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

Corollary 10.18 In particular, the kernels defined in eqn. (10.10), (10.21) arerelated by˜K λ = 1 L − 1λL 2 KP c ⋆ KcPBy deriving,D s KcP = δ 0 − 1 Lwhere δ 0 is the Dirac’s delta; so we obtain the relationsWe also recall this Lemma.D s ˜Kλ = − 1λL 2 KP cD ss ˜Kλ = − 1λL 2 δ 0 + 1λL 3 .(10.23)Lemma 10.19 (De la Vallée-Poussin) Suppose that f : S 1 → lR n is integrableand satisfies∫k · f ds = 0cfor all k ∈ D c M smooth; then f is almost everywhere equal to a constant.For the proof, see Chapter 13 in [1], or Lemma VIII.1 in [4].We now compute the gradient of the centroid energy (2.9).Proposition 10.20 (Gradient of the centroid-based energy) Let in thefollowing again, for simplicity of notation, c = avg c (c) be the center of mass ofthe curve. LetE(c) = 1 2 |c − v|2 .The ˜H 1 gradient of E is∇˜H1E(c) = c − v + 1λL 2 P cΠ c(Ds c〈(c − v), (c − c)〉 ) . (10.24)Proof. We already computed the Gâteaux differential of E; but this time weprefer to use the first form of eqn. (2.10), so as to write∫DE(c; h) = (c − v) · h + 〈c − v, c − c〉(D s h, D s c) ds .Let f = ∇˜H1E(c) be the ˜H 1 gradient of E. The equalitycDE(c; h) = 〈h, f〉 ˜H1∀hbecomes〈c − v − f, h〉 +∫cD s h · (〈c− v, c − c〉D s c − λL 2 D s f ) ds = 0, ∀h .70