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

• During the flow, the value of the energy changes with rate∂ t E(C) = −‖∇˜H1E(C)‖ 2˜H1 =using (10.37), we can bound= −|C − v| 2 1−λ len(C)∫C2 〈C − v, C − C〉 2 ds|∂ t E(C)| ≤ E(C)(2 + 1/λ)so from Gronwall inequality we obtain that E(C) does not blow up; consequentlyby (10.38), ‖C‖ 0 does not blow up as well.• LetH := D s C〈(C − v), (C − C)〉 ,from the above we obtain that H does not blow up as well, since‖H(t, ·)‖ 0 ≤ len(c 0 ) √ 2E(C) (10.39)• The parameterization of the curves changes according to the lawsothis proves thatand∂ t log(|∂ θ C| 2 ) = 2〈D s ∂ t C, D s C〉 = 2H · D sCλ len(c 0 ) 2|∂ t log(|∂ θ C| 2 )| ≤ 2√ 2E(C)λ len(c 0 )lim supt↘t −lim inft↘t − I(t) > 0sup |∂ θ C(t, θ)| < ∞ .θ(As a consequence, ∂ θ C(t, θ) ≠ 0 at all times: curves will always beimmersed)• So by (10.29)does not blow up as well.‖P C Π C H‖ 1 ≤ (2π + 2)‖C ′ ‖ 0 ‖H‖ 0• Since both ‖C‖ 0 and ‖∂ θ C‖ 0 do not blow up, then ‖C‖ 1 does not blow up.• Decomposing F = −∂ t C = ∇˜H1E(C) (as was done in (10.36)) we write‖ ˜F ‖ 1 =1λ len(c 0 ) 2 ‖P CΠ C H‖ 1and from all above ‖ ˜F ‖ 1 does not blow up; but also|F | = |avg C (C) − v| ≤ √ 2E(C)as well; but then ‖∇˜H1E(C)‖ 1 itself does not blow up, as we wanted toprove.81