Flat flow solution to the mean curvature flow with volume constraint

Theorem 1.

Assume that E 0 ⊂ R n + 1 is an open and bounded set with finite perimeter and let ( E t h ) t ≥ 0 be an approximative flat flow solution to (1.1) stating from E 0 (see Definition 2.1). Then there exists a family of bounded sets of finite perimeter ( E t ) t ≥ 0 and a subsequence h k → 0 such that

and for every 0 < t < s it holds | E t | = | E 0 | , P ⁢ ( E t ) ≤ P ⁢ ( E 0 ) and

where C depends on the dimension and on E 0 . Moreover, if the initial set E 0 is C 1 , 1 -regular, then any such limit flow ( E t ) t ≥ 0 agrees with the unique classical solution of (1.1) as long as the latter exists.

Theorem 2.

Let ( E t h ) t ≥ 0 be an approximative flat flow solution to (1.1) and let ( E t h k ) t ≥ 0 be the converging subsequence in Theorem 1. Assume further that it holds

Then for n ≤ 6 the flat flow ( E t ) t ≥ 0 is a distributional solution to (1.1) (see Definition 4.5).

Remark 1.1.

Assume that E 0 ⊂ ℝ n + 1 , with n ≤ 2 , is as in Theorem 1 and let ( E t ) t ≥ 0 be a limit flat flow. When n = 1 , the flow E t converges to a union of disjoint balls exponentially fast and when n = 2 the flow converges to a union of disjoint balls up to a possible translation of the components.

Definition 2.1.

Let E 0 ⊂ ℝ n + 1 be an open and bounded set of finite perimeter and fix h ∈ ( 0 , 1 ) . Define the sequence of sets ( E k h ) k = 0 ∞ iteratively as E 0 h = E 0 and E k + 1 h is a minimizer of the problem

Moreover, define an approximative flat flow ( E t h ) t ≥ 0 for (1.1) starting from E 0 as

Proposition 2.2.

Let F ⊂ R n + 1 be an open and bounded set, fix h ∈ ( 0 , 1 ) and let E be a minimizer of (2.2). Then E can be chosen to be open, which topological boundary is C 2 , α -regular up to a relatively closed singular set which Hausdorff dimension is at most n - 7 . The regular part is exactly the reduced boundary ∂ * ⁡ E .

The Euler–Lagrange equation

where λ ∈ R is the Lagrange-multiplier, holds point-wise on ∂ * ⁡ E and in a distributional sense on ∂ ⁡ E . The quadratic form associated with the second variation of the energy is non-negative, i.e., for all φ ∈ H 1 ⁢ ( ∂ * ⁡ E ) with ∫ ∂ * ⁡ E φ ⁢ d H n = 0 it holds

where B E ⁢ ( x ) denotes the second fundamental form of E at x ∈ ∂ * ⁡ E .

Proof.

Since the argument is standard, I will only give the outline. As I already mentioned, the minimizer E is also a minimizer of problem (2.3) for some large constant Λ ~ , which depends on h and on E itself. This implies that the set E is a Λ-minimizer of the perimeter and thus the reduced boundary ∂ * ⁡ E is relatively open, C 1 , α -regular hypersurface and the singular set ∂ ⁡ E ∖ ∂ * ⁡ E has dimension at most n - 7 (see [15]). The C 2 , α -regularity then follows from the Euler–Lagrange equation and from standard Schauder-estimates for elliptic PDEs.

One may obtain the second variation condition (2.5) by using the argument from [1]. Indeed, given a function φ ∈ C 0 2 ⁢ ( ∂ * ⁡ E ) with ∫ ∂ * ⁡ E φ ⁢ d ℋ n = 0 , we may construct a family of diffeomorphisms Φ t such that Φ 0 = id , | Φ t ⁢ ( E ) | = | E | and ∂ ∂ ⁡ t | t = 0 ⁢ Φ t ⁢ ( x ) ⋅ ν E = φ . Then the inequality follows from the minimality of E as

and following the standard calculation of the second variation (see, e.g., [1]). Finally, one obtains (2.5) for all φ ∈ H 1 ⁢ ( ∂ * ⁡ E ) by approximation argument and by the fact that the singular set has zero capacity. ∎

Proposition 3.1.

Let F ⊂ R n + 1 be an open and bounded set of finite perimeter, fix h ∈ ( 0 , 1 ) and let E be a minimizer of (2.2). Then there is a constant c > 0 , which depends on the dimension n, | F | and on P ⁢ ( F ) such that for all r ≤ h and all x ∈ ∂ ⁡ E it holds

and for all r ≤ C 0 ⁢ h , where C 0 ≥ 1 , it holds

where C 1 depends also on C 0 . Moreover, the following estimates hold

Proposition 3.2.

Let F , E ⊂ R n + 1 be as in Proposition 3.1. Then there are constants C ≥ 1 , c > 0 and h 0 > 0 , depending on the dimension, | F | and P ⁢ ( F ) such that E is the ( Λ , r ) -minimizer of the perimeter for Λ = C h , r = c ⁢ h and for h < h 0 . To be more precise, for sets G ⊂ R n + 1 with Δ ⁡ E G ⊂ B c ⁢ h ⁢ ( x 0 ) it holds

Proof.

The argument is standard but I recall it for the reader’s convenience. Let me first show that there is x ∈ E and c ~ > 0 such that for ρ = c ~ ⁢ h it holds B ρ ⁢ ( x ) ⊂ E . Fix ρ and apply the Besicovitch covering theorem to find disjoint balls { B ρ ⁢ ( x i ) } i = 1 N such that x i ∈ E and

I claim that for some i = 1 , 2 , … , N it holds B ρ 2 ⁢ ( x i ) ⊂ E . Indeed, if this is not the case, then Proposition 3.1 implies

for all i. Since the balls are disjoint, one has by the above and by (3.1) that

This is a contradiction when h is small enough.

Fix x 0 and G as in the claim. Note that in general the set G does not have the same measure as E and one needs to modify it to G ~ with | G ~ | = | E | , e.g., by using the argument from [9] as follows. Assume that | G | < | E | (the case | G | > | E | follows from similar argument). Since B ρ ⁢ ( x ) ⊂ E , by decreasing ρ and r if needed, it holds B ρ ⁢ ( x ) ⊂ G . By continuity there is z ∈ ℝ n + 1 such that | z - x 0 | ≥ 2 ⁢ ρ and | G ∪ B ρ ⁢ ( z ) | = | E | . Define G ~ = G ∪ B ρ ⁢ ( z ) . Then by the minimality of E and Proposition 3.1 it holds

Arguing as in [9], one then deduces

and the claim follows as | Δ ⁡ G ~ E | ≤ 2 ⁢ | Δ ⁡ G E | . ∎

Lemma 3.3.

Assume E ⊂ R n + 1 is a set of finite perimeter with distributional mean curvature H E which satisfies ∥ H E ∥ L ∞ ⁢ ( B 2 ⁢ R ⁢ ( x 0 ) ) ≤ Λ . Then for all x ∈ B R ⁢ ( x 0 ) which are on the boundary of E and r ≤ min ⁡ { R , Λ - 1 } it holds

for a dimensional constant c n > 0 .

Lemma 3.4.

Let F ⊂ R n + 1 be an open and bounded set of finite perimeter, fix h ∈ ( 0 , 1 ) and let E be a minimizer of (2.2). Then for all x ∈ ∂ ⁡ E and r ≤ C 0 ⁢ h it holds

for a constant which depends on the dimension and on C 0 > 0 . In, particular it holds P ⁢ ( E , B r ⁢ ( x ) ) ≤ C ⁢ r n .

Proof.

Fix h ∈ ( 0 , 1 ) , x and r > 0 as in the claim and without loss of generality assume that x = 0 . One may also consider only the case | E ∩ B 2 ⁢ r | ≤ | B 2 ⁢ r ∖ E | as the other case is similar. In particular, it holds | E ∩ B 2 ⁢ r | ≤ 1 2 ⁢ | B 2 ⁢ r | . Since

there is ρ ∈ ( r , 2 ⁢ r ) such that

Consider first the set E 1 = E ∖ B ¯ ρ . In order to have a competing set with the volume of E, define ρ ~ ≤ ρ to be a radius such that | B ρ ~ | = | E ∩ B ρ | and define E 2 = E 1 ∪ B ρ ~ . Then it holds by construction that | E 2 | = | E | , ρ ~ < ρ and

By the minimality of E we have

Estimate the perimeter of E 2 using (3.2) and (3.3) as

Use then Δ ⁡ E E 2 ⊂ B 2 ⁢ r , | E | = | E 2 | and the fact that the signed distance function is 1-Lipschitz to estimate

Therefore one obtains by combining the three above inequalities and r ≤ C 0 ⁢ h

Lemma 3.5.

Let F , E and h be as in Proposition 3.1. Then it holds

where the constant C 1 depends on the dimension and on | F | and P ⁢ ( F ) .

Proof.

The proof relies on the second variation inequality in Proposition 2.2. I would like to point out that in the case of the mean curvature flow, when there is no volume constraint, the proof is considerable easier as one could choose constant function in (2.5). In the volume preserving case I will choose a cut-off function for a test function.

To this end, use first [11, Proposition 2.3] (see also [17, Lemma 2.1]) to find a point x 0 ∈ ℝ n + 1 and a radius r ∈ ( c , 1 ) , where c = c ⁢ ( n , | F | , P ⁢ ( F ) ) , such that

Note that the minimality of E yields P ⁢ ( E ) ≤ P ⁢ ( F ) ≤ C . Moreover, by the isoperimetric inequality it holds P ⁢ ( E ) ≥ c n ⁢ | E | n n + 1 = c n ⁢ | F | n n + 1 ≥ c . These estimates are used repeatedly from now on without mentioning. Without loss of generality assume that x 0 = 0 . Choose ρ < r such that | B r ∖ B ρ | = 1 4 ⁢ | B r | . Note that then r - ρ ≥ c n > 0 and

Define first a cut-off function ζ ∈ C 0 1 ⁢ ( ℝ n + 1 ) such that 0 ≤ ζ ≤ 1 , ζ = 1 in B ρ , ζ = 0 outside B r and | ∇ ⁡ ζ | ≤ C n . Choose then φ = ζ - ζ ¯ , where ζ ¯ = ⨍ ∂ * ⁡ E ζ ⁢ d ℋ n , as a test function in (2.5), use | 〈 ∇ ⁡ d ¯ F , ν E 〉 | ≤ 1 and | φ | ≤ 1 , and obtain

Since H E = Trace ⁡ ( B E ) , it holds point-wise on ∂ * ⁡ E

Recall that 0 ≤ ζ ≤ 1 . Moreover, by the isoperimetric inequality and by (3.4) it holds P ⁢ ( E , B ρ ) ≥ c n ⁢ | E ∩ B ρ | n n + 1 ≥ c . Therefore ζ ¯ ≥ c for c = c ⁢ ( n , | F | , P ⁢ ( F ) ) . In particular, it holds | ζ ⁢ ( x ) - ζ ¯ | ≥ c for x ∈ ∂ * ⁡ E ∖ B r . Hence, we have by (3.5),

We repeat the same argument by defining a cut-off function ζ ∈ C 1 ⁢ ( ℝ n + 1 ) as ζ = 0 in B r , ζ = 1 outside B R and | ∇ ⁡ ζ | ≤ C n , where R > r is such that | B R ∖ B r | = 1 4 ⁢ | B r | . Using φ = ζ - ζ ¯ in (2.5) and arguing as above yields

and the claim follows. ∎

Lemma 3.6.

Let F , E and h be as in Proposition 3.1. Then for the Lagrange multiplier in (2.4), i.e.,

it holds

where the constant C 2 depends on the dimension, on | F | and on P ⁢ ( F ) .

Proof.

Let Λ ≥ 0 be such that | λ | = Λ h . Below all the constants depend on n , | F | and P ⁢ ( F ) . I only treat the case when λ is positive as in the negative case the proof is similar. Define the set

I claim that we may choose C ^ > 2 such that it depends on n , | F | , P ⁢ ( F ) and on C 1 from Lemma 3.5 and it holds

Indeed, by Lemma 3.5 and by C ^ > 2 it holds

By choosing C ^ large enough one then obtains ℋ n ⁢ ( ∂ * ⁡ E ∖ Σ ) < P ⁢ ( E ) 2 and (3.7) follows.

By the Besicovitch covering theorem one finds disjoint balls of radius h , denote them by { B h ⁢ ( x i ) } i = 1 N , with x i ∈ Σ such that

By the Euler–Lagrange equation (2.4) and by the definition of the set Σ it holds for all x ∈ ∂ * ⁡ E ∩ B h ⁢ ( x i ) , with x i ∈ Σ , that

Therefore by Lemma 3.3 it holds

On the other hand, applying Lemma 3.4 first with r = h 2 yields

and then with r = h yields h n 2 ≥ c ⁢ P ⁢ ( E , B h ⁢ ( x i ) ) . In conclusion, it holds

for all balls in the cover.

The minimality of E implies

Note that by (3.9), by λ = Λ h and by the Euler–Lagrange equation (2.4) we have for all x ∈ B h ⁢ ( x i ) that

Therefore either Λ ≤ 4 ⁢ C ^ , in which case the claim follows trivially, or

I assume the latter and show that also in this case Λ is bounded. Indeed, by the above discussion the balls B h ⁢ ( x i ) are in the exterior of F. Therefore we estimate by (3.10) and by (3.8) that

Since P ⁢ ( E ) ≥ c n ⁢ | E | n n + 1 = c n ⁢ | F | n n + 1 , the above and (3.11) gives a bound for Λ and the claim follows. ∎

Proof of Proposition 3.1.

By Lemma 3.3, Lemma 3.4, Lemma 3.6 and by the Euler–Lagrange equation (2.4) it is enough to prove

Argue by contradiction and assume that there is x 0 ∈ ∂ * ⁡ E such that

for large Λ ≫ 1 . Without loss of generality assume that x 0 = 0 and consider only the case d ¯ F ⁢ ( x 0 ) > 0 as the case d ¯ F ⁢ ( x 0 ) < 0 is similar.

By the Euler–Lagrange equation (2.4), by Lemma 3.3 and Lemma 3.6 it holds for r 0 = h 2 ⁢ Λ that

Define radii r k = k ⁢ h + r 0 for k = 0 , 1 , … , n + 1 , where n + 1 is the dimension of the ambient space. For every k = 0 , 1 , … , n + 1 choose a cut-off function ζ k ∈ C 0 1 ⁢ ( ℝ n + 1 ) such that 0 ≤ ζ ≤ 1 , ζ k = 1 in B r k , ζ k + 1 = 0 in ℝ n + 1 ∖ B r k + 1 and | ∇ ⁡ ζ k | ≤ 2 h . Choose

where ζ ¯ k = ⨍ ∂ * ⁡ E ζ k ⁢ d ℋ n , as a test function in the second variation condition (2.5), use | 〈 ∇ ⁡ d ¯ F , ν E 〉 | ≤ 1 and (3.6), and obtain

Since ζ k = 0 outside B r k + 1 , one may estimate

Moreover, since 0 ≤ ζ k ≤ 1 , it holds

When Λ is large enough, for all x ∈ ∂ * ⁡ E ∩ B r k + 1 and all k ≤ n + 1 it holds d ¯ F ⁢ ( x ) ≥ Λ 2 ⁢ h . By the Euler–Lagrange equation (2.4), by d ¯ F ⁢ ( x ) ≥ Λ 2 ⁢ h , and by Lemma 3.6 it holds

when Λ is large. Therefore, by using the notation from Lemma 3.6, it holds

when Λ is large. Then one may deduce from (3.7) that P ⁢ ( E , B r k + 1 ) ≤ 1 2 ⁢ P ⁢ ( E ) . This yields

Therefore it holds

Combining the three above estimates with (3.14) yields

For Λ large enough this implies

Use (3.15) ( n + 1 ) -times from k = 0 to k = n , use then (3.13) and recall that r 0 = h 2 ⁢ Λ and obtain finally that

But now since r n + 1 = ( n + 1 ) ⁢ h + r 0 ≤ 2 ⁢ ( n + 1 ) ⁢ h , one obtains from Lemma 3.4 with r = 2 ⁢ ( n + 1 ) ⁢ h that

which contradicts (3.16) when Λ is large. ∎