Integral estimates

eγⁱ(x, t) = γⁱ(θⁱ(x), t)these gives a C^2+2α,1+α([0, 1]× [0, T⁰))curvature flow of the initial network eS⁰which satisfies the compatibility conditions of order 2, hence, by the above argument, T⁰ = T and the maps eγⁱandeγⁱ only differ by reparametrization by some maps ϕⁱ ∈ C^2+2α,1+α([0, 1]× [0, T ))with ϕⁱ(x, 0) = x, that is

eγⁱ(x, t) =eγⁱ(ϕⁱ(x, t), t) . It follows that

γⁱ(x, t) =eγⁱ([θⁱ]⁻¹(x), t) =eγⁱ(ϕⁱ([θⁱ]⁻¹(x), t), t) = γⁱ(θⁱ(ϕⁱ([θⁱ]⁻¹(x), t)), t)

which shows that the two flows γⁱ and γⁱ of the initial network S⁰ coincide up to the time–

dependent reparametrizations (x, t)7→ (θⁱ(ϕⁱ([θⁱ]⁻¹(x), t)), t).  An immediate consequence is the following.

COROLLARY3.23. For any initial, regular geometrically smooth networkS⁰ = Sn

i=1σⁱ([0, 1])in a smooth, convex, open set Ω ⊂ R², there exists a geometrically unique solution of Problem (1.5) in the class of maps C^2+2α,1+α([0, 1]× [0, T )) in a maximal positive time interval [0, T ). Moreover, such solution is C^∞.

REMARK3.24. Notice that it follows that any curvature flow as in the hypotheses of the above theorem and corollary is a reparametrization (of class C^2+2α,1+α in the first case and C^∞in the latter) of the special curvature flow given by Theorem 3.7 (which is C^∞under the hypotheses of the corollary, by Theorem 3.17).

We conclude this section stating the following natural open problem, that we already men-tioned, related to geometric uniqueness of the flow.

OPENPROBLEM3.25. Show that for any initial, regular C^2+2αnetworkS⁰ =Sn

i=1σⁱ([0, 1]), with α ∈ (0, 1/2), which is 2–compatible, the solution given by Theorem 3.8 is the geometrically unique curvature flow ofS⁰. That is, the maps γⁱgive the geometrically unique solution of Prob-lem (1.5) in the class of continuous maps γⁱ : [0, 1]× [0, T ) → R²which are of class at least C²in space and C¹in time in [0, 1]× (0, T ).

The difficulty in getting such a uniqueness result is connected to the lack of some sort of (possibly geometric) maximum principle for this evolution problem.

4. Integral estimates

In this section we work out some integral estimates for a special flow by curvature of a smooth regular network. These estimates were previously proved for the case of the special curvature flow of a regular smooth triod with fixed end–points, in [71]. We extend now them to the case of a regular smooth network with “controlled” behavior of its end–points. We advise the reader that when the computations are exactly the same we will refer directly to [71, Section 3], where it is possible to find every detail.

In all this section we will assume that the special flow by curvature is given by a C^∞solution γⁱof system (3.3), that is, there holds

γⁱ_t(x, t) = γ_xxⁱ (x, t)

|γxⁱ(x, t)|²,

(see Remark 1.8 and Definition 1.9 for the case of an initial C²network). Some of the estimates hold also for any smooth flow (the ones where we do not use the special form of the functions λⁱ given by this equation), not necessarily special. To use these estimates for a general smooth flow, because of geometric uniqueness (see Corollary 3.23 and Remark 3.24), one must reparametrize such flow in order it becomes special, then carry back the geometric (invariant by reparametriza-tion) estimates to the original flow.

We will see that such special flow of a regular smooth network with “controlled” end–points exists smooth as long as the curvature stays bounded and none of the lengths of the curves goes to zero (Theorem 4.14).

4. INTEGRAL ESTIMATES 50

We suppose that the special solution maps γⁱ above exist and are of class C^∞ in the time interval [0, T ) and that they describe the flow of a regular C^∞networkS^tin Ω, composed by n curves γⁱ(·, t) : [0, 1] → Ω with m 3–points O¹, O², . . . , O^mand l end–points P¹, P², . . . , P^l. We will assume that either such end–points are fixed or that there exist uniform (in time) constants C_j, for every j∈ N, such that

|∂^jsk(P^r, t)| + |∂s^jλ(P^r, t)| ≤ C^j, (4.1) for every t∈ [0, T ) and r ∈ 1, 2, . . . , l.

The very first computation we are going to show is the evolution in time of the total length of a network under the curvature flow.

PROPOSITION4.1. The time derivative of the measure ds on any curve γⁱof the network is given by the measure (λⁱs− (kⁱ)²) ds. As a consequence, we have

where, with a little abuse of notation, λ(P^r, t)is the tangential velocity at the end–point P^rof the curve of the network getting at such point, for any r∈ {1, 2, . . . , l}.

In particular, if the end–points P^rof the network are fixed during the evolution, we have dL(t)

dt =− Z

St

k²ds ,

thus, in such case, the total length L(t) is decreasing in time and uniformly bounded above by the length of the initial networkS⁰.

PROOF. The formula for the time derivative of the measure ds follows easily by the commu-tation formula (2.1). Then, Adding these relations for all the curves, the contributions of λ^piat every 3–point O^pvanish, by relation (2.6), and the formula of the statement follows. If the end–points are fixed all the terms

λ(P^r, t)are zero and the last formula follows. 

The following notation will be very useful for the next computations in this section.

DEFINITION4.2. We will denote with pσ(∂_s^jλ, ∂^h_sk)a polynomial with constant coefficients in λ, . . . , ∂^jsλand k, . . . , ∂^hsksuch that every monomial it contains is of the form

Moreover, if one of the two arguments of pσdoes not appear, it means that the polynomial does not contain it, for instance, pσ(∂^h_sk)does not contain neither λ nor its derivatives.

We will denote with qσ(∂_t^jλ, ∂_s^hk)a polynomial as before in λ, . . . , ∂t^jλand k, . . . , ∂s^hksuch that all its monomials are of the form

C

4. INTEGRAL ESTIMATES 51

Finally, when we will write pσ(|∂^jsλ|, |∂s^hk|) (or q^σ(|∂t^jλ|, |∂s^hk|)) we will mean a finite sum of terms like

C Yj l=0

|∂s^lλ|^αl· Yh l=0

|∂s^lk|^βl with Xj l=0

(l + 1)α_l+ Xh l=0

(l + 1)β_l= σ,

where C is a positive constant and the exponents αl, βlare non negative real values (analogously for qσ).

Clearly we have pσ(∂_s^jλ, ∂_s^hk)≤ p^σ(|∂s^jλ|, |∂s^hk|).

By means of the commutation rule (2.1), the relations in the next lemma are easily proved by induction (Lemmas 3.7 and 3.8 in [71]).

LEMMA4.3. The following formulas hold for every curve of the evolving networkS^t:

∂t∂_s^jk = ∂^j+2_s k + λ∂_s^j+1k + pj+3(∂_s^jk) for every j∈ N,

∂_s^jk = ∂_t^j/2k + q_j+1(∂^j/2−1_t λ, ∂_s^j−1k) if j ≥ 2 is even,

∂_s^jk = ∂_t^(j−1)/2k_s+ q_j+1(∂_t^(j−3)/2λ, ∂_s^j−1k) if j ≥ 1 is odd,

∂_t∂_s^jλ = ∂_s^j+2λ− λ∂s^j+1λ− 2k∂s^j+1k + p_j+3(∂_s^jλ, ∂_s^jk) for every j∈ N,

∂_s^jλ = ∂_t^j/2λ + p_j+1(∂_s^j−1λ, ∂_s^j−1k) if j ≥ 2 is even,

∂_s^jλ = ∂_t^(j−1)/2λ_s+ p_j+1(∂_s^j−1λ, ∂_s^j−1k) if j ≥ 1 is odd.

REMARK4.4. Notice that, by relations (2.8) at any 3–point O^p of the network there holds

∂_t^jλ^pi = (S∂_t^jK)^pi, that is, the time derivatives of λ^pi are expressible as time derivatives of the functions k^pi. Then, by using repeatedly such relation and the first formula of Lemma 4.3, we can express these latter as space derivatives of k^pi. Hence, we will have the relation

X3 i=1

qσ(∂_t^jλ^pi, ∂_s^hk^pi) 

at the 3–point O^p

= p_σ(∂_s^max{2j,h}K^p) 

at the 3–point O^p

with the meaning that this last polynomial contains also product of derivatives of different k^pi’s, because of the action of the linear operator S.

We will often make use of this identity in the computations in the sequel in the following form, X3

i=1

q_σ(∂_t^jλ^pi, ∂_s^hk^pi)  

at the 3–point O^p

≤ kp^σ(|∂s^max{2j,h}k|)k^L∞.

REMARK4.5. We state the following calculus rules which will be used extensively in the se-quel,

p_α(∂_s^jλ, ∂_s^hk)· p^β(∂_s^lλ, ∂^m_s k) = p_α+β(∂_s^max{j,l}λ, ∂_s^max{h,m}k) , qα(∂_t^jλ, ∂_s^hk)· q^β(∂_t^lλ, ∂^m_s k) = qα+β(∂_t^max{j,l}λ, ∂^max{h,m}_s k) .

We already saw that the time derivatives of k and λ can be expressed in terms of space derivatives of k at any 3–point, the same holds for the space derivatives of λ, arguing by induction using the last two formulas in Lemma 4.3. Hence, it follows that

∂_s^lpα(∂^j_sλ, ∂_s^hk) = pα+l(∂_s^j+lλ, ∂_s^h+lk) , ∂_t^lpα(∂_s^jλ, ∂_s^hk) = pα+2l(∂_s^j+2lλ, ∂_s^h+2lk)

∂_t^lq_α(∂_t^jλ, ∂_s^hk) = q_α+2l(∂_t^j+lλ, ∂_s^h+2lk) , q_α(∂^j_tλ, ∂_s^hk) = p_α(∂_s^2jλ, ∂max{h,2j−1}

s k) .

We are now ready to compute, for j∈ N,

4. INTEGRAL ESTIMATES 52

where we integrated by parts the first term on the second line and we estimated the contributions given by the end–points P^rby means of assumption (4.1).

In the case that we consider the end–points P¹, P², . . . , P^lto be fixed, we can assume that the terms CjC_j+1are all zero in the above conclusion, by the following lemma.

LEMMA4.6. If the end–points P^rof the network are fixed, then there holds ∂^jsk = ∂_s^jλ = 0, for every even j∈ N.

PROOF. The first case j = 0 simply follows from the fact that the velocity v = λτ + kν is always zero at the fixed end–points P^r.

We argue by induction, we suppose that for every even natural l≤ j − 2 we have ∂s^lk = ∂_s^lλ = 0, then, by using the first equation in Lemma 4.3, we get

∂_s^jk = ∂t∂^j−2_s k− λ∂s^j−1k− p^j+1(∂_s^j−2k) least one of the terms of this sum has to be odd, hence at least for one index l, the product (l +1)αl

is odd. It follows that at least for one even l the exponent αlis nonzero. Hence, at least one even derivatives is present in every monomial of pj+1(∂_s^j−2k), which contains only derivatives up to the order (j− 2).

Again, by the inductive hypothesis we then conclude that at the end–points ∂_s^jk = 0.

We can deal with λ similarly, by means of the relations in Lemma 4.3.  In the very special case j = 0 we get explicitly

d

where the two constants C0and C1come from assumption (4.1).

Then, recalling relation (2.9), we haveP3

i=1k^pik^pi_s + λ^pi|k^pi|²

hence, we lowered the maximum order of the space derivatives of the curvature in the 3–point terms, particular now it is lower than the one of the “nice” negative integral.

4. INTEGRAL ESTIMATES 53

As we have just seen for the case j = 0, also for the general case we want to simplify the term P3

i=12∂_s^jk^pi∂_s^j+1k^pi+ λ^pi|∂s^jk^pi|²at the 3–point O^p, in order to control it.

Using formulas in Lemma 4.3, we have (see [71, pp. 258–259], for details) 2∂_s^jk ∂_s^j+1k + λ|∂s^jk|²

Now, the key tool to estimateR

Stp2j+4(∂_s^jk) dsandP3

i=1q2j+3(∂_t^j/2λ^pi, ∂_s^jk^pi)at the 3–point O^pwill be the following Gagliardo–Nirenberg interpolation inequalities (see [77, Section 3], pp. 257–263).

PROPOSITION4.7. Let γ be a C^∞, regular curve inR²with finite length L. If u is a C^∞function

and the constants Cn,m,pand Bn,m,pare independent of γ. In particular, if p = +∞,

k∂sⁿukL^∞ ≤ C^n,mk∂s^muk^σL²kuk^1−σL² +Bn,m

L^mσkukL² with σ = n + 1/2

m . (4.6)

After estimating the integral of every monomial of p2j+4(∂_s^jk)by means of the H ¨older in-equality, one uses the Gagliardo–Nirenberg estimates on the result, concluding that

Z

where the constant C depends only on j∈ N and the lengths of the curves of the network (see [71, pp. 260–262], for details).

Any termP3

i=1q2j+3(∂_t^j/2λ^pi, ∂_s^jk^pi)

at the 3–point O^pcan be estimated similarly.

4. INTEGRAL ESTIMATES 54

Hence, for every even j≥ 2 we can finally write d

Recalling the computation in the special case j = 0, this argument gives the same final esti-mate without the contributions coming from the 3–points:



Integrating (4.7) in time on [0, t] and estimating we get Z where in the last passage we used Remark 4.4. The constant C depends only on j∈ N and on the networkS⁰.

Interpolating again by means of inequalities (4.6), one gets

kp^2j+1(|∂s^j−1k|)k^L∞ ≤ 1/2k∂s^jkk²L²+ Ckkk^4j+2L² . Hence, putting all together, for every even j∈ N, we conclude

Z Passing from integral to L^∞estimates, by using inequalities (4.6), we have the following propo-sition.

PROPOSITION4.8. If assumption (4.1) holds, the lengths of all the curves are uniformly positively bounded from below and the L²norm of k is uniformly bounded on [0, T ), then the curvature ofS^tand all its space derivatives are uniformly bounded in the same time interval by some constants depending only on the L²integrals of the space derivatives of k on the initial networkS⁰.

By using the relations in Lemma 4.3 , one then gets also estimates for every time and space derivatives of λ which finally imply estimates on all the derivatives of the maps γⁱ, stated in the next proposition (see [71, pp. 263–266] for details).

PROPOSITION4.9. IfS^tis a C^∞special evolution of the initial networkS⁰ =Sn

i=1σⁱ, satisfying assumption (4.1), such that the lengths of the n curves are uniformly bounded away from zero and the L² norm of the curvature is uniformly bounded by some constants in the time interval [0, T ), then

• all the derivatives in space and time of k and λ are uniformly bounded in [0, 1] × [0, T ),

• all the derivatives in space and time of the curves γⁱ(x, t)are uniformly bounded in [0, 1]×[0, T ),

4. INTEGRAL ESTIMATES 55

• the quantities |γⁱx(x, t)| are uniformly bounded from above and away from zero in [0, 1] × [0, T ).

All the bounds depend only on the uniform controls on the L²norm of k, on the lengths of the curves of the network from below, on the constants Cj in assumption (4.1), on the L^∞norms of the derivatives of the curves σⁱ and on the bound from above and below on|σxⁱ(x, t)|, for the curves describing the initial networkS⁰.

Before proceeding with another set of estimates we want to stress a particular case of (4.7) (we consider a single triod with moving end-points and j = 0) putting it into a separate lemma.

This result crucial in proving Theorem 8.20.

LEMMA4.10. Let F :T × [0, T ) → R², with T <∞, be a triod moving by curvature with moving end–points Qⁱ: [0, T )→ Ω such that the lengths of the three curves are uniformly bounded from below away from zero by L > 0.

Then, for some constants C > 0, eC > 0, independent of the triod, the following estimate holds:

d PROOF. We start writing explicitly Equation (4.2) for j = 0 and for a network with only one triple junction with moving end–points: where we applied the “orthogonality” relation 2.9.

Letting L to be the minimum of the length of the three curves of the triod, by Proposition 4.7 (applied to u = k and having set p = 4, n = 0, m = 1, σ = 1/4) and Peter–Paul inequality, for any ε > 0 we have the interpolation estimate

Z

and again by Peter–Paul inequality, we obtain kkk³L^∞ ≤ ε where the constant C3depend on ε.

Substituting in the equation above, after taking ε < 1, we get the thesis. 

4. INTEGRAL ESTIMATES 56

Now, we work out a second set of estimates where everything is controlled – still under the assumption (4.1) – only by the L² norm of the curvature and the inverses of the lengths of the curves at time zero.

As before we consider the C^∞special curvature flowS^tof a smooth networkS⁰in the time interval [0, T ), composed by n curves γⁱ(·, t) : [0, 1] → Ω with m 3–points O¹, O², . . . , O^mand l end–points P¹, P², . . . , P^l, satisfying assumption (4.1).

PROPOSITION 4.11. For every M > 0 there exists a time TM ∈ (0, T ), depending only on the structure of the network and on the constants C0and C1in assumption (4.1), such that if the square of the L²norm of the curvature and the inverses of the lengths of the curves ofS⁰are bounded by M , then the square of the L²norm of k and the inverses of the lengths of the curves ofS^tare smaller than 2(n+1)M +1, for every time t∈ [0, T^M].

PROOF. The evolution equations for the lengths of the n curves are given by dLⁱ(t)

dt = λⁱ(1, t)− λⁱ(0, t)− Z

γⁱ(·,t)

k²ds , then, recalling computation (4.4), we have

d where we used Young inequality in the last passage.

Interpolating as before (and applying again Young inequality) but keeping now in evidence the terms depending on Lⁱin inequalities (4.5), we obtain

d

with a constant C depending only on the structure of the network and on the constants C0and C1in assumption (4.1).

4. INTEGRAL ESTIMATES 57

This means that the positive function f (t) = R

Stk²ds +Pn i=1

1

Lⁱ(t) + 1satisfies the differential inequality f⁰≤ Cf³, hence, after integration

f²(t)≤ f²(0)

By means of this proposition we can strengthen the conclusion of Proposition 4.9.

COROLLARY4.12. In the hypothesis of the previous proposition, in the time interval [0, TM]all the bounds in Proposition 4.9 depend only on the L²norm of k onS⁰, on the constants Cjin assumption (4.1), on the L^∞norms of the derivatives of the curves σⁱ, on the bound from above and below on|σxⁱ(x, t)| and on the lengths of the curves of the initial networkS⁰.

From now on we assume that the L²norm of the curvature and the inverses of the lengths of the curves are bounded in the interval [0, TM].

Considering j∈ N even, if we differentiate the function Z

St

k²+ tk_s²+t²k²_ss

2! +· · · +t^j|∂s^jk|² j! ds ,

and we estimate with interpolation inequalities as before (see [71, pp. 268–269], for details), we obtain of the curvature, the constants in assumption (4.1) and the inverses of the lengths of the n curves ofS⁰.

We proceed as we did before for the computation of_dt^d R

St|∂^jsk|²ds. First we deal with the last line,

X3

4. INTEGRAL ESTIMATES 58

By formulas in Lemma 4.3 and by Remark 4.4, for any termP3

i=1t^h−1∂_s^h−1kⁱ∂_s^hkⁱ (see [71, p. 270], for details).

The term t^h−1kp^2h+1(|∂s^h−1k|)k^L∞ is controlled as before by a small fraction of t^h−1R

St|∂^hsk|²ds and a possibly large multiple of t^h−1times some power of the L²norm of k (which is bounded), whereas t^h−1k∂s^hkk^L∞kp^h(|∂s^h−2k|)k^L∞is the critical term.

We apply this argument for every even h from 2 to j, choosing accurately small values εj. Hence, we can continue estimate (4.9) as follows,

d

Integrating this inequality in time on [0, t] with t≤ T^M and taking into account Remark 4.4, we get Z Now we absorb all the polynomial terms, after interpolating each one of them between the cor-responding “good” integral in the left member and some power of the L²norm of k, as we did in showing Proposition 4.8, hence we finally obtain for every even j∈ N,

Z

St

k²+ tk²_s+t²k²_ss

2! +· · · +t^j|∂s^jk|²

j! ds≤ C^j

4. INTEGRAL ESTIMATES 59

with t∈ [0, T^M]and a constant Cjdepending only on the constants in assumption (4.1) and the bounds onR

S0k²dsand on the inverses of the lengths of the curves of the initial networkS⁰. This family of inequalities clearly implies

Z

St

|∂s^jk|²ds≤Cjj!

t^j for every even j∈ N. (4.10)

Then, passing as before from integral to L^∞estimates by means of inequalities (4.6), we have the following proposition.

PROPOSITION4.13. For every µ > 0 the curvature and all its space derivatives ofS^tare uniformly bounded in the time interval [µ, TM](where TMis given by Proposition 4.11) by some constants depending only on µ, the constants in assumption (4.1) and the bounds onR

S0k²dsand on the inverses of the lengths of the curves of the initial networkS⁰.

By means of these a priori estimates we can now work out some results about the smooth flow of an initial regular geometrically smooth networkS⁰. Notice that these are examples of how to use the previous estimates on special smooth flows in order to get conclusion on general flows or even only C^∞flows, as we mentioned at the beginning of this section.

THEOREM 4.14. If [0, T ) is the maximal time interval of existence of a C^∞ curvature flow of an initial geometrically smooth networkS⁰, with T < +∞, then

(1) either the inferior limit of the length of at least one curve ofS^tgoes to zero when t→ T , (2) or lim supt→T

R

Stk²ds = +∞.

Moreover, if the lengths of the n curves are uniformly positively bounded from below, then this superior limit is actually a limit and there exists a positive constant C such that

Z

St

k²ds≥ C

√T− t, for every t∈ [0, T ).

PROOF. We can C^∞reparametrize the flowS^tin order that it becomes a special smooth flow eS^tin [0, T ).

If the lengths of the curves ofS^t are uniformly bounded away from zero and the L² norm of kis bounded, the same holds for the networks eS^t, then, by Proposition 4.9 and Ascoli–Arzel`a Theorem, the network eS^tconverges in C^∞ to a smooth network eS^T as t → T . Then, applying Theorem 3.18 to eS^T we could restart the flow obtaining a C^∞special curvature flow in a longer time interval. Reparametrizing back this last flow, we get a C^∞“extension” in time of the flow S^t, hence contradicting the maximality of the interval [0, T ).

Now, considering again the flow eS^t, by means of differential inequality (4.8), we have d which, after integration between t, r∈ [0, T ) with t < r, gives

 1

4. INTEGRAL ESTIMATES 60

for some positive constant C and limt→T

R

eStk²ds = +∞.

By the invariance of the curvature by reparametrization, this last estimate implies the same

esti-mate for the flowS^t. 

This theorem obviously implies the following corollary.

COROLLARY 4.15. If [0, T ), with T < +∞, is the maximal time interval of existence of a C^∞ curvature flow of an initial geometrically smooth networkS⁰and the lengths of the curves are uniformly bounded away from zero, then

max

St

k²≥ C

√T− t → +∞ , (4.11)

as t→ T .

REMARK4.16. In the case of the evolution γtof a single closed curve in the plane there exists a constant C > 0 such that if at time T > 0 a singularity develops, then

max

γt

k²≥ C T− t for every t∈ [0, T ) (see [48]).

If this lower bound on the rate of blowing up of the curvature (which is clearly stronger than the one in inequality (4.11)) holds also in the case of the evolution of a network is an open problem (even if the network is a triod).

We conclude this section with following estimate from below on the maximal time of smooth existence.

PROPOSITION4.17. For every M > 0 there exists a positive time TM such that if the L²norm of the curvature and the inverses of the lengths of the geometrically smooth networkS⁰are bounded by M , then the maximal time of existence T > 0 of a C^∞curvature flow ofS⁰is larger than TM.

PROOF. As before, considering again the reparametrized special curvature flow eS^t, by Propo-sition 4.11 in the interval [0, min{T^M, T}) the L²norm of ekand the inverses of the lengths of the curves of eS^tare bounded by 2M²+ 6M.

Then, by Theorem 4.14, the value min{T^M, T} cannot coincide with the maximal time of existence

of eS^t(hence ofS^t), so it must be T > TM. 

The last part of the section is devoted to estimate the L^∞–norm of the curvature of the net-work trying to avoid as much as possible interpolation inequalities that introduce in the estimates coefficients depending on the length of the curves of the network.

LEMMA4.18. Let Ω be a convex open regular set andS⁰a tree with end–points P¹, P², . . . , P^l(not necessarily fixed during its motion) on ∂Ω. LetS^tbe a smooth evolution by curvature for t∈ [0, T ) of the networkS⁰such that the square of the curvature at the end–points ofS^tis uniformly bounded in time by

LEMMA4.18. Let Ω be a convex open regular set andS⁰a tree with end–points P¹, P², . . . , P^l(not necessarily fixed during its motion) on ∂Ω. LetS^tbe a smooth evolution by curvature for t∈ [0, T ) of the networkS⁰such that the square of the curvature at the end–points ofS^tis uniformly bounded in time by

In document Minimal and Evolving Networks (Page 49-65)