Metrics of curves in shape optimization and analysis - Andrea Carlo ...

By using (i) and (ii) from lemma 10.25 once again, we can conclude that‖ ˜f 1 − ˜f 2 ‖ 1 ≤ a 2 ‖c 1 − c 2 ‖ 1for a constant a 2 > 0. The Cauchy–Lipschitz theorem is now invoked to guaranteethat the gradient descent flow does exist and is unique for small times. Let thenC(t, θ) be the solution, that will exist for t ∈ (t − , t + ), the maximal interval.In the following, given any g = g(t, θ) we will simply write ‖g‖ 0 instead ofand similarly‖g(t, ·)‖ 0 = sup |g(t, θ)|θ‖g‖ 1 = ‖g‖ 0 + ‖∂ θ g‖ 0 = sup(|g(t, θ)| + |∂ θ g(t, θ)|) ;θwe will also write len(C) = len(C(t, ·)) for the length of the curve at time t.We want to prove that the maximal interval is actually lR, that is, t + =−t − = ∞. The base and rough idea of the proof is assuming that t + or t − arefinite, and derive a contradiction by showing that ∇E(C) does not blow up whent ↘ t − or t ↗ t + .More precisely, we will show that, if t − > −∞ thenlim sup ‖∇E(C)‖ 1 < ∞ (∗∗)t↘t −this implies that the flow C(t, ·) admits a limit (inside the Banach space C 1 ) ast → t − , and then it may continued (contradicting the fact that t − is the lowesttime limit of existence of the flow). A similar result may be derived when t ↗ t +(but we will omit the proof, that is actually simpler).One key step in showing that (∗∗) holds is to consider the two fundamentalquantitiesI(t) := infθ |∂ θC(t, θ)| , N(t) := ‖C‖ 1and prove thatlim inf I(t) > 0 , lim sup N(t) < ∞when t − > −∞ and t ↘ t − . 26 . We proceed in steps.• By remark 10.30, we know that len(C) is constant (for as long as the flowis defined), and then equal to len(c 0 ).• At any fixed time, by (10.5),|C − C| ≤ len(C)/2 = len(c 0 )/2 (10.37)where C is the center of mass of C(t, ·); and then|C| ≤ |C − C| + |v − C| + |v| ≤ len(c 0 )/2 + √ 2E(C) + |v| (10.38)26 Actually, by tracking the first part of the proof in detail, it possible to prove that all otherconstants a 1 , a 2 , a 3 , a 4 , P, Q may be bounded in terms of these two quantities I(t), N(t)80