3.3 Volume preserving mean curvature flow
Let us consider here the so called volume preserving mean curvature flow with outer boundary contact, which is a direct generalization of the previous Section 3.2. With the same notations as before, we assume the special representation of the evolving hypersurface as a graph from Section 3.1 and linearize the resulting equation. For the stability analysis we can then refer to the last section, where we used methods that are also applicable in this case.
With the same notations as in the section of the mean curvature flow we consider here the problem of finding an evolving hypersurface
Γ = [
t∈[0,T)
{t} ×Γ(t) with Γ(t)⊂Rn+1, (3.36)
as in Definition 2.31, evolving due to the volume preserving mean curvature flow, such that Γ(t) lies in a fixed bounded region Ω ⊂Rn+1 and the boundary∂Γ(t) of each of the hypersurfaces intersects the boundary∂Ω of the fixed region at a right angle.
In formulas, the problem reads as follows. Find Γ as in (3.36), such that
V = H−H in Γ(t) for all t >0,
Γ(t) ⊂ Ω for all t >0,
∂Γ(t) ⊂ ∂Ω for all t >0, n(t)·µ = 0 on∂Γ(t) for all t >0,
Γ(0) = Γ0.
(3.37)
Here V,H,n andµare the normal velocity, the mean curvature, a unit normal of the evolving hypersurface Γ and the outer unit normal to∂Ω as explained in Sections 3.1 and 3.2.
H is the mean value of mean curvature, that is H(t) =
Z
−Γ(t)H dHn. (3.38)
Γ0 is a given starting surface, which lies in Ω and intersects the boundary∂Ω at a right angle.
We observe that stationary surfaces of this flow satisfy H ≡H, so they are hypersurfaces with constant mean curvature, so-called H-hypersurfaces.
With the notations of Section 2.4 for area A(t) and volume V ol(t) of Γ(t) , we can justify the name of the flow in (3.37).
Lemma 3.18. For a solution Γ of the flow (3.37) the following estimates hold true (i) dtdA(t)≤0 and
(ii) dtdV ol(t) = 0.
Therefore the flow is area minimizing and also volume preserving, as the name of the flow already indicated.
CHAPTER 3. EVOLUTION EQUATIONS WITH BOUNDARY CONTACT
Proof. ad (i): With the result of Lemma 2.46 we get d
where the last inequality follows from the Cauchy-Schwarz inequality. In fact, we have Z ad (ii): Again with the result of Lemma 2.46, we get
d
and the proof is finished.
As in the previous Section 3.2 we consider special solutions Γ of (3.37). Recalling the notation, we fix a stationary solution Γ∗ of (3.37) and consider hypersurfaces Γρ(t) given as in Section 3.1 with the help of a function
ρ: [0, T)×Γ∗ −→(−d, d) through a diffeomorphism onto its image
Φρt : Γ∗−→Ω by
Γρ(t) = Φρ(t,Γ∗). For the details we refer again to Section 3.1.
The corresponding equation to (3.12) for ρ on the fixed stationary hypersurface Γ∗ is given through
Here the dependence of the mean value of mean curvatureH on ρ reads as follows H(ρ, t) =
Z
Γρ(t)
− HdHn,
3.3. VOLUME PRESERVING MEAN CURVATURE FLOW
such that this is an additional nonlocal term compared to mean curvature flow.
For the linearization of (3.39) around ρ ≡0, which is our notation for linearization of (3.37) around the given stationary state Γ∗, we can use the results and notation of the previous sec-tion. In particular, we use the linearization of normal velocity, mean curvature and the angle condition. For the mean value of mean curvature, we have the following result.
Lemma 3.19. The linearization of the mean value of mean curvature is given through d
Proof. For fixed t, we use the mapping from the proof of the linearization of mean curvature in Lemma 3.5 in the previous section
Φερt : Γ∗ →Ω , q7→Φερt (q) = Ψ(q, ερ(t, q)),
where εis small. This mapping is a diffeomorphism onto its image and we get evolving hyper-surfaces inεthrough eΓ(ε) = im(Φερ) = Γερ(t) and in particular Γ(0) = Γe ρ≡0(t) = Γ∗. With this notation, we can write the mean value of mean curvature as
H(ερ, t) = notation to describe an evolving hypersurface inε.
For the derivative with respect to εat ε≡0 we calculate with Lemma 2.46 d
where we used the resultVe(0, p) =ρ(t, p) from the proof of Lemma 3.5 from the previous section and the fact, that the mean curvatureH∗ of the stationary surface Γ∗ is a constant.
Furthermore we get with the help of the Transport theorem 2.44 and the formula (2.3) for the
CHAPTER 3. EVOLUTION EQUATIONS WITH BOUNDARY CONTACT
normal time derivative for mean curvature d
Altogether, we get for the linearization of the mean value of mean curvature d
Here we omitted the volume form dHn for reasons of shortness.
So together with the results of the previous section we get for the linearization of (3.39) around the stationary hypersurface Γ∗ represented throughρ≡0 the following equations.
We give a remark concerning a term in the linearization of the mean value of mean curvature.
3.3. VOLUME PRESERVING MEAN CURVATURE FLOW
Remark 3.20. It is also possible to write the termR
Γ∗
A solvability condition for solutions of the linearized problem (3.40) gives here R
Γ∗ρ≡0.
Lemma 3.21. Solutions of the linearized problem (3.40) fulfill Z
Γ∗
ρdHn ≡ 0. (3.41)
Proof. Integrating the first line in (3.40) gives Z t where the left side equals 0 and the right side gives
Z t
which shows the claim.
We introduce the same bilinear form on H1,2(Γ∗) as in the previous chapter I(ρ1, ρ2) =
Due to the solvability condition, we introduce the space V := H1(Γ∗)∩ {ρ|R
Γ∗ρ= 0}. (3.43)
and supply it with theL2-inner product.
In analogy to the previous section, we want to show that the linearized problem (3.40) is the L2-gradient flow of the functionalE(ρ) := 12I(ρ, ρ) defined onV, which means exactly the iden-tity stated in the next lemma.
CHAPTER 3. EVOLUTION EQUATIONS WITH BOUNDARY CONTACT
Lemma 3.22. The time dependent function ρ with values in V is a solution of the linearized equation (3.40) if and only if
(∂tρ(t), ξ)L2 = −∂E(ρ(t))(ξ)
where the last equality follows with the same calculation as in the case of mean curvature flow.
On the other hand, let ρ(t)∈V fulfill the identity
(ρ(t), ξ)L2 = −∂E(ρ(t))(ξ)
Γ∗ξ = 0. Regularity theory for weak solutions of (3.44) leads to ρ∈H2(Γ∗). After integration by parts, we observe