Measuring families of curves by approximation modulus Jan MAL Y a - - PowerPoint PPT Presentation

Measuring families of curves by approximation modulus

Jan MAL´ Y

a joint work with Olli Martio and Vendula Honzlov´ a Exnerov´ a

Faculty of Mathematics and Physics, Charles University, Prague

Geometric Measure Theory Mathematics Institute, University of Warwick July 10-14, 2017

Jan Mal´ y (Prague) Approximation modulus 1 / 17

Modulus

Modulus is an outer measure on the class of curves, it is efficient for description of small families of curves. It has been introduced by Ahlfors and Beurling 1950, its theory has been developed by Fuglede 1957 and widely exploited in the quasiconformaly theory and its generalizations by the Finnish school and others.

Jan Mal´ y (Prague) Approximation modulus 2 / 17

Modulus

Modulus is an outer measure on the class of curves, it is efficient for description of small families of curves. It has been introduced by Ahlfors and Beurling 1950, its theory has been developed by Fuglede 1957 and widely exploited in the quasiconformaly theory and its generalizations by the Finnish school and others. Seminal results: Any Sobolev function is absolutely continuous along a.e. curve (Fuglede 1957).

Jan Mal´ y (Prague) Approximation modulus 2 / 17

Modulus

Modulus is an outer measure on the class of curves, it is efficient for description of small families of curves. It has been introduced by Ahlfors and Beurling 1950, its theory has been developed by Fuglede 1957 and widely exploited in the quasiconformaly theory and its generalizations by the Finnish school and others. Seminal results: Any Sobolev function is absolutely continuous along a.e. curve (Fuglede 1957). Modulus of the family of all curves which reaches E from the boundary of Ω is the capacity of E in Ω (Ziemer 1969).

Jan Mal´ y (Prague) Approximation modulus 2 / 17

Modulus

Modulus is an outer measure on the class of curves, it is efficient for description of small families of curves. It has been introduced by Ahlfors and Beurling 1950, its theory has been developed by Fuglede 1957 and widely exploited in the quasiconformaly theory and its generalizations by the Finnish school and others. Seminal results: Any Sobolev function is absolutely continuous along a.e. curve (Fuglede 1957). Modulus of the family of all curves which reaches E from the boundary of Ω is the capacity of E in Ω (Ziemer 1969). Recent alternative: Probability measures on a space of curves (Ambrosio, Gigli and Savar´ e 2013, 2014). The two approaches are mutually dual (Ambrosio, Di Marino and Savar´ e 2015).

Jan Mal´ y (Prague) Approximation modulus 2 / 17

Modulus

Modulus is an outer measure on the class of curves, it is efficient for description of small families of curves. It has been introduced by Ahlfors and Beurling 1950, its theory has been developed by Fuglede 1957 and widely exploited in the quasiconformaly theory and its generalizations by the Finnish school and others. Seminal results: Any Sobolev function is absolutely continuous along a.e. curve (Fuglede 1957). Modulus of the family of all curves which reaches E from the boundary of Ω is the capacity of E in Ω (Ziemer 1969). Recent alternative: Probability measures on a space of curves (Ambrosio, Gigli and Savar´ e 2013, 2014). The two approaches are mutually dual (Ambrosio, Di Marino and Savar´ e 2015). A similar object: Alberti representation - invented for another purposes.

Jan Mal´ y (Prague) Approximation modulus 2 / 17

Definition

Let Γ be a family of nowhere constant rectifiable parametric curves into a metric measure space (X, | · − · |, m). We say that a function ρ: X → [0, ∞] is an admissible function for Γ if

• γ

ρ ds ≥ 1 for each γ ∈ Γ. Let p ∈ [1, ∞). We define the Lp,q-modulus of Γ as MLp,q(Γ) = inf

• ρp

Lp,q : ρ is admissible for Γ

• .

We simplify the symbol MLp,p as Mp.

Jan Mal´ y (Prague) Approximation modulus 3 / 17

Definition

Let Γ be a family of nowhere constant rectifiable parametric curves into a metric measure space (X, | · − · |, m). We say that a function ρ: X → [0, ∞] is an admissible function for Γ if

• γ

ρ ds ≥ 1 for each γ ∈ Γ. Let p ∈ [1, ∞). We define the Lp,q-modulus of Γ as MLp,q(Γ) = inf

• ρp

Lp,q : ρ is admissible for Γ

• .

We simplify the symbol MLp,p as Mp. Recall: f Lp,q =

• p

∞ αq−1m({f > α})

q p dα

1

q ;

in particular f Lp,p = f Lp.

Jan Mal´ y (Prague) Approximation modulus 3 / 17

Approximation modulus

Definition (Martio 2016)

Let Γ be a family of nowhere constant rectifiable parametric curves into a metric measure space (X, | · − · |, m). We say that a sequence of functions (ρj)j, ρj : X → [0, ∞] is an admissible sequence for Γ if lim inf

j

• γ

ρj ds ≥ 1 for each γ ∈ Γ. Let p ∈ [1, ∞). We define the Lp,q-approximation modulus of Γ as AMLp,q(Γ) = inf

• lim inf

j

ρjp

Lp,q : (ρj)j is admissible for Γ

• .

We simplify the symbols AMLp,p as AMp and AM1 as AM.

Jan Mal´ y (Prague) Approximation modulus 4 / 17

More generally, modulus and approximation modulus can be investigated in the framework of Banach function spaces; in this context, a remarkable

• bservation is the following

Jan Mal´ y (Prague) Approximation modulus 5 / 17

More generally, modulus and approximation modulus can be investigated in the framework of Banach function spaces; in this context, a remarkable

• bservation is the following

Theorem (HEMM)

Let F be a reflexive Banach function space. Then the MF-modulus and the AMF-modulus are the same.

Jan Mal´ y (Prague) Approximation modulus 5 / 17

More generally, modulus and approximation modulus can be investigated in the framework of Banach function spaces; in this context, a remarkable

• bservation is the following

Theorem (HEMM)

Let F be a reflexive Banach function space. Then the MF-modulus and the AMF-modulus are the same. We focus our attention to Lorentz spaces, as the AMLp,1 moduli fit to measuring families of curves related to (n−p)-dimensional sets.

Jan Mal´ y (Prague) Approximation modulus 5 / 17

More generally, modulus and approximation modulus can be investigated in the framework of Banach function spaces; in this context, a remarkable

• bservation is the following

Theorem (HEMM)

Let F be a reflexive Banach function space. Then the MF-modulus and the AMF-modulus are the same. We focus our attention to Lorentz spaces, as the AMLp,1 moduli fit to measuring families of curves related to (n−p)-dimensional sets.

Remark

The AM-modulus has been originally introduced by Martio 2016 et 2016 as a modulus corresponding to BV spaces. Actually, he developed an alternative to Miranda BV-spaces on metric measure spaces. (Martio’s spaces are bigger; it is not investigated whether they are equivalent.) AMF-moduli for other function spaces were investigated later by Honzlov´ a Exnerov´ a, M. and Martio.

Jan Mal´ y (Prague) Approximation modulus 5 / 17

Theorem (Martio)

Let u ∈ BV (Rn) (a precise representative). Then for AM1 almost every Lipschitz curve γ, the composition u ◦ γ is of bounded variation.

Jan Mal´ y (Prague) Approximation modulus 6 / 17

Theorem (Martio)

Let u ∈ BV (Rn) (a precise representative). Then for AM1 almost every Lipschitz curve γ, the composition u ◦ γ is of bounded variation. The result above fails if we replace AM1 by M1.

Jan Mal´ y (Prague) Approximation modulus 6 / 17

Theorem (Martio)

Let u ∈ BV (Rn) (a precise representative). Then for AM1 almost every Lipschitz curve γ, the composition u ◦ γ is of bounded variation. The result above fails if we replace AM1 by M1.

Example (HEMM)

Let u be a characteristic function of the unit cube. Then the M1 modulus

• f the family of Lipschitz curves

Γ = {γ : u ◦ γ fails to be of bounded variation} is infinity.

Jan Mal´ y (Prague) Approximation modulus 6 / 17

Estimates of moduli of concrete curve families

In the subsequent results, the size of sets in consideration is measured by the lower Minkowski content, or by the Dirichlet-Lorentz capacity (like the Newtonian-Lorentz capacity, but without the lower order term).

Jan Mal´ y (Prague) Approximation modulus 7 / 17

Estimates of moduli of concrete curve families

In the subsequent results, the size of sets in consideration is measured by the lower Minkowski content, or by the Dirichlet-Lorentz capacity (like the Newtonian-Lorentz capacity, but without the lower order term). For simplicity, we can imagine that E is a nonempty part of a k-dimensional C 1-surface in Rn, where k ∈ {0, 1, . . . , n − 1}, p = n − k, and E is compact (for the upper estimate) and relatively open (for the lower estimate).

Jan Mal´ y (Prague) Approximation modulus 7 / 17

We suppose that the measure m on X is doubling.

Definition

Let E ⊂ X. Define Γ(E) = {γ : γ meets E}, Γ∞(E) = {γ : γ meets E infinitely times}, Γτ(E) =

• γ : γ(0) ∈ E, limt→0+

d(γ(t),E) t

= 0

• (“tangential”).

Jan Mal´ y (Prague) Approximation modulus 8 / 17

We suppose that the measure m on X is doubling.

Definition

Let E ⊂ X. Define Γ(E) = {γ : γ meets E}, Γ∞(E) = {γ : γ meets E infinitely times}, Γτ(E) =

• γ : γ(0) ∈ E, limt→0+

d(γ(t),E) t

= 0

• (“tangential”).

Theorem (HEMM)

Let E ⊂ X has a finite lower Minkowski content of co-dimension p ≥ 1. Then AMLp,1(Γ(E)) < ∞ and AMLp,1(Γ∞(E)) = AMLp,1(Γτ(E)) = 0.

Jan Mal´ y (Prague) Approximation modulus 8 / 17

We suppose that the measure m on X is doubling.

Definition

Let E ⊂ X. Define Γ(E) = {γ : γ meets E}, Γ∞(E) = {γ : γ meets E infinitely times}, Γτ(E) =

• γ : γ(0) ∈ E, limt→0+

d(γ(t),E) t

= 0

• (“tangential”).

Theorem (HEMM)

Let E ⊂ X has a finite lower Minkowski content of co-dimension p ≥ 1. Then AMLp,1(Γ(E)) < ∞ and AMLp,1(Γ∞(E)) = AMLp,1(Γτ(E)) = 0. For the “ordinary” moduli the situation is different:

Theorem (HEMM)

Let E ⊂ X has a nonzero Dirichlet-Lorentz capacity w.r.t. DLp,1, p > 1. Then MLp,1(Γ(E)) = MLp,1(Γ∞(E)) = MLp,1(Γτ(E)) = ∞.

Jan Mal´ y (Prague) Approximation modulus 8 / 17

Application to Stokes theorem

Let G be an n-dimensional C 1 manifold and ϕ be a parametric surface in G of co-dimension 1. How to recognize that the integration over ϕ induces a boundary of a full-dimensional integral current?

Jan Mal´ y (Prague) Approximation modulus 9 / 17

Example

In the planar case, a simple sufficient condition is that ϕ is a “closed curve”, a more sofisticated and n-dimensional criterion is that the boundary of ϕ in the sense of current is trivial. However, both conditions fail in the manifold setting:

Jan Mal´ y (Prague) Approximation modulus 10 / 17

Example

In the planar case, a simple sufficient condition is that ϕ is a “closed curve”, a more sofisticated and n-dimensional criterion is that the boundary of ϕ in the sense of current is trivial. However, both conditions fail in the manifold setting: Let G be a 2-dimensional anuloid, We can imagine it as embedded into R3: G =

• x ∈ R3 : (
• x2

1 + x2 2 − 2)2 + x2 3 = 1

• .

Let γ be a closed Jordan curve. Sometimes γ represents a boundary, γ =

• x ∈ G : (x1 − 2)2 + x2

2 = 1 9, x3 > 0

• ,

sometimes γ does not represent a boundary, γ =

• x ∈ G : x2 = 0, x1 > 0
• .

Jan Mal´ y (Prague) Approximation modulus 10 / 17

Setting

For simplicity, we present our results in the Euclidean case G = Rn. In fact, this is the difficult part; the generalization to the manifold setting is quite straightforward although useful.

Jan Mal´ y (Prague) Approximation modulus 11 / 17

Setting

For simplicity, we present our results in the Euclidean case G = Rn. In fact, this is the difficult part; the generalization to the manifold setting is quite straightforward although useful. We define a parametric surface as a Lipschitz mapping ϕ of a measurable set E ⊂ Rn−1, 0 < |E| < ∞, to Rn. Let us emphasize that E is only measurable, so that the image of ϕ can be wildly fragmented.

Jan Mal´ y (Prague) Approximation modulus 11 / 17

Setting

For simplicity, we present our results in the Euclidean case G = Rn. In fact, this is the difficult part; the generalization to the manifold setting is quite straightforward although useful. We define a parametric surface as a Lipschitz mapping ϕ of a measurable set E ⊂ Rn−1, 0 < |E| < ∞, to Rn. Let us emphasize that E is only measurable, so that the image of ϕ can be wildly fragmented. Then ϕ induces a vector valued measure νϕ which acts on Cc(Rn, Rn) as

• Rn
• ψ · d

νϕ =

• E

det

• ψ(ϕ(t)), D1ϕ(t), . . . , Dn−1ϕ(t)
• dt,
• ψ ∈ Cc(Rn, Rn).

Jan Mal´ y (Prague) Approximation modulus 11 / 17

Divergence theorem

Our question is: Under which conditions there exists an integer valued “multiplicity” function u : Rn → Rn such that − νϕ is the derivative of u in the sense of distributions?

Jan Mal´ y (Prague) Approximation modulus 12 / 17

Divergence theorem

Our question is: Under which conditions there exists an integer valued “multiplicity” function u : Rn → Rn such that − νϕ is the derivative of u in the sense of distributions? In other words, we are interested in validity of the divergence theorem

• Rn
• ψ · d

νϕ =

• Rn u div

ψ dx,

• ψ ∈ C∞

c (Rn, Rn).

Jan Mal´ y (Prague) Approximation modulus 12 / 17

The idea

The idea of the criterion is the following: Whenever a closed curve γ crosses the surface “in”, it must be compensated by another crossing which is “out”.

Jan Mal´ y (Prague) Approximation modulus 13 / 17

The idea

The idea of the criterion is the following: Whenever a closed curve γ crosses the surface “in”, it must be compensated by another crossing which is “out”. However, it cannot be tested on all curves, as the qualitative behavior of a curve with respect to a parametric surface can be very complicated.

Jan Mal´ y (Prague) Approximation modulus 13 / 17

The idea

The idea of the criterion is the following: Whenever a closed curve γ crosses the surface “in”, it must be compensated by another crossing which is “out”. However, it cannot be tested on all curves, as the qualitative behavior of a curve with respect to a parametric surface can be very complicated. Therefore we use the approximation modulus defined above to eliminate bad curves showing that the offending families of curves are “negligible”. We define the “crossing number” of a pair (ϕ, γ), where ϕ is the surface and γ is a testing curve. Then the bad curves are of two kinds

Jan Mal´ y (Prague) Approximation modulus 13 / 17

The idea

The idea of the criterion is the following: Whenever a closed curve γ crosses the surface “in”, it must be compensated by another crossing which is “out”. However, it cannot be tested on all curves, as the qualitative behavior of a curve with respect to a parametric surface can be very complicated. Therefore we use the approximation modulus defined above to eliminate bad curves showing that the offending families of curves are “negligible”. We define the “crossing number” of a pair (ϕ, γ), where ϕ is the surface and γ is a testing curve. Then the bad curves are of two kinds The crossing number does not make sense.

Jan Mal´ y (Prague) Approximation modulus 13 / 17

The idea

The idea of the criterion is the following: Whenever a closed curve γ crosses the surface “in”, it must be compensated by another crossing which is “out”. However, it cannot be tested on all curves, as the qualitative behavior of a curve with respect to a parametric surface can be very complicated. Therefore we use the approximation modulus defined above to eliminate bad curves showing that the offending families of curves are “negligible”. We define the “crossing number” of a pair (ϕ, γ), where ϕ is the surface and γ is a testing curve. Then the bad curves are of two kinds The crossing number does not make sense. The crossing number makes sense but it does not give a valuable result.

Jan Mal´ y (Prague) Approximation modulus 13 / 17

The crossing number - smooth case

Let γ : [0, ℓ] → Rn be a regular smooth curve. If γ is closed, we require γ′

+(0) = γ′ −(ℓ). If (y, t) ∈ E × [0, ℓ) we define

σ(y, t) =

• sgn
• det(γ′(t), D1ϕ(y), . . . , Dn−1ϕ(y))
• ,

ϕ(y) = γ(t),

• therwise

Jan Mal´ y (Prague) Approximation modulus 14 / 17

The crossing number - smooth case

Let γ : [0, ℓ] → Rn be a regular smooth curve. If γ is closed, we require γ′

+(0) = γ′ −(ℓ). If (y, t) ∈ E × [0, ℓ) we define

σ(y, t) =

• sgn
• det(γ′(t), D1ϕ(y), . . . , Dn−1ϕ(y))
• ,

ϕ(y) = γ(t),

• therwise

Then the crossing number of the pair (ϕ, γ) is defined as ⊗(ϕ, γ) =

• (y,t)∈E×[0,ℓ)

σ(y, t).

Jan Mal´ y (Prague) Approximation modulus 14 / 17

The crossing number - smooth case

Let γ : [0, ℓ] → Rn be a regular smooth curve. If γ is closed, we require γ′

+(0) = γ′ −(ℓ). If (y, t) ∈ E × [0, ℓ) we define

σ(y, t) =

• sgn
• det(γ′(t), D1ϕ(y), . . . , Dn−1ϕ(y))
• ,

ϕ(y) = γ(t),

• therwise

Then the crossing number of the pair (ϕ, γ) is defined as ⊗(ϕ, γ) =

• (y,t)∈E×[0,ℓ)

σ(y, t). However, this definition is only illustrative; we are able to define the crossing number just for Lipschitz curves, although the crossing points where the derivative of the curve is not defined cannot be neglected.

Jan Mal´ y (Prague) Approximation modulus 14 / 17

When ⊗(ϕ, γ) fails to make sense?

Jan Mal´ y (Prague) Approximation modulus 15 / 17

When ⊗(ϕ, γ) fails to make sense? The number of couples (y, t) with ϕ(y) = γ(t) is infinite.

Jan Mal´ y (Prague) Approximation modulus 15 / 17

When ⊗(ϕ, γ) fails to make sense? The number of couples (y, t) with ϕ(y) = γ(t) is infinite. The curve crosses the surface “tangentially”

Jan Mal´ y (Prague) Approximation modulus 15 / 17

When ⊗(ϕ, γ) fails to make sense? The number of couples (y, t) with ϕ(y) = γ(t) is infinite. The curve crosses the surface “tangentially” The normal to ϕ at the point where the curve crosses the surface is undefined.

Jan Mal´ y (Prague) Approximation modulus 15 / 17

When ⊗(ϕ, γ) fails to make sense? The number of couples (y, t) with ϕ(y) = γ(t) is infinite. The curve crosses the surface “tangentially” The normal to ϕ at the point where the curve crosses the surface is undefined. When ⊗(ϕ, γ) does not give an applicable result? This is more difficult to describe, so we mention only some indications. Let us have already a “multiplicity function” u ∈ BV . Then, according to the result on fine behavior of BV functions, there is an exceptional set N such that u has a Lebesgue point or a jump at each z / ∈ N.

Jan Mal´ y (Prague) Approximation modulus 15 / 17

When ⊗(ϕ, γ) fails to make sense? The number of couples (y, t) with ϕ(y) = γ(t) is infinite. The curve crosses the surface “tangentially” The normal to ϕ at the point where the curve crosses the surface is undefined. When ⊗(ϕ, γ) does not give an applicable result? This is more difficult to describe, so we mention only some indications. Let us have already a “multiplicity function” u ∈ BV . Then, according to the result on fine behavior of BV functions, there is an exceptional set N such that u has a Lebesgue point or a jump at each z / ∈ N. This means that there exists a set A(z) of Lebesgue density 0 at z such that the function u is controlled

• utside A(z). Then a good curve

Jan Mal´ y (Prague) Approximation modulus 15 / 17

When ⊗(ϕ, γ) fails to make sense? The number of couples (y, t) with ϕ(y) = γ(t) is infinite. The curve crosses the surface “tangentially” The normal to ϕ at the point where the curve crosses the surface is undefined. When ⊗(ϕ, γ) does not give an applicable result? This is more difficult to describe, so we mention only some indications. Let us have already a “multiplicity function” u ∈ BV . Then, according to the result on fine behavior of BV functions, there is an exceptional set N such that u has a Lebesgue point or a jump at each z / ∈ N. This means that there exists a set A(z) of Lebesgue density 0 at z such that the function u is controlled

• utside A(z). Then a good curve

must avoid N

Jan Mal´ y (Prague) Approximation modulus 15 / 17

When ⊗(ϕ, γ) fails to make sense? The number of couples (y, t) with ϕ(y) = γ(t) is infinite. The curve crosses the surface “tangentially” The normal to ϕ at the point where the curve crosses the surface is undefined. When ⊗(ϕ, γ) does not give an applicable result? This is more difficult to describe, so we mention only some indications. Let us have already a “multiplicity function” u ∈ BV . Then, according to the result on fine behavior of BV functions, there is an exceptional set N such that u has a Lebesgue point or a jump at each z / ∈ N. This means that there exists a set A(z) of Lebesgue density 0 at z such that the function u is controlled

• utside A(z). Then a good curve

must avoid N must approach z not through A(z).

Jan Mal´ y (Prague) Approximation modulus 15 / 17

Necessity

Theorem (HEMM)

Let ϕ: E → Rn be an (n−1)-dimensional Lipschitz parametric surface and u be a BV function on Rn. Assume that Du = − νϕ. Then u(γ(ℓ) − u(γ(0) = ⊗(ϕ, γ) for AM-a.e. curve γ : [0, ℓ] → Rn. In particular, u is integer-valued and the crossing number is zero if the curve is closed.

Jan Mal´ y (Prague) Approximation modulus 16 / 17

Necessity

Theorem (HEMM)

Let ϕ: E → Rn be an (n−1)-dimensional Lipschitz parametric surface and u be a BV function on Rn. Assume that Du = − νϕ. Then u(γ(ℓ) − u(γ(0) = ⊗(ϕ, γ) for AM-a.e. curve γ : [0, ℓ] → Rn. In particular, u is integer-valued and the crossing number is zero if the curve is closed. The proof consists in showing that all kinds of “bad curves” can be neglected.

Jan Mal´ y (Prague) Approximation modulus 16 / 17

Sufficiency

Theorem (HEMM)

Let ϕ: E → Rn be an (n−1)-dimensional Lipschitz parametric surface. Suppose that ⊗(ϕ, γ) = 0 for AM-a.e. closed curve γ. Then there exists a BV function u on Rn such that Du = − νϕ.

Jan Mal´ y (Prague) Approximation modulus 17 / 17

Sufficiency

Theorem (HEMM)

Let ϕ: E → Rn be an (n−1)-dimensional Lipschitz parametric surface. Suppose that ⊗(ϕ, γ) = 0 for AM-a.e. closed curve γ. Then there exists a BV function u on Rn such that Du = − νϕ. Idea of the proof: We define a mapping Φ : E × [0, ℓ(γ)] → Rn as Φ(y, t) = γ(t) − ϕ(y) Then JΦ(y, t) = det

• − D1ϕ(y), · · · , −Dn−1ϕ(y), γ′(t)
• and
• {(y,t): Φ(y,t)=x}

sgn JΦ(y, t) = ⊗(ϕ, γ − x).

Jan Mal´ y (Prague) Approximation modulus 17 / 17

Sufficiency

Theorem (HEMM)

Let ϕ: E → Rn be an (n−1)-dimensional Lipschitz parametric surface. Suppose that ⊗(ϕ, γ) = 0 for AM-a.e. closed curve γ. Then there exists a BV function u on Rn such that Du = − νϕ. Idea of the proof: We define a mapping Φ : E × [0, ℓ(γ)] → Rn as Φ(y, t) = γ(t) − ϕ(y) Then JΦ(y, t) = det

• − D1ϕ(y), · · · , −Dn−1ϕ(y), γ′(t)
• and
• {(y,t): Φ(y,t)=x}

sgn JΦ(y, t) = ⊗(ϕ, γ − x). Consider a standard family of mollifiers (ωδ)δ>0. We show that

• γ(ωδ ∗

νϕ) · d τγ = 0 for each smooth closed curve γ. Then it follows that ωδ ∗ νϕ is a gradient and letting δ → 0 we infer that νϕ is a gradient.

Jan Mal´ y (Prague) Approximation modulus 17 / 17