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

• The length functional (from C 1 to lR) is Lipschitz, sincelen(c 1 ) − len(c 2 ) ≤ 2π‖c ′ 1 − c ′ 2‖ 0 ;(ii)• if h 1 , h 2 : S 1 → lR n are continuous fields, by (i)|∫h 1 (s) ds −c 1∫h 2 (s) ds|c 2≤ π(‖h 1 ‖ 0 + ‖h 2 ‖ 0 )‖c ′ 1 − c ′ 2‖ 0 ++ π(‖c ′ 1‖ 0 + ‖c ′ 2‖ 0 )‖h 1 − h 2 ‖ 0 ; (iii)• similarly|∫h 1 (s) ds −c 1∫h 2 (s) ds| ≤ A‖c ′ 1 − c ′ 2‖ 0 + ‖h 1 − h 2 ‖ 0c 2(iv)whereA := 2π2 (‖h 1 ‖ 0 + ‖h 2 ‖ 0 )(‖c ′ 1‖ 0 + ‖c ′ 2‖ 0 )len(c 0 ) len(c 1 ). (v)• Let then Π c be the projection operator (as in definition (10.17)); note that(Π c h) ′ = h ′ ; from all above inequalities we obtain that‖Π c1 h 1 − Π c2 h 2 ‖ 0 ≤ A‖c ′ 1 − c ′ 2‖ 0 + 2‖h 1 − h 2 ‖ 0‖Π c1 h 1 − Π c2 h 2 ‖ 1 = ‖Π c1 h 1 − Π c2 h 2 ‖ 0 + ‖(Π c1 h 1 ) ′ − (Π c2 h 2 ) ′ ‖ 0 ≤≤ A‖c ′ 1 − c ′ 2‖ 0 + 2‖h 1 − h 2 ‖ 0 + ‖h ′ 1 − h ′ 2‖ 0≤ A‖c 1 − c 2 ‖ 1 + 2‖h 1 − h 2 ‖ 1 . (vi)We will now prove the local Lipschitz regularity of the operators P c (h), D c (h)(that were defined in 10.14) in both the two variables h and c. Unfortunatelyto prove the theorem 10.29 we will need to also compare the action of P c fordifferent values of c; since P c h is properly defined only when h ∈ D c M (and ingeneral D c1 M ≠ D c2 M), we will actually need to study the composite operatorP c Π c h.Lemma 10.26 Let Π c the projector from T c M to D c M (that we defined ineqn. (10.17)). Let c, c 1 , c 2 be C 1 immersed curves, and h, h 1 , h 2 be continuousfields; then‖P c Π c h‖ 1 ≤ (2π + 2)‖c ′ ‖ 0 ‖h‖ 0 (10.29)and‖P c1 Π c1 h 1 − P c2 Π c2 h 2 ‖ 1 ≤ P ‖c ′ 1 − c ′ 2‖ 0 + (10.30)+ Q ‖h 2 − h 0 ‖ 0whereQ := (1/2 + π)(‖c ′ 2‖ 0 + ‖c ′ 1‖ 0 ) .74