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

Lemma 10.1 (Poincaré inequality) Pick h : [0, l] → lR n , weakly differentiable,with h(0) = h(l) (so h is periodically extensible); let ĥ = ∫ 1 llh(x) dx;0thensup |h(u) − ĥ| ≤ 1u2∫ l0|h ′ (x)| dx . (10.5)This is proved as Lemma 18 in [35]; it is one of the main ingredients for thefollowing propositions, whose full proofs are in [35].Proposition 10.2 The H j and ˜H j distances are equivalent:√1 + (2π)d ˜Hj≤ d H j ≤2j λ(2π) 2j λd ˜Hjwhereas d H j ≤ d H k for j < k.Proposition 10.3 The H j and ˜H j distances are lower bounded (with appropriateconstants depending on λ) by the Fréchet distance (defined in 7.2).Proposition 10.4 c ↦→ len(c) is Lipschitz in M with H j metric, that is,| len(c 0 ) − len(c 1 )| ≤ d H j (c 0 , c 1 )Theorem 10.5 (Completion of B i w.r.to H 1 ) let d H 1 be the distance inducedby H 1 ; the metric completion of the space of curves is equal to the spaceof all rectifiable curves.Theorem 10.6 (Completion of B i w.r.to H 2 ) Let E(c) := ∫ |D 2 sc| 2 ds bedefined on non-constant smooth curves; then E is locally Lipschitz in w.r.to d H 2.Moreover the completion of C ∞ (S 1 ) according to the metric H 2 is the space ofcurves that admit curvature D 2 sc ∈ L 2 (S 1 ).The hopeful consequence of the above theorem would be a possible solution tothe problem 4.12; it implies that the space of geometric curves B with the H 2Riemannian metric completes onto the usual Hilbert space H 2 . 16 This resultin turn would ease a (possible) proof of existence geodesics.Concluding, it seems that, to have a complete Riemannian manifold ofgeometric (freely) immersed curves, a metric should penalize derivatives of 2ndorder (at least).10.3 Sobolev metrics in shape optimizationWe first present a definition.16 The detailed proof has not yet been written...63