\). We define for such \(\rho \) a comparison set \(F_\rho \) by setting
and we make the following technical observations: first, since \(E^h_{\min } \cap E\) is open and contained in \(F_\rho \), then \(\mathcal {H}^n(\partial ^* F_\rho \cap (E^h_{\min } \cap E))=0\). Second, \(\partial ^* F_\rho \subset \overline{E^h_{\min }} \cup {\overline{E}}\). With help of these, (3.14) and (3.16) we estimate
Thus, the inequality \({\mathcal {F}}_h(E^h_{\min },E) \le \mathcal F_h(F_\rho ,E)\), (3.13), (3.15), \(|E^h_{\min }|=m_0\) and the definition of \(F_\rho \) yield
Recall that for the fixed point \(x \in \partial E^h_{\min }\) it holds \(x=d_E(x)e_{n+1}\) with \(|d_E(x)| \le Ch\). Thus we may assume \(B_\rho (x) \subset \textbf{C}(0,\rho ,r_0/2)\) for \(0<\rho <r_0/4\). Hence, the above estimate yields
Moreover, it holds \(\Vert H_{E^h_{\min }}\Vert _{L^\infty (\partial ^* E_{\min }^h)} \le C\) by Proposition 3.1, \(P(E^h_{\min }) \le P(E)\) and \(P(E) \le C\) by Lemma 2.10. Therefore,
We infer from the previous estimate and (3.17) the existence of numbers \(\tilde{h}\) and \( \rho \) satisfying (3.11) and (3.12).
Step 2: We assume that \(h \le h_1\) and fix \(x \in \partial E^h_{\min }\). We may assume that \(x = 0\) and \(\nu _{E^h_{\min }}(0) = e_{n+1}\). According to Step 1, up to a possible rotation of the coordinates, there is \(f \in C^3(B^n_{\rho _1}(x'))\) with \(f(0) = \nabla f(0)= 0\) satisfying (3.9) and (3.10). We use Schauder estimate in a quantitative manner to prove there is a positive \(h_0=h_0(n,m_0,r_0) \le h_1\) such that \(h \le h_0\) implies
Once we have proven (3.18) then the claim that \(E^h_{\min }\) satisfies UBC with radius \(c_0 h^{1/3}\) follows in a straightforward manner as we discussed in Remark 2.8.
Thus, we are left to prove (3.18). We may write \(H_{E^h_{\min }}\) in local coordinates as the mean curvature of the subgraph \(\{(y',y_{n+1}: y' \in B^n_{\rho }, \ y_{n+1} < f(y')\}\), that is,
It follows from (3.10) that \({\mathcal {A}}\) is uniformly elliptic and bounded in the \(C^{0,1/3}\)-sense. To be more precise, we have
Thus, by using standard Schauder interior estimate [24], (3.10) and (3.19), we obtain
where \(u: B^n_{\rho _1} \rightarrow \mathbb {R}^n\) is given by \(u(y') =H_{E^h_{\min }} (y',f(y'))\). We may assume h is chosen sufficiently small so that via Proposition 3.1 we have \(\Vert u\Vert _{L^\infty ( B^n_{\rho } )} \le C\). Again, (3.10) implies \(|\nabla u (y')| \le C |\nabla _\tau H_{E^h_{\min }} (y',f(y'))|\) for every \(y' \in B^n_{\rho }\). On the other hand, by (tangentially) differentiating the Euler-Lagrange equality (3.4) we obtain \(|\nabla _\tau H_{E^h_{\min }} (y',f(y'))| \le 1/h\) for every \(y' \in B^n_{\rho }\). Hence, \(\Vert \nabla u\Vert _{L^\infty (B^n_{\rho })} \le C/h\) and since \(\Vert u\Vert _{L^\infty ( B^n_{\rho } )} \le C\), assuming \(h \le 1\) yields \(\Vert u\Vert _{C^1( B^n_{\rho })} \le C/h\). Again, Lemma 2.1 yields \(\Vert u\Vert _{C^{0,1/3}( B^n_{\rho } )} \le C h^{-1/3}\) and hence, by recalling (3.20), we conclude the existence of \(h_0 = h_0(n,m_0,r_0)\) satisfying (3.18) for all \(h \le h_0\). \(\quad \square \)
Remark 3.3
We may replace the exponent 1/3 with a generic \(0<\alpha <1\) in the proof of Lemma 3.2. Then, naturally, \(h_0\) and \(c_0\) also depend on \(\alpha \). UBC with radius \(r_0\) for E and UBC with radius \(c_0 h^{1/3}\) for \(E^h_{\min }\) imply together with the distance estimate of Proposition 3.1 and (2.22) that there is \(h_0 = h_0(n,m_0,r_0)\) such that if \(h \le h_0\), then \(\nabla d_E \cdot \nu _{E^h_{\min }}> 0\) on \(\partial E^h_{\min }\) and the projection \(\pi _{\partial E}\) is injective on \(\partial E^h_{\min }\).
4 Uniform Ball Condition for Short-Time
In this section, we adopt the two-point function method to prove that if the initial set \(E_0\) satisfies UBC with radius \(r_0\), then there are positive numbers \(h_0\) and \(T_0\) such that
where the approximative flow \((E_t^h)_{t \ge 0}\) starting from \(E_0\) is defined as in (3.1). For more precise statement, see Theorem 4.7 at the end of the section. As we have seen in Lemma 3.2, UBC for an initial set is crucial, as it guarantees that the corresponding minimizer of the energy (3.2) has improved regularity and an initial quantitative bound on UBC although the latter depends on h. In this section, we improve the previous non-sharp estimate on UBC for the minimizer by showing the minimizer satisfies almost the same UBC as the initial set.
The original idea of the two-point function goes back to [27], where it is used to study the regularity of the classical solution to the mean curvature flow. We refer to [10] for a comprehensive overview of the topic and mention also the works [4, 11, 18] which have inspired us. Here we will show that the method can be applied to the approximative flat flow at the level of discrete time scale. We will assume that the approximative flat flow is related to the volume preserving mean curvature flow but the arguments hold with essentially no modifications also in the case of the mean curvature flow.
4.1 Two-Point Function Method
The main idea is to double the variables and, given a set \(E \subset \mathbb {R}^{n+1}\) satisfying UBC, to study the function \(S_E\) defined for \((x,y) \in \partial E \times \partial E\) with \(x \ne y\) as
It is known, but we will include the proof below, that the maximum value of \(|S_E|\) is explicitly related to the maximal UBC for E. In other words, doubling the variables allows us to quantify the maximal UBC via the function \(S_E\). It is interesting that the idea of doubling the variables is also used in [29] to study regularity of solutions of nonlinear PDEs.
For the next lemma we note that if a set E satisfies UBC with radius r, then it satisfies UBC with every \(0<\rho < r\). We define \(r_E\) to be the supremum of such radii and recalling our previous discussion we may write this as
Note that \(r_E>0\). We use the abbreviation \(\Vert S_E\Vert _{L^\infty }:= \sup \{ |S_E(x,y)|: x,y \in \partial E, \ x \ne y \}\).
Lemma 4.1
Let \(E \subset \mathbb {R}^{n+1}\) be an open and bounded set satisfying UBC. Then it holds that
where \(r_E\) is defined in (4.3). In the case E is \(C^2\)-regular, we also have \(|H_E|, |B_E| \le 2n \Vert S_E\Vert _{L^\infty }\) on \(\partial E\).
Proof
Let us first show \(2\Vert S_E\Vert _{L^\infty } \ge 1/r_E\). First of all, we infer from the boundedness of E that \(r_E < \infty \). Since E does not satisfy UBC with given a radius \(r \in (r_E, \infty )\), there is \(z \in {\mathcal {N}}_r(\partial E)\) such that \(d_E\) is not differentiable at z. Hence, there are distinct points \(x,y \in \partial E\) such that \(|z-x| = |d_E(z)| = |z-y|\). Without loss of generality, we may assume \(z=0\) which implies \(|x|=|y| < r\) and \(\nu _E (x) = \pm x/|x|\). Thus,
and we conclude the inequality \(2 \Vert S_E\Vert _{L^\infty } \ge 1 /r_E\).
To conclude the opposite estimate, we choose \(0<r<r_E\). Let \(x,y \in \partial E\) be distinct points. Since E satisfies UBC with r, we have \(|d_E(x \pm r \nu _E(x))| = r\) and, hence,
By subtracting and dividing terms we obtain \(\pm 2 S_E(x,y) \le 1/r\). We let \(r \rightarrow r_E\) to obtain \(2 \Vert S_E\Vert _{L^\infty } \le 1 /r_E\). Thus, \(2 \Vert S_E\Vert _{L^\infty } =1 /r_E\). The rest of the claim is a direct consequence of the previous identity, (2.30) and Proposition 2.7. \(\quad \square \)
An obvious consequence of Lemma 4.1 is that for every open and bounded set \(E\subset \mathbb {R}^{n+1}\) it holds
for a positive constant \(c_0 = c_0(n,|E|)\).
We will also use the regularized version of \(S_E\), which we define for any \(\varepsilon \in \mathbb {R}_+\) as \(S_{E,\varepsilon }: \partial E \times \partial E \rightarrow \mathbb {R}\),
As in the case of \(S_E\), we use the abbreviation \(\Vert S_{E,\epsilon }\Vert _{L^\infty } = \max \{ |S_{E,\varepsilon }(x,y)|: (x,y) \in \partial E \times \partial E\}\). The idea behind considering \(S_{E,\varepsilon }\) instead of \(S_E\) is that, on the one hand, \(S_{E,\epsilon } \rightarrow S\) pointwise in \(\partial E \times \partial E {\setminus } \{(x,x): x \in \partial E\}\) as \(\epsilon \) tends to zero (in particular, \(\Vert S_{E,\epsilon }\Vert _{L^\infty } \uparrow \Vert S_E\Vert _{L^\infty }\)) and, on the other hand, we may differentiate \(S_{E,\epsilon }\) on the product \(\partial E \times \partial E\) provided that E is sufficiently regular. The following calculations are similar to [4, 18] but we give them in order to be self-consistent.
Let us first differentiate \(S_{E,\varepsilon }\) in the case E is \(C^2\)-regular. In the computations, the notations \(\nabla ^x_\tau \) and \(\nabla ^y_\tau \) stand for the tangential differentiation along \(\partial E\) with respect to x and y -variables respectively. Recalling the basic identities (2.4) as well as observing \(B_E \nu _E = 0\) and \(\nabla _\tau \textrm{id}= P_{\partial E}\) on \(\partial E\) we compute
and
for every \((x,y) \in \partial E \times \partial E\). We immediately obtain the following identities at critical points:
Lemma 4.2
Let \(E \subset \mathbb {R}^{n+1}\) be a bounded and \(C^2\)-regular set. Assume \((x,y) \in \partial E \times \partial E\) is a local maximum or a local minimum point of \(S_{E,\varepsilon }\) defined in (4.5). Then it holds that
Moreover, the condition \(r_E> \sqrt{\varepsilon }\) implies
Proof
Since (x, y) is a critical point for the functions \(S_{E,\varepsilon }( x, \ \cdot \ )\) and \(S_{E,\varepsilon }( \ \cdot \, y )\), then the equality (4.8) follows from (4.6) and the equality (4.9) follows from (4.7). Using \(P_{\partial E}(y) = I - \nu _E(y) \otimes \nu _E(y)\) and (4.9) we have
The equality (4.10) thus follows once we show that
We argue by contradiction and assume \(\nu _E(x)=2 S_{E,\varepsilon }(x,y) (x-y)\). Then it holds \(S_{E,\varepsilon }(x,y) \ne 0\) and the definition of \( S_{E,\varepsilon }(x,y)\) implies
Therefore, we have \(|x-y| = \sqrt{\varepsilon }\). On the other hand, the contradiction assumption, the definition of \(S_{E,\varepsilon }\) and Lemma 4.1 together yield that
which is impossible, by the assumption that \(r_E > \sqrt{\varepsilon }\). \(\quad \square \)
If E has higher regularity and \(\varepsilon \) is sufficiently small, we may naturally extract more information at local extreme points. Indeed, if E is \(C^3\)-regular, then by maximum principle at a local maximum (minimum) point \((x,y) \in \partial E \times \partial E\) of \(S_{E,\varepsilon }\) it holds that
We calculate the LHS of (4.12) in the next lemma.
Lemma 4.3
Let \(E \subset \mathbb {R}^{n+1}\) be a bounded and \(C^3\)-regular set with \(r_E > \sqrt{\varepsilon }\). At a local maximum (minimum) point \((x,y) \in \partial E \times \partial E\) of \(S_{E,\varepsilon }\) it holds that
Proof
First, we compute the terms on the LHS of (4.12) by taking tangential divergences of (4.6) and (4.7) with respect to x and y -variables. In the computations, we use the identities (2.5) and the fact that the gradients \(\nabla _\tau ^x S_{E,\varepsilon }(x,y)\) and \(\nabla _\tau ^y S_{E,\varepsilon }(x,y)\) vanish. Omitting all the details we obtain by straightforward calculation
and
Collecting the terms and applying the inequality (4.12), we obtain that at a local maximum (minimum) point it holds that
The claim follows once we show that the last line above vanishes, i.e., that
Since \(r_E > \sqrt{\varepsilon }\), this follows by first applying the equalities (4.8) and (4.10) in Lemma 4.2 and recalling \(B_E(x) \nu _E (x)= 0\):
Then we use (4.10) to deduce that
and (4.13) follows. \(\quad \square \)
In conclusion, by combining Lemma 4.1 and Lemma 4.3, we obtain that if a bounded \(C^3\)-regular set \(E \subset \mathbb {R}^{n+1}\) satisfies \(r_E > \sqrt{\varepsilon }\), then at a local maximum (minimum) point \((x,y) \in \partial E \times \partial E\) of \(S_{E,\varepsilon }\) it holds that
4.2 Short-Time Uniform Ball Estimate
Let us turn our focus on how to prove (4.1) for an approximative flat flow solution \((E^h_t)_{t\ge 0}\) defined in (3.1) when the initial set \(E_0\) satisfies UBC with given a radius \(r_0\). Assuming we may control the evolution of the quantity \(\Vert S_{E^h_t}\Vert _{L^\infty }\), then thanks to Lemma 4.1 we also control (from below) the maximal UBC for \(E_t^h\).
We motivate ourselves by considering first the continuous and embedded setting. Assume \((E_t)_t\) is a smooth flow and let \(\nu _t\) and \(V_t\) denote the outer unit normal of \(E_t\) and the normal velocity of the flow on \(\partial E_t\) respectively. Then one may use the fact that for fixed t there is a smooth normal parametrization \((\Phi ^t_s)_s\) of the flow such that \(\Phi ^t_0 = \textrm{id}\) and \(\partial _s \Phi ^t_s = [V_s \, \nu _s] \circ \Phi ^t_s\). This follows essentially from [5, Thm 8]. It is straightforward to calculate that for such a parametrization
In the case of volume preserving mean curvature flow, we have \(V_s = -(H_s - \bar{H_s})\), where \(H_s\) is the scalar mean curvature on \(\partial E_s\) and \(\bar{H_s}\) its integral average over \(\partial E_s\). If x and y are distinct points on \(\partial E_t\), then by using (4.15) and the previous identity, we may compute
where the remainder term \(R_t(x,y)\) has a bound \(|R_t(x,y)| \le C_n \Vert S_{E_t}\Vert _{L^\infty }^3\). Suppose that \(\Vert S_{E_t}\Vert _{L^\infty } = \pm S_{E_t}(x,y)\) and the function \(s \mapsto \Vert S_{E_{t+s}}\Vert _{L^\infty }\) is differentiable at \(s=0\), then we deduce
Again, the estimate (4.14) also holds for \(S_E\) when the points are distinct. Thus, by possibly increasing \(C_n\), we infer from above and (4.16) that
provided that \(s \ne 0\) is sufficiently small.
The idea is to mimic the previous argument in the discrete setting for an approximative flat flow \((E^h_t)_{t\ge 0}\). To this end, we need to approximate the two-point functional by its \(\varepsilon \)-regularized version. We consider the element \(E^h_t\) and its consequent set \(E^h_{t+s}\). For sake of brevity, we use the shorthand notations \(E_1 = E^h_t\) and \(E_2 = E^h_{t+s}\) for the rest of the subsection. First, we want to find a discrete version of the equalities in (4.15). Suppose that an element \(E_1\) satisfies UBC and h is so small that by the discussion of the previous section we have that \(E_2\) is \(C^1\)-regular set, \(\partial E_2 \subset {\mathcal {N}}_{r_{E_1}}(\partial E_1)\) and \(\nabla d_{E_1} \cdot \nu _{E_2}>0\) on \(\partial E_2\) are satisfied.
Then it is natural to project the boundary \(\partial E_2\) to \(\partial E_1\) by the projection \(\pi _{\partial E_1}\) and, hence, using the identities in (2.22) we have
which can be seen as a discrete time counterpart of the first identity in (4.15). In the next simple but crucial lemma, we derive a relation between \(\nu _{E_2}\) and \(\nu _{E_1} \circ \pi _{\partial E_1}\) for \(x \in \partial E_2\).
Lemma 4.4
Assume that \(E_1 \subset \mathbb {R}^{n+1}\) is an open set satisfying UBC, \(E_2\) is a \(C^1\)-regular set such that \(\partial E_2 \subset {\mathcal {N}}_{r_{E_1}}(\partial E)\) and \( \nabla d_{E_1} \cdot \nu _{E_2}>0\) on \(\partial E_2\). Then
Proof
By using the second identity of (2.22) for \(d_{E_1}\), as well as the definition of a tangential gradient, the following holds on \(\partial E_2\):
Since \(|\nu _{E_1} \circ \pi _{\partial E_1}|=1=|\nu _{E_2}|\) and \(\nabla _{\tau _2} d_{E_1} \cdot \nu _{E_2} = 0\), then the previous decomposition implies \(| \nabla d_{E_1} \cdot \nu _{E_2}|= \sqrt{1-|\nabla _{\tau _2} d_{E_1}|^2}\). Thus, the claim follows from the assumption \(\nabla d_{E_1} \cdot \nu _{E_2}>0\) on \(\partial E_2\). \(\quad \square \)
The equality in the statement of Lemma 4.4 gives us a discrete analog for the second equality in (4.15) as
or equivalently
which will be useful later. We need yet one technical lemma related to the projection \(\pi _{\partial E_1}\) on the consequent boundary \(\partial E_2\).
Lemma 4.5
Let \(E_1,E_2 \subset \mathbb {R}^{n+1}\) be open and bounded sets satisfying UBC. If \(\partial E_2 \subset \mathcal N_{r_{E_1}/2}(\partial E_1)\), then for any \(x,y \in \partial E_2\) satisfying \(\pi _{\partial E_1}(x) \ne \pi _{\partial E_1}(y)\) it holds that
where \(C_0\ge 1\) is a universal constant.
Proof
First, we obtain from (2.22) and the definition of \(S_{E_1}\) that
Thus,
The normal \(\nu _{E_1}\) is \(1/r_{E_1}\)-Lipschitz continuous by Proposition 2.7 and \(\pi _{\partial E_1}\) is 2-Lipschitz continuous in \({\mathcal {N}}_{r_{E_1}/2}(\partial E_1)\) by Lemma 2.9. On the other hand, recalling Lemma 4.1 we conclude \(\Vert d_{E_1}\Vert _{L^\infty (\partial E_2)}\Vert S_{E_1}\Vert _{L^\infty } \le 1/4\). Hence, we infer from the previous estimate that
Thus, we are left to estimate the term \(|d_{E_1}(x) -d_{E_1}(y)|^2\) on the boundary \(\partial E_2\).
We divide this into two cases. First, suppose that \(|x-y|\ge r_{E_2}/2\). Then using Lemma 4.1 we obtain
Suppose then \(|x-y|<r_{E_1}/2\). We define a \(C^1\)-extension \(\tilde{d}_{E_1}: {\mathcal {N}}_{r_{E_2}}(\partial E_2) \rightarrow \mathbb {R}\) of the restriction \(d_{E_1}|_{\partial E_2}\) by setting \({\tilde{d}}_{E_1} = d_{E_1} \circ \pi _{\partial E_2}\). Then \(\nabla {\tilde{d}}_{E_1} = \nabla \pi _{\partial E_2} \nabla _{\tau _2} d_{E_1} \circ \pi _{\partial E_2}\) and by Lemma 2.9\(|\nabla \pi _{\partial E_2}|_\textrm{op}\le 2\) in \({\mathcal {N}}_{r_{E_2}/2}(\partial E_2)\) so that \(|\nabla {\tilde{d}}_{E_1}| \le 2 \Vert \nabla _{\tau _2} {\tilde{d}}_{E_1}\Vert _{L^\infty (\partial E_2)}\). Since the line segment \(J_{yx}\) belongs to \(\mathcal N_{r_{E_2}/2}(\partial E_2)\), we have
By Lemma 2.9 we have \(|\nabla ^2 d_{E_1}|_\textrm{op}\le 2/r_{E_1}\) in \({\mathcal {N}}_{r_{E_1}}(\partial E_1)\). Therefore, by using Lemma 2.11 and Lemma 4.1 we get an estimate
Thus, we gather the estimate as claimed from (4.20), (4.21), (4.22) and the estimate above. \(\quad \square \)
We are now ready prove an analogous estimate to (4.17) in the discrete setting.
Lemma 4.6
Assume that \(E_1 \subset \mathbb {R}^{n+1}\) is an open and bounded set, with \(|E_1| = m_0\), which satisfies UBC with radius \(r_0 \in \mathbb {R}_+\). Let \(E_2\) be any minimizer of the energy \(\mathcal {F}_h( \ \cdot \, E_1)\) defined in (3.2). Then there is \(h_0=h_0(n,m_0,r_0)\) such that for \(h \le h_0\) \(E_2\) is \(C^3\)-regular and
If in addition \(E_1\) is \(C^k\)-regular, then \(E_2\) is \(C^{k+2}\)-regular.
Proof
As previously, \(C=C(n,m_0,r_0) >0\) may change from line to line. We find \(h_0=h_0(n,m_0,r_0)\in \mathbb {R}_+\) such that assuming \(h \le h_0\) implies that the conclusions of Proposition 3.1, Lemma 3.2 and Remark 3.3 are valid. Let us quickly summarize what we have achieved so far. First, \(E_2\) is open and bounded, \(C^3\)-regular set, or \(C^{k+2}\)-regular set provided that \(E_1\) is \(C^k\)-regular, and it satisfies UBC with radius \(c_0h^{1/3}\) for a constant \(c_0=c_0(n,m_0,r_0) >0\). Hence, by Lemma 4.1 we have apriori estimate
Second, \(\partial E_2\) is “close" to \(\partial E_1\). To be more precise, we have \(\Vert d_{E_1}\Vert _{L^\infty (E_2)} \le C_n h /r_0\) and we may assume that \(\partial E_2 \subset {\mathcal {N}}_{r_0/2}(\partial E_1)\). Moreover, it holds that \(\nabla d_{E_1} \cdot \nu _{E_2} > 0\) on \(\partial E_2\) and \(\pi _{\partial E_1}\) is injective on \(\partial E_2\). Third, we have the Euler-Lagrange equation (3.4) on \(\partial E_2\) in the classical sense.
Thus, we assume that \(h \le h_0\). We might need to shrink \(h_0\) but always in a way that we preserve the dependency \(h_0=h_0(n,m_0,r_0)\). By combining the estimate \(\Vert d_{E_1}\Vert _{L^\infty (E_2)} \le C_n h /r_0\) from Proposition 3.1 with Lemma 4.1 and (4.24) and by possibly shrinking \(h_0\) we obtain
Then, by (3.4), Lemma 4.1 and the first estimate in (4.25), the Lagrange multiplier \(\lambda ^h\) can be controlled as
The claim follows once we show
Indeed, assuming the above holds true we have by Lemma 4.1 and (4.24)
and, hence, recalling (4.4) and shrinking \(h_0\), if neccessary, we obtain \(\Vert S_{E_2}\Vert _{L^\infty } \le 2\Vert S_{E_1}\Vert _{L^\infty }\). Thus, reiterating the previous inequality via (4.27) yields the claim.
To prove (4.27), we initially fix any \(\varepsilon < r_{E_2}^2\) and choose \((x,y) \in \partial E_2 \times \partial E_2\) such that \(|S_{E_2,\varepsilon }(x,y)|=\Vert S_{E_2,\varepsilon }\Vert _{L^\infty }\). Since \(\Vert S_{E_2,\varepsilon }\Vert _{L^\infty }>0\), then \(x \ne y\) and, hence, the injectivity of \(\pi _{\partial E_1}\) on \(\partial E_2\) ensures that \(\pi _{\partial E_1}(x) \ne \pi _{\partial E_1}(x)\). In order to simplify our notations, we write \(\pi =\pi _{\partial E_1}\) and \(H_2 = H_{E_2}\) for short. By using the definition in (4.5), the identities (2.22) and (4.18) as well as the Euler-Lagrange equation we may decompose the difference quotient as
Next, we estimate the last four terms on the RHS. First, since \(\partial E_2 \subset {\mathcal {N}}_{r_0/2}(\partial E) \subset \mathcal N_{r_{E_1}/2}(\partial E_1)\), we have the estimate (4.23) for \(\Vert \nabla _{\tau _2} d_{E_1}\Vert _{L^\infty (\partial E_2)}^2\) and, hence, recalling the first estimate in (4.25) we have
For the next term, we use Lemma 4.1, the first estimate in (4.25) and (4.26) to obtain
By Proposition 2.7\(\nu _{E_1}\) is \(1/r_0\)-Lipschitz and by Lemma 2.9\(\pi \) is 2-Lipschitz continuous in \({\mathcal {N}}_{r_0/2}(\partial E_1)\). Thus, by Lemma 4.1 and the first inequality in (4.25), we estimate the second last term as
Finally, by using Lemma 4.5 and the identities in (4.25), we have
We infer from (4.28), (4.29), (4.30), (4.31) and (4.32) the expression
where for the remainder term it holds \(|R| \le C_n\left( \Vert S_{E_1}\Vert _{L^\infty }^3 + \Vert S_{E_2}\Vert _{L^\infty }^3\right) \). Since (x, y) is a maximum (or minimum) point for \(S_{E_2,\varepsilon }\), then we conclude from (4.14)
Since now \(\Vert S_{E_i,\varepsilon }\Vert _{L^\infty } \uparrow \Vert S_{E_i,\varepsilon }\Vert _{L^\infty }\) for \(i=1,2\) as \(\varepsilon \) tends to zero, the above yields (4.27) and we conclude the proof. \(\quad \square \)
We may now prove the main result of this section which is the UBC estimate for the approximative flat flow.
Theorem 4.7
Let \(E_0 \subset \mathbb {R}^{n+1}\) be an open and bounded set which satisfies UBC with radius \(r_0\in \mathbb {R}_+\) and let \(m_0\) denote its volume. There are \(h_0=h_0(n,m_0,r_0) \in \mathbb {R}_+\) and \(T_0=T_0(n,r_0) \in \mathbb {R}_+\) such that if \(h \le h_0\), then any approximative flat flow \((E_t^h)_{t \ge 0}\) of (1.1) starting from \(E_0\) satisfies UBC with radius \(r_0/2\) for all \(t \le T_0\). Moreover, \(E^h_t\) is \(C^{1+2\lfloor t/h\rfloor }\)-regular for every \(0 \le t \le T_0\).
Proof
By a slight abuse of notation, we set \(h_0\) to be as in Lemma 4.6 for the parameters n, \(m_0\) and \(r_0/2\). Then we choose
where the dimensional constant is the same as in Lemma 4.6. We assume that \(h \le h_0\) and consider an approximative flat flow \((E_t^h)_{t \ge 0}\) starting from \(E_0\) obtained via the minimizing movements scheme (3.1). We may assume \(h \le T_0\), since otherwise the proof is trivial. Since \(E_0\) satisfies UBC with radius \(r_0\), we have by Lemma 4.1 that \(\Vert S_{E_0}\Vert _{L^\infty } =1/(2r_0)\). Then we set
Note that if \(E_{kh}^k\) is a bounded set satisfying UBC with \(\Vert S_{E_k^h}\Vert _{L^\infty } \le 1 /r_0\), then thanks to Lemma 4.1 we know that it satisfies UBC with radius \(r_0/2\). Thus, it follows from the construction of \((E^h_t)_{t \ge 0}\), the choice of \(h_0\), and Lemma 4.6 that \(E_{(k+1)h}^h\) is a bounded \(C^3\)-regular set satisfying
Since \(h_0 \le T_0\), then the choices in (4.33) imply that K is well-defined. By summing the above from \(k=0\) to \(k=K\) we obtain
This yields \(K \ge \lfloor T_0/h \rfloor \) and, hence, it follows from the construction (3.1) that \(E^h_t\) satisfies UBC with radius \(r_0/2\) for every \(0 \le t \le T_0\). The last claim then follows directly from Lemma 4.6. \(\quad \square \)
5 Higher Regularity
In this section we utilize the short-time UBC from previous section and prove the full regularity of the flat flow solution of (1.1). It is well known that the classical solution for the mean curvature flow is well defined as long as the second fundamental form stays bounded [39]. For the volume preserving flow this is not enough as the flow may develop singularities even if it stays regular [40, 41]. However, if the flow in addition satisfies UBC then these singularities do not occur. In this section we show that the approximative flat flow becomes instantaneously smooth and stays smooth as long as it satisfies UBC. We will prove this via energy estimates.
Our starting point is the formula in Lemma 4.4, which for sets \(E_1\) and \(E_2\) as in the lemma, gives the formula which relates their normals as
Recall that \(\nabla _{\tau _2}\) denotes the tangential gradient on \(\partial E_2\). Assume now further that \(E_2\) is a minimizer of the functional \(\mathcal {F}_h(\ \cdot \, E_1) \) defined in (3.2). We may use the Euler-Lagrange equation (3.4) and have
This identity is simple enough for us to differentiate multiple times and this in turn gives us formula which is the discrete analog of the identity (see e.g. [38, Lemma 3.5])
Let us, for the sake of clarification, show how we obtain the discrete version of (5.2) for \(k =0\) from (5.1), which reads as follows
where the function \(a_1(\cdot )\) and the matrix field \(A_2(\cdot )\) depend smoothly on \(d_{E_1}\), \(\nu _{E_1}\circ \pi _{\partial E_1}\) \(\nu _{E_2}\), \(B_{E_1}\circ \pi _{\partial E_1}\) and \(B_{E_2}\). In particular, since \(E_1\) and \(E_2\) satisfy UBC with radius \(r_0/2\), then \(a_1(\cdot )\) and \(A_2(\cdot )\) are uniformly bounded.
Indeed, by applying the tangential divergence on (5.1) we have
In order to calculate the LHS, we use (2.22), (2.32) and (2.33) to obtain
which holds in the tubular neighborhood \({\mathcal {N}}_{r_0}(\partial E_1)\), where we also used the fact
Again from (5.1) we have
on \(\partial E_2\). Using the above identities and the fact \(B_{E_1} \nu _{E_1}= 0\) on \(\partial E_1\), we have the following equality on \(\partial E_2\)
The equation (5.3) then follows from the previous calculations and from the identity
which is a direct consequence of Lemma 4.4.
We may differentiate the equality (5.3) further and obtain a discrete version of (5.2) for every order k. This will produce several nonlinear error terms which have rather complicated structure. However, by introducing sufficiently efficient notation we are able to identify the structure of these error terms and by using UBC and the interpolation inequality from Proposition 2.2 we are able to reproduce the argument from [22] in the discrete setting. The following proposition is the core of the proof for the higher order regularity.
Proposition 5.1
Assume that \(E_1 \subset \mathbb {R}^{n+1}\) is an open and bounded set, with \(|E_1| = m_0\), which satisfies UBC with radius \(r_0\) and let \(E_2\) be any minimizer of \(\mathcal {F}_h( \ \cdot \, E_1) \) defined in (3.2). There is \(h_0=h_0(n,m_0,r_0)\) such that if \(h \le h_0\) and \(E_1\) is \(C^{2m+3}\)-regular for \(m = 0,1,2, \dots \) then
on \(\partial E_2\) and the error term \(R_{l}\) for \(l = 0,1,2, \dots \) satisfies the estimate
where \(C_l = C_l(l,n,m_0,r_0)\).
We note that so far we have not used any results from differential geometry. In fact, we need the notation from geometry only to prove Proposition 5.1. Therefore, instead of giving the proof of Proposition 5.1, which is technically challenging, we show first how we may use it to obtain the regularity estimate (1.2) in the statement of Theorem 1.1. Here is the main result of this section.
Theorem 5.2
Let \(E_0\) be an open and bounded set, with \(|E_0|=m_0\), and let \((E_t^h)_{t \ge 0}\) be an approximative flat flow starting from \(E_0\) defined in (3.1). For given \(r_0 \in \mathbb {R}_+\) there is \(h_0=h_0(n,m_0,r_0) \in \mathbb {R}_+\) such that if \(h \le h_0\), \(E_t^h\) satisfies UBC with radius \(r_0\) in [0, T] and if \((l+2)h \le T\) for a given \(l \in \mathbb {N}\cup \{0\}\), then we have
for a constant \(C = C(l,n,m_0,r_0,T)\).
Proof
In the proof, C and \(C_m\) denote a positive real number which may change their values but always in a manner that we have the dependencies \(C=C(n,m_0,r_0)\) and \(C_m=C_m(m,n,m_0,r_0, T)\). We use the abbreviation \(E_k = E^h_{kh}\) for \(k=0,1,2,\ldots \)
First, by Proposition 3.1, Lemma 3.2, Remark 3.3 and Theorem 4.7, we find \(h_0=h_0(n,m_0,r_0) >0\) such that if \(h \le h_0\) and \(E_k\) is \(C^{2k+1}\)-regular, bounded set of volume \(m_0\), which satisfies uniform ball condition with radius \(r_0\), then the consequent set \(E_{k+1}\) is \(C^{2k+3}\)-regular, bounded and of volume \(m_0\), with
Moreover, \(E_{k+1}\) satisfies UBC with radius \(r_0/2\) and the projection \(\pi _{\partial E_k}: \partial E_{k+1} \rightarrow \partial E_k\) is injective. We may then prove that, for \(k\ge 1\), \(\pi _{\partial E_k}: \partial E_{k+1} \rightarrow \partial E_k\) is a diffeomorphism with
where the tangential Jacobian \(J_{\tau _{k+1}}{\pi _{\partial E_k}}\) of \(\pi _{\partial E_k}\) on \(\partial E_{k+1}\) is defined in (2.3). Indeed, since \(\partial E_{k+1} \subset {\mathcal {N}}_{r_0/2}(\partial E_k)\), then \(\pi _{\partial E_k}\) is \(C^1\)-regular map on \(\partial E_{k+1}\). Recalling the injectivity of the projection we are remain to prove (5.5). By (2.31) we may write
Thus, it follows from the definition in (2.3) and \(\nabla ^2 d_{E_k} \nabla d_{E_k}=0\) in \({\mathcal {N}}_{r_0}(\partial E_k)\) that for given a point \(x \in \partial E_{k+1}\) there is an orthonormal basis \(v_1, \dots , v_n\) of \(G_x \partial E_{k+1}\) such that
Since \(\partial E_{k+1} \subset {\mathcal {N}}_{r_0/2}(\partial E_k)\), then Lemma 2.9 yields \(\sup _{\partial E_{k+1}} |\nabla ^2 d_{E_k}|_{\textrm{op}} \le C\). Further, since \(E_{k+1}\) satisfies UBC with radius \(r_0/2\), then by Lemma 2.11 and by the previous estimates we deduce
Therefore, by combining the previous observations and shrinking \(h_0\), if needed, we obtain (5.5). Again, by possibly shrinking \(h_0\), we may assume that the implications of Proposition 5.1 hold true for the parameters \(m_0\) and \(r_0/2\).
Let us from now on assume that the sets \(E^h_t\) satisfy UBC with radius \(r_0\) for every \(t \in [0,T]\). Let us denote \(K = \lfloor T/h\rfloor \). Then the previous discussion holds for every \(E_k\) and \(k=0,1,2,\ldots ,K\). For the sake of presentation, we use abbreviations \(\Vert B_{E_k}\Vert _{L^2}=\Vert B_{E_k}\Vert _{L^2(\partial E_k)}\), \( \Vert B_{E_k}\Vert _{H^{2m}} = \Vert B_{E_k}\Vert _{H^{2m}(\partial E_k)}\) etc.
After the initialization, we prove the claim by induction and to this aim we begin by proving the main regularity estimates. We claim that for every \(m =0,1,2, \dots \), with \(m \le K-2\), and every \(k =m+1, m+2, \dots , K\) it holds that
and
We first prove (5.6) and fix m. Recall that for \(k \ge m+1\) the set \(E_k\) is \(C^{2m+3}\)-regular. Therefore, by Proposition 5.1, it holds for every \(k =m+1, m+2, \dots , K\) that
where the remainder term \( R_{2m,k}\) satisfies
Again, since \(E_k\) and \(E_{k+1}\) satisfy UBC with radius \(r_0/2\) and \(|E_k|=m_0=|E_{k+1}|\), then \(\Vert B_{E_k}\Vert _{L^\infty },\Vert B_{E_k}\Vert _{L^\infty } \le C\) by (2.30) and \(P(E_k),P(E_{k+1}) \le C\) by Lemma 2.10. Therefore, we may use Proposition 2.6 and Young’s inequality to deduce that
We also observe that \( \Vert \Delta _{\tau _{k+1}}^m H_{E_{k+1}}\Vert _{L^2}^2 \le C_m \Vert B_{E_{k+1}}\Vert _{H^{2m}}\). Let \(\varepsilon \in (0,1)\) be a number which we will choose later. By using the previous observations, (5.5), Young’s inequality, and integration by parts we estimate as follows:
By choosing \(\varepsilon = (1+C_m)^{-1}/2\), the previous estimate yields
Since \(\Vert B_{E_{k+1}}\Vert _{L^\infty }, P(E_{k+1}) \le C\), we may use Proposition 2.2 to find \(\theta = \theta (m,n) \in (0,1)\) such that
for any \(\varepsilon \in (0,1)\), where the last inequality follows from Young’s inequality and the curvature bound. Thus, by combing the above with (5.9) and (5.8) the estimate (5.6) follows with a suitable choice of \(\varepsilon \).
Let us then prove (5.7). The argument is similar than above and we only point out the main differences. Now Proposition 5.1 gives for every \(k=m+1, m+2, \dots , K\) the formula
where
and, again, by using Proposition 2.6 and Young’s inequality we have estimates
We use the previous observations, the Cauchy-Schwarz inequality, the estimate \(\Vert \nabla _{\tau _{k+1}} \Delta _{\tau _{k+1}}^m H_{E_{k+1}}\Vert _{L^2} \le C_m \Vert B_{E_{k+1}}\Vert _{H^{2m+1}}^2\) and argue as in proving (5.6) to deduce that
Again, Proposition 2.2 implies that there is \(\theta = \theta (m,n) \in (0,1)\) such that
and we may proceed as previously to obtain (5.7).
Let us then prove the claim by induction. To be more precise, under the assumption \(h \le h_0\), we claim that, for every \(l \in \mathbb {N}\cup \{0\}\) it holds that
for \(C_l = C_l(l,n,m_0,r_0,T)\), provided that \( (l+2) h \le T\). Since \(t - lh \le 3 \lfloor t / h\rfloor h -3(l+1)h \) for every \(t \ge (l+2)h\), then by multiplying (5.10) by \(3^l\) and recalling the definition for the approximative solution in (3.1), we obtain the statement of the theorem.
Let us consider first the case \(l =0\). Since \(P(E_k),\Vert B_{E_k}\Vert _{L^\infty }\le C\), then \(\Vert H_{E_k}\Vert _{L^2} \le C\) for every \(k=0,1,\ldots ,K\). By combining this with (5.6) gives us that for every \(k=1,2,\ldots ,K-1\)
We sum over \(k=1,2,\dots , K-1\) and use \(\Vert H_{E_k}\Vert _{L^2} \le C\) as well as \(K h \le T\) to obtain
Thus, we conclude that (5.10) holds in the case \(l=0\).
Let us then assume that (5.10) holds for \(l-1\), where \(l \in \mathbb {N}\). We assume that \((l+2)h\le T\) and prove (5.10) for l. To this aim, we denote \(K' = K - l\) and \(E'_k = E_{k+l}\). Again, let \(\tau _k\) denote the tangential differentiation along \(\partial E'_k\). Thus, the induction assumption reads as
We divide the argument into two cases depending whether l is even or odd.
Let us first assume that l is even and thus is of the form \(l = 2m\) for \(m=1,2,\dots \). By binomial expansion it holds \((k+1)^{2m} - k^{2m} \le 2m(k+1)^{2m-1}\). Therefore, by multiplying (5.6) by \(k^{2m} h^{2m}\) we deduce, for every \(k=0,1,2,\ldots , K'\),
Fix any \(j=2,\ldots K'\). Summing the previous estimate from \(k =0\) to \(k = j-1\) and using the fact \(K'h \le T\) yields
Thus, reordering the previous estimate and using the induction assumption (5.11) gives us
After substituting \(E'_k=E_{k+l}\) and reindexing we have for every \(j=l+2,\ldots ,K\)
Since we have \(\Vert B_{E_k}\Vert _{L^\infty },P(E_k) \le C\) for every \(k \le K\), then by combining the estimates of Proposition 2.6 with the previous estimate and using \(Kh \le T\) we obtain (5.10).
The case when l is odd is similar. In this case, we have \(l = 2m+1\) for some \(m \in \mathbb {N}\cup \{0\}\). Thus, by using (5.7) in the place of (5.6) we may proceed as in the previous case. \(\quad \square \)
Let us then focus on Proposition 5.1. We will begin by proving two technical lemmas which involve high order derivatives of \(d_E\) and \(\pi _{\partial E}\). To overcome the technicalities we adopt the notation where \(A_i\) denotes a generic tensor field, which depends on the distance function, the normal and the second fundamental form in a smooth way, i.e.,
We also adopt here the notation \(S \star T\) to denote a tensor formed by contraction on some indexes of tensors S and T. If the set E satisfies UBC, then the quantities \(d_E, \nu _E\) and \( B_E \circ \pi _{\partial E}\) are uniformly bounded in \(\mathcal {N}_{r/2}(\partial E)\), we may treat \(A_i\) in (5.12) as a bounded coefficient.
It is immediate that it holds for \(x \in \partial E\) and \(u \in C^2(\partial E)\) that
Let us then derive related formulas for points \(x \in \mathcal {N}_{r/2}(\partial E)\) outside \(\partial E\).
Lemma 5.3
Assume \(E \subset \mathbb {R}^{n+1}\), with \(\Sigma = \partial E\), is bounded and \(C^3\)-regular set which satisfies UBC with radius r. Then it holds for \(u \in C^2(\partial E)\) in \( \mathcal {N}_{r/2}(\partial E)\) that
and
where \(A_1, A_2\) are tensor fields as in (5.12). Moreover, if \(\Sigma \) is in addition \(C^{k+2}\)-regular and \(u \in C^k(\Sigma )\) for \(k \in \mathbb {N}\), then for all \(x \in \mathcal {N}_{r/2}(\partial E)\) we may estimate
Here \(\tilde{\nabla }_{\Sigma }\) denotes the covariant derivative on \(\Sigma \).
Proof
Let us denote \({\hat{u}} = u\circ \pi _{\partial E}\) and \(\pi = \pi _{\partial E}\) for short. Since \(\pi \) is projection it holds
for all \(x \in \mathcal {N}_{r/2}(\partial E)\). By differentiating this we obtain
The first claim then follows from (2.31) and from \(\nabla {\hat{u}} \cdot (\nu _E \circ \pi ) = 0\). The second claim follows by differentiating the first and by writing \(\nabla ^2 d_E(x), \nabla ^3 d_E(x)\) and \(\nabla \pi \) in a geometric way by using (2.33) and (2.34).
In order to prove the third claim we observe that we may write the second equality simply as
By differentiating this \((k-2)\)-times and by using (2.12) and (2.34) we deduce that
The claim follows once we show that for all \(y \in \Sigma \) it holds that
which is the opposite estimate as to that of Lemma 2.4.
We argue as in the proof of Lemma 2.4 and assume \(y = 0\), \(\nu _E(0) = e_{n+1}\) and write the surface \(\Sigma \) locally as a graph of f, i.e., \(\Sigma \cap B_r \subset \{ (x',f(x')): x' \in \mathbb {R}^n\}\) and extended f to \(\mathbb {R}^{n+1}\) trivially as \(f(x', x_{n+1}) = f(x')\). We may then write the metric tensor and the Christoffel symbols in coordinates as
Since \(\nu _{E} = \frac{(-\nabla _{\mathbb {R}^n} f,1)}{\sqrt{1 + |\nabla _{\mathbb {R}^n}f |^2}}\) and \(\nabla {\hat{u}} \cdot (\nu _E \circ \pi ) = 0\), we have
Let us denote the lth order differential of the function \(x' \rightarrow {\hat{u}}(x',0)\) as \(\nabla _{\mathbb {R}^n}^l {\hat{u}}\). Then by applying first (5.14) and (2.12), and then (2.14) we deduce that
Denote the local chart given by the coordinate parametrization by \(\Phi \), i.e., \(\Phi ^{-1}(x') = (x', f(x'))\) and note that \({\hat{u}} (\Phi ^{-1}(x')) = u(\Phi ^{-1}(x'))\). Fix an index vector \(\beta = (\beta _1, \dots , \beta _n, 0) \) with \(|\beta | = m\). Then by (2.12) and (2.14) we obtain after straightforward calculations
From here we deduce by an inductive argument that
Finally using the definition of the covariant derivative and the expression of the Christoffel symbols we obtain arguing as in the proof of Lemma 2.4 that
Hence, we have (5.13) and the third claim follows. \(\quad \square \)
Let us from now on assume \(E_1, E_2\subset \mathbb {R}^{n+1}\) are as in Proposition 5.1. We write the equality (5.3) by using the Euler-Lagrange equation (3.4) as
on \( \partial E_2\), where the error function is of the form
Here and in the rest of the section \(A_i(\cdot )\) denotes a tensor field which depends smoothly on \(d_{E_1}, \nu _{E_1} \circ \pi _{\partial E_1}\), \(\nu _{E_2}\), \(B_{E_1} \circ \pi _{\partial E_1}\) and on \(B_{E_2}\). i.e.,
The following lemma is a consequence of Lemma 5.3.
Lemma 5.4
Assume that the sets \(E_1, E_2\subset \mathbb {R}^{n+1}\) are as in Proposition 5.1. Then it holds for \(u \in C^2(\partial E_1)\) on \( \partial E_2 \)
Proof
Let us denote \({\hat{u}} = u\circ \pi _{\partial E_1}\) and \(\pi = \pi _{\partial E_1}\) for short. Recall that we may write the Laplace-Beltrami on \(\partial E_2\) as
where \(\Delta _{\mathbb {R}^{n+1}} {\hat{u}} = \text {Tr}(\nabla ^2 {\hat{u}})\) denotes the Euclidean Laplacian. Recall that \(P_{\partial E_1}= I - \nu _{E_1} \otimes \nu _{E_1}\) stands for the projection on the (geometric) tangent space. We deduce by applying the trace on the second equality in Lemma 5.3, by \(\nabla ^2 d_{E_1} \nabla d_{E_1} = 0\), and by the Euler-Lagrange equation (3.4) that it holds on \(\partial E_2\)
Similarly, we have
We write
and
We then use (4.19) to write \(\nu _{E_2}- \nu _{E_1} \circ \pi \) as
for functions \(a_1\) and \(a_2\) which depend on \(|\nabla _{\tau _2} d_{E_1}(x)|^2\). Therefore we may write (5.20) by the Euler-Lagrange equation (3.4) as
We use the first equality in Lemma 5.3, (4.19) and the Euler-Lagrange equation (3.4) to write on \(\partial E_2\)
The claim then follows from (5.18), (5.19), (5.21) and (5.22). \(\quad \square \)
We may now prove Proposition 5.1.
Proof of Proposition 5.1
We prove only the first equality since the second follows by differentiating the first. We point out that since \(E_1\) is \(C^{2m+3}\)-regular, then by Lemma 3.2 the set \(E_2\) is \(C^{2m+5}\)-regular. In particular, we have the necessary regularity for the proceeding calculations. To that aim we recall that by (5.15) it holds
where
We differentiate (5.23), use Lemma 5.4 and have on \(\partial E_2\)
where
We continue and deduce by an iterative argument that it holds on \(\partial E_2\)
where \(\rho _0\) is defined in (5.16) and \(\rho _{2k}\) for \(k \ge 1\) is
We have thus derived a formula for the error terms in the statement of Proposition 5.1, i.e., we have
We need to estimate the norm \(\Vert R_{2m}\Vert _{L^2(\Sigma _2)}\), where \(\Sigma _2 = \partial E_2\). The idea is that the total amount of derivatives acting on the curvature terms in \(\Delta _{\tau _2}^{m -k} \rho _{2k}\) is for most of the terms at most 2m. The only difference is the second row in the definition of \(\rho _{2k}\), which total amount of derivatives is higher but it has an extra h as a coefficient. Therefore we need to treat this term more carefully.
Recall that the tensor fields \(A_i(\cdot )\) depend on \(d_{E_1}, \nu _{E_1} \circ \pi _{\partial E_1}\), \(\nu _{E_2}\), \(B_{E_1} \circ \pi _{\partial E_1}\) and on \(B_{E_2}\) as stated in (5.17). Denote \(\pi = \pi _{\partial E_1}\) for short. We use repeatedly (2.12), Lemma 2.4 and the last inequality in Lemma 5.3 and obtain after long but straightforward calculations the following pointwise estimate for all \(x \in \partial E_2\):
We use the uniform curvature bounds \(\Vert B_{\Sigma _1}\Vert _{L^\infty }, \Vert B_{\Sigma _2}\Vert _{L^\infty } \le C\) and Proposition 2.3 to estimate
and
We are left with the last term in (5.24). As we already mentioned, this term has different scaling with respect to h. We use the Euler-Lagrange equation (3.4), (4.23) and \(\Vert d_{E_1}\Vert _{L^\infty (\partial E_2)} \le Ch\) from Proposition 3.1 to deduce that
Therefore we have, by Proposition 2.3, that
when \(h \le 1\), and the claim follows. \(\quad \square \)
Let us conclude this section by discussing briefly how we obtain Theorem 1.1 and Corollary 1.2 from the results in Sects. 4 and 5. We obtain first from Lemma 4.6 and from Theorem 4.7 that the approximative flow \((E_t^h)_k\) satisfies UBC with radius \(r_0/2\) for \(t \le T_0\) and we have
Then we use Theorem 5.2 to deduce that for \(t \in [\delta ,T_0]\) the sets \(E_t^h\) are uniformly \(C^3\)-regular when h is small enough. By Ascoli-Arzela theorem we may pass the estimate (5.25) to the limit as \(h \rightarrow 0\) and conclude that the function \(t \mapsto \sup _{s \le t} \Vert S_{E_{s}}\Vert _{L^\infty }\) is locally Lipschitz continuous and satisfies
for almost every \(t \ge 0\) as long as \(\sup _{s \le t} \Vert S_{E_{s}}\Vert _{L^\infty }\) remains bounded. The inequality (5.26) implies that UBC is an open condition in time. To be more precise if the flat flow \((E_t^h)_t\), starting from \(E_0\), satisfies \(\sup _{t \le T} \Vert S_{E_{t}}\Vert _{L^\infty } \le C\), then by (5.26) there is \(\delta >0\) such that
This together with the estimate in Theorem 5.2 implies Theorem 1.1.
The consistency principle follows from the regularity in a rather straightforward way. Indeed, we obtain by the uniform regularity of the approximate flat flow \((E_t^h)_{t\in [0,T]}\) and by the Euler-Lagrange equation (3.4) that the signed distance function satisfies
for \(t \le T\) and for x in a neighborhood of \(\partial E_t\), where f(t) is a bounded function ot time. From here we may conclude that the flat flow satisfies
Since the flat flow preserves the volume then necessarily 
and thus it is a solution to (1.1).
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Acerbi, E., Fusco, N., Morini, M.: Minimality via second variation for a nonlocal isoperimetric problem. Comm. Math. Phys. 322, 515–557, 2013
Allard, W.K.: On the first variation of a Varifold. Annals of Math. 95, 417–491, 1972
Almgren, F., Taylor, J., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Optim. 31, 387–438, 1993
Andrews, B.: Noncollapsing in mean-convex curvature flow. Geom. Topol. 16, 1413–1418, 2012
Ambrosio, L., Dancer, N.: Calculus of variations and partial differential equations: topics on geometrical evolution problems and degree theory. Springer-Verlag, Berlin-Heidelberg (2000)
Aubin, T.: Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin (1999)
Bellettini, G.: Lecture notes on mean curvature flow, barriers and singular perturbations. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), 12. Edizioni della Normale, Pisa, 2013
Bellettini, G., Caselles, V., Chambolle, A., Novaga, M.: The volume preserving crystalline mean curvature flow of convex sets in \(\mathbb{R} ^N\). J. Math. Pure Appl. 92, 499–527, 2009
Brakke, K.A.: The Motion of a Surface by its Mean Curvature. Math. Notes 20, Princeton Univ. Press, Princeton, NJ 1978
Brendle, S.: Two-point functions and their applications in geometry. Bull. Amer. Math. Soc. (N.S.) 51, 581–596 2014
Brendle, S.: A sharp bound for the inscribed radius under mean curvature flow. Invent. Math. 202, 217–237, 2015
Chambolle, A.: An algorithm for mean curvature motion. Interfaces and free Boundaries 6, 195–218, 2004
Chambolle, A., Morini, M., Ponsiglione, M.: Minimizing movements and level set approaches to nonlocal variational geometric flows. Geometric partial differential equations, 93–104, CRM Series, 15, Ed. Norm., Pisa, 2013
Chambolle, A., Morini, M., Ponsiglione, M.: Existence and uniqueness for a crystalline mean curvature flow. Comm. Pure Appl. Math. 70, 1084–1114, 2017
Chen, Y.G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature. Proc. Japan Acad. Ser. A Math. Sci. 65, 207–210 1989
De Gennaro, D., Kubin, A.: Long time behaviour of the discrete volume preserving mean curvature flow in the flat torus. Calc. Var. Partial Differ. Equ. 62 (2023), no. 3, Paper No. 103
De Philippis, G., Laux, T.: Implicit time discretization for the mean curvature flow of mean convex sets. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 21, 911–930, 2020
De Philippis, G., Goldman, M.: A two-point function approach to connectedness of drops in convex potentials, Comm. Anal. Geom. 30 (2022), no. 4, 815–841.
Escher, J., Simonett, G.: The volume preserving mean curvature flow near spheres. Proc. Am. Math. Soc. 126, 2789–2796, 1998
Evans, L.C., Spruck, J.: Motion of level sets by mean curvature I. J. Differential Geom. 33, 635–681, 1991
Evans, L.C., Spruck, J.: Motion of level sets by mean curvature IV. Journal of Geometric Analysis 5(1), 77–114, 1995
Fusco, N., Julin, V., Morini, M.: The Surface Diffusion Flow with Elasticity in Three Dimensions. Arch. Rational. Mech. Anal. 237, 1325–1382, 2020
Fusco, N., Julin, V., Morini, M.: Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane, J. Geom. Anal. 32 (2022), no. 2, Paper No. 53.
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin (2001)
Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17, 255–306, 1982
Hensel, S., Laux, T.: A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-string uniqueness. Preprint 2021
Huisken, G.: A distance comparison principle for evolving curves. Asian J. Math. 2, 127–133, 1998
Huisken, G.: The volume preserving mean curvature flow. J. Rein. Angew. Math 382, 35–48, 1987
Ishii, H., Lions, P.-L.: Viscosity solutions of fully nonlinear secondorder elliptic partial differential equations. J. Differential Equations 83, 26–78, 1990
Julin, V., La Manna, D. A.: A priori estimates for the motion of charged liquid drop: A dynamic approach via free boundary Euler equations. Preprint 2021
Julin, V., Morini, M., Ponsiglione, M., Spadaro, E.: The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions. Preprint 2021
Julin, V., Niinikoski, J.: Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow. Anal. PDE 16, 679–710, 2023
Kim, I., Kwon, D.: Volume preserving mean curvature flow for star-shaped sets. Comm. Partial Diffenretial Equations 45, 414–455, 2020
Laux, T.: Weak-Strong uniqueness for volume-preserving mean curvature flow. Preprint 2022
Lee, J. M.: Riemannian manifolds. An introduction to curvature. Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997
Luckhaus, S., Stürzenhecker, T.: Implicit time discretization for the mean curvature flow equation. Calc. Var. PDEs 3, 253–271, 1995
Maggi, F.: Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory. Cambridge Studies in Advanced Mathematics, 135. Cambridge University Press, Cambridge 2012
Mantegazza, C.: Smooth geometrc evolutions of hypersurfaces. Geom. Funct. Anal. 12, 138–182, 2002
Mantegazza, C.: Lecture notes on Mean Curvature Flow. Progress in Mathematics 290. Birkhäuser/ Springer, Basel 2011
Mayer, U.F.: A singular example for the average mean curvature flow. Experimental Mathematics 10, 103–107, 2001
Mayer, U.F., Simonett, G.: Self-intersections for the surface diffusion and the volume-preserving mean curvature flow. Differential and Integral Equations 13, 1189–1199, 2000
Morini, M., Ponsiglione, M., Spadaro, E.: Long time behaviour of discrete volume preserving mean curvature flows. J. Reine Angew. Math. 784, 27–51, 2022
Mugnai, L., Seis, C., Spadaro, E.: Global solutions to the volume-preserving mean-curvature flow. Calc. Var. Partial. Diff. Eq. 55, Art. 18, 23 pp. 2016
Niinikoski, J.: Volume preserving mean curvature flows near strictly stable sets in flat torus. J. Differential Equations 276, 149–186, 2021
Simon, L.: Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1, 281–326, 1992
Simon, L.: Introduction to Geometric Measure Theory. Tsinghua Lectures 2014
Swartz, D., Yip, N.K.: Convergence of diffusion generated motion to motion by mean curvature. Comm. Partial Differential Equations 42, 1598–1643, 2017
Topping, P.: Relating diameter and mean curvature for submanifolds of Euclidean space. Comment. Math. Helv. 83, 539–546, 2008
Funding
Open Access funding provided by University of Jyväskylä (JYU). V.J. was supported by the Academy of Finland Grant 314227. J.N. was partially supported by ERC-CZ grant LL2105 and the University Centre UNCE/SCI/023 of Charles University.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflicts of interest.
Additional information
Communicated by G. Dal Maso.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Julin, V., Niinikoski, J. Consistency of the Flat Flow Solution to the Volume Preserving Mean Curvature Flow. Arch Rational Mech Anal 248, 1 (2024). https://doi.org/10.1007/s00205-023-01944-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00205-023-01944-y