Metrics of curves in shape optimization and analysis - Andrea Carlo ...
Metrics of curves in shape optimization and analysis - Andrea Carlo ... Metrics of curves in shape optimization and analysis - Andrea Carlo ...
• If the second request is waived, then ‖ · ‖ is a seminorm.• If the third request holds only for λ ≥ 0, then the norm is asymmetric;in this case, it may happen that ‖x‖ ≠ ‖ − x‖.Each (semi/asymmetric)norm ‖ · ‖ defines a (semi/asymmetric)distanceSo a norm induces a topology.d(x, y) := ‖x − y‖.3.2.1 Examples of spaces of functionsWe present some examples and definitions.Definition 3.7 A locally-convex topological vector space E (shortenedas l.c.t.v.s. in the following) is a vector space equipped with a collection ofseminorms ‖ · ‖ k (with k ∈ K an index set); the seminorms induce a topology,such that c n → c iff ‖c n − c‖ k → n 0 for all k; and all vector space operationsare continuous w.r.to this topology.The simplest example of l.c.t.v.s. is obtained when there is only one norm; thisgives raise to two renowned examples of spaces.Definition 3.8 (Banach and Hilbert spaces) • A Banach space is avector space E with a norm ‖ · ‖ defining a distance d(x, y) := ‖x − y‖ suchthat E is metrically complete.• A Hilbert space is a space with an inner product 〈f, g〉, that defines anorm ‖f‖ := √ 〈f, g〉 such that E is metrically complete.(Note that a Hilbert space is also a Banach space).Example 3.9 Let I ⊂ lR k be open; let p ∈ [1, ∞]. A standard example ofBanach space is the L p space of functions f : I → lR n . If p ∈ [1, ∞), the L pspace contains all functions such that |f| p is Lebesgue integrable, and is equippedwith the norm√ ∫‖f‖ L p := p |f(x)| p dx .IFor the case p = ∞, L ∞ contains all Lebesgue measurable functions f : I →lR n such that there is a constant c ≥ 0 for which |f(x)| ≤ c for all x ∈ I \ N,where N is a set of measure zero; and L ∞ is equipped with the norm‖f‖ L ∞ := supess I |f(x)|that is the lowest possible constant c.If p = 2, L 2 is a Hilbert space by inner product∫〈f, g〉 := f(x) · g(x) dx .INote that, in these spaces, by definition, f = g iff the set {f ≠ g} hasLebesgue measure zero.22
3.2.2 Fréchet spaceThe following citations [24] are referred to the first part of Hamilton’s 1982survey on the Nash&Moser theorem.Definition 3.10 A Fréchet space E is a complete Hausdorff metrizable l.c.t.v.s.;where we define that the l.c.t.v.s. E iscomplete when, for any sequence (c n ), the fact thatfor all k implies that c n converges;lim ‖c m − c n ‖ k = 0m,n→∞Hausdorff when, for any given c, if ‖c‖ k = 0 for all k then c = 0;metrizable when there are countably many seminorms associated to E.The reason for the last definition is that, if E is metrizable, then we can define adistance∞∑ 2 −k ‖x − y‖ kd(x, y) :=1 + ‖x − y‖ kk=0that generates the same topology as the family of seminorms ‖ · ‖ k ; and the viceversa is true as well, see [48].3.2.3 More examples of spaces of functionsExample 3.11 Let I ⊂ lR m be open and non empty.• The Banach space C j (I → lR n ), with associated norm‖f‖ := supi≤jsup |f (i) (t)| ;t∈Iwhere f (i) is the i-th derivative.uniformly for all i ≤ j.In this space f n → f iff f (i)n→ f (i)• The Sobolev space H j (I → lR n ), with scalar product∫〈f, g〉 H n := f(t) · g(t) + · · · + f (j) · g (j) dtIwhere f (j) is the j-th derivative 6 .6 The derivatives are computed in distributional sense, and must exists as Lebesgue integrablefunctions.23
- Page 1 and 2: Metrics of curves in shape optimiza
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- Page 5 and 6: • F (c) = F (c ◦ φ) for all cu
- Page 7 and 8: κ > 0HNNHNHκ < 0Figure 1: Example
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- Page 11 and 12: (a) (b) (c) (d) (e)Figure 3: Segmen
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- Page 15 and 16: 2.4.1 Example: geometric heat flowW
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- Page 24 and 25: • The Fréchet space of smooth fu
- Page 26 and 27: Since φ k are homeomorphisms, then
- Page 28 and 29: 3.6.1 Riemann metric, lengthDefinit
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- Page 32 and 33: Example 3.38 Let M = C ∞ ([−1,
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- Page 42 and 43: 6.1.1 Length induced by a distanceI
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- Page 48 and 49: of a small ball from A. The motion
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3.2.2 Fréchet spaceThe follow<strong>in</strong>g citations [24] are referred to the first part <strong>of</strong> Hamilton’s 1982survey on the Nash&Moser theorem.Def<strong>in</strong>ition 3.10 A Fréchet space E is a complete Hausdorff metrizable l.c.t.v.s.;where we def<strong>in</strong>e that the l.c.t.v.s. E iscomplete when, for any sequence (c n ), the fact thatfor all k implies that c n converges;lim ‖c m − c n ‖ k = 0m,n→∞Hausdorff when, for any given c, if ‖c‖ k = 0 for all k then c = 0;metrizable when there are countably many sem<strong>in</strong>orms associated to E.The reason for the last def<strong>in</strong>ition is that, if E is metrizable, then we can def<strong>in</strong>e adistance∞∑ 2 −k ‖x − y‖ kd(x, y) :=1 + ‖x − y‖ kk=0that generates the same topology as the family <strong>of</strong> sem<strong>in</strong>orms ‖ · ‖ k ; <strong>and</strong> the viceversa is true as well, see [48].3.2.3 More examples <strong>of</strong> spaces <strong>of</strong> functionsExample 3.11 Let I ⊂ lR m be open <strong>and</strong> non empty.• The Banach space C j (I → lR n ), with associated norm‖f‖ := supi≤jsup |f (i) (t)| ;t∈Iwhere f (i) is the i-th derivative.uniformly for all i ≤ j.In this space f n → f iff f (i)n→ f (i)• The Sobolev space H j (I → lR n ), with scalar product∫〈f, g〉 H n := f(t) · g(t) + · · · + f (j) · g (j) dtIwhere f (j) is the j-th derivative 6 .6 The derivatives are computed <strong>in</strong> distributional sense, <strong>and</strong> must exists as Lebesgue <strong>in</strong>tegrablefunctions.23
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