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

We can eventually estimate the difference |k 1 (τ) − k 2 (τ)| by using e.g. theinequality|A 2 B 2 C 2 − A 1 B 1 C 1 | ≤ |A 1 B 1 | |C 2 − C 1 | + |A 1 C 2 | |B 2 − B 1 | + |B 2 C 2 | |A 2 − A 1 |on the difference of the integrands( 12 − l 2(τ) − l 2 (θ))h 2 (θ) |c ′L} {{ 2 } {{ } }2(θ)|{{ }}B 2 C 2A 2( 1−2 − l 1(τ) − l 1 (θ))L 1} {{ }A 1to obtain that the above term is less or equal than‖h 1 ‖ 0 ‖c ′ 2 − c ′ 1‖ 0 + ‖c ′ 2‖ 0 ‖h 2 − h 0 ‖ 0 + ‖h 2 ‖ 0 ‖c ′ 2‖ 0∣ ∣∣∣ l 2 (θ) − l 2 (τ)L 2h 1 (θ) |c ′} {{ } }1(θ)|{{ }B 1 C 1− l ∣1(θ) − l 1 (τ) ∣∣∣L 1since |A i | ≤ 1/2. In turn (since the formulas defining k 1 (τ), k 2 (τ) the parametersare bound by τ − 2π ≤ θ ≤ τ) then|A 2 − A 1 | =l 2 (τ) − l 2 (θ)∣− l ∣ 1(τ) − l 1 (θ) ∣∣∣ ∫ τ∫ τ∣ θ=|c′ 2(x)| dxθ−|c′ 1(x)| dx ∣∣∣∣≤L 2L 1 ∣ L 2L 1≤ π L 1 + L 2‖c ′ 1 − c ′L 1 L2‖ 0 + π ‖c′ 1‖ 0 + ‖c ′ 2‖ 0|L 1 − L 2 | ≤2 L 1 L 2≤ 4π 2 ‖c′ 1‖ 0 + ‖c ′ 2‖ 0L 1 L 2‖c ′ 1 − c ′ 2‖ 0(by equations (ii) and (i) in lemma 10.25). Summarizing(|k 1 (τ) − k 2 (τ)| ≤ 2π ‖h 1 ‖ 0 + ‖h 2 ‖ 0 ‖c ′ 2‖ 0 4π 2 ‖c′ 1‖ 0 + ‖c ′ )2‖ 0‖c ′ 1 − c ′L 1 L2‖ 0 +2+ 2π‖c ′ 2‖ 0 ‖h 2 − h 0 ‖ 0 .If we derive, k i ′(θ) = h i(θ)|c ′ i (θ)|, so|k ′ 2(θ)−k ′ 2(θ)| ≤ |h 2(θ)| + |h 1 (θ)|2Symmetrizing we prove (10.30).|c ′ 2(θ)−c ′ 1(θ)| + |c′ 2(θ)| + |c ′ 1(θ)||h ′22(θ)−h ′ 1(θ)| .Lemma 10.27 Conversely, let c 1 , c 2 be two C 1 immersed curves, and h 1 , h 2be two differentiable fields; then (by using once again eqn. (i) from the lemma10.25)‖D c1 h 1 −D c2 h 2 ‖ 0 ≤ ‖c 1‖ 1 + ‖c 2 ‖ 12ε 2 ‖h 1 −h 2 ‖ 1 + ‖h 1‖ 1 + ‖h 2 ‖ 12ε 2 ‖c 1 −c 2 ‖ 1 (10.35)where ε = min(inf S 1 |c ′ 1|, inf S 1 |c ′ 2|).76