Geometric Measure Theory

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Gláucio Terra

Departamento de Matemática IME - USP

October 21, 2019

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Recall: Lipschitz constant

Given metric spacesX andY andf :X →Y Lipschitz, Lipf :=sup{d_Y f(x),f(y)

d_X(x,y) |x 6=y ∈X}, is calledLipschitz constantoff.

Theorem (McShane’s lemma; 5.1)

Let A⊂Rⁿand f :A→Ra Lipschitz map. Define F :Rⁿ→Rby:

F(x) :=inf{f(a) +Lipf· kx−ak |a∈A}. (1) Then F extends f andLipF =Lipf .

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Recall: Lipschitz constant

Given metric spacesX andY andf :X →Y Lipschitz, Lipf :=sup{d_Y f(x),f(y)

d_X(x,y) |x 6=y ∈X}, is calledLipschitz constantoff.

Theorem (McShane’s lemma; 5.1)

Let A⊂Rⁿand f :A→Ra Lipschitz map. Define F :Rⁿ→Rby:

F(x) :=inf{f(a) +Lipf· kx−ak |a∈A}. (1) Then F extends f andLipF =Lipf .

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Theorem (Kirszbraun’s theorem; 5.2)

Let A⊂Rⁿand f :A→R^ma Lipschitz map. Then there exists a Lipschitz extension f :Rⁿ→R^mof f such thatLipF =Lipf .

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Motivation

IfΩis an open subset ofRⁿandX ∈C¹_c(Ω,Rⁿ), Z

Ω

divXdLⁿ =0.

Ifu∈C¹(Ω)andϕ∈C¹_c(Ω,Rⁿ), the previous equality applied to X =uϕyields theelementary Gauss-Green’s identity in divergence form:

Z

Ω

h∇u, ϕidLⁿ=− Z

Ω

udivϕdLⁿ.

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Motivation

IfΩis an open subset ofRⁿandX ∈C¹_c(Ω,Rⁿ), Z

Ω

divXdLⁿ =0.

Ifu∈C¹(Ω)andϕ∈C¹_c(Ω,Rⁿ), the previous equality applied to X =uϕyields theelementary Gauss-Green’s identity in divergence form:

Z

Ω

h∇u, ϕidLⁿ=− Z

Ω

udivϕdLⁿ.

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Definition (weak derivatives and gradients; 5.3)

LetΩbe an open subset ofRⁿandu ∈L¹_loc(Lⁿ|_Ω). We say that:

1 For 1≤i≤n,uhasweak i-th partial derivative v_i ∈L¹_loc(Lⁿ|_Ω)if

∀ϕ∈C^∞_c (Ω), Z

Ω

v_iϕdLⁿ =− Z

Ω

u∂ϕ

∂x_i dLⁿ.

2 uhasweak gradient v ∈L¹_loc(Lⁿ|_Ω,Rⁿ)if∀ϕ∈C^∞_c (Ω,Rⁿ), Z

Ω

hv, ϕidLⁿ =− Z

Ω

udivϕdLⁿ. (2)

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Definition (weak derivatives and gradients; 5.3)

LetΩbe an open subset ofRⁿandu ∈L¹_loc(Lⁿ|_Ω). We say that:

1 For 1≤i≤n,uhasweak i-th partial derivative v_i ∈L¹_loc(Lⁿ|_Ω)if

∀ϕ∈C^∞_c (Ω), Z

Ω

v_iϕdLⁿ =− Z

Ω

u∂ϕ

∂x_i dLⁿ.

2 uhasweak gradient v ∈L¹_loc(Lⁿ|_Ω,Rⁿ)if∀ϕ∈C^∞_c (Ω,Rⁿ), Z

Ω

hv, ϕidLⁿ =− Z

Ω

udivϕdLⁿ. (2)

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Exercise (weak gradients, bis; 5.4)

Weak gradients may be also characterized by means of Gauss-Green identity in gradient form. That is, letΩbe an open subset ofRⁿand u ∈L¹_loc(Lⁿ|_Ω); then u admits weak gradient v ∈L¹_loc(Lⁿ|_Ω,Rⁿ) iff∀ϕ∈C^∞_c (Ω),

Z

Ω

ϕvdLⁿ=− Z

Ω

u∇ϕdLⁿ. (3)

Exercise (5.5)

LetΩbe an open subset ofRⁿ, u∈L¹_loc(Lⁿ|_Ω)and1≤i≤n. If there exists ^∂_∂x^w^u

i ∈L¹_loc(Lⁿ|_Ω), then∀ϕ∈C¹_c(Ω), Z

Ω

∂^wu

∂x_i ϕdLⁿ=− Z

Ω

u∂ϕ

∂x_i dLⁿ.

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Exercise (weak gradients, bis; 5.4)

Weak gradients may be also characterized by means of Gauss-Green identity in gradient form. That is, letΩbe an open subset ofRⁿand u ∈L¹_loc(Lⁿ|_Ω); then u admits weak gradient v ∈L¹_loc(Lⁿ|_Ω,Rⁿ) iff∀ϕ∈C^∞_c (Ω),

Z

Ω

ϕvdLⁿ=− Z

Ω

u∇ϕdLⁿ. (3)

Exercise (5.5)

LetΩbe an open subset ofRⁿ, u∈L¹_loc(Lⁿ|_Ω)and1≤i≤n. If there exists ^∂_∂x^w^u

i ∈L¹_loc(Lⁿ|_Ω), then∀ϕ∈C¹_c(Ω), Z

Ω

∂^wu

∂x_i ϕdLⁿ=− Z

Ω

u∂ϕ

∂x_i dLⁿ.

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Proposition (vanishing weak gradient)

LetΩ⊂Rⁿbe a connected open set and u∈L¹_loc(Lⁿ|_Ω)such that

∀ϕ∈C^∞_c (Ω),R

Ωu∇ϕdLⁿ=0. Then u coincidesLⁿ-a.e. with a constant function.

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Definition (5.8)

LetΩbe an open subset ofRⁿ,u : Ω→Rand 1≤p≤ ∞. We say that

1 uis a(1,p)-Sobolev functionifu ∈L^p(Lⁿ|_Ω)and,∀1≤i ≤n,u has weak partial derivatives _∂x^∂u

i ∈L^p(Lⁿ|_Ω). Notation: W^1,p(Ω).

2 uis alocal(1,p)-Sobolev functionifu∈L^p_loc(Lⁿ|_Ω)and,

∀1≤i ≤n,u has weak partial derivatives _∂x^∂u

i ∈L^p_loc(Lⁿ|_Ω).

Notation: W^1,p_loc(Ω).

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Notation

Letu:Rⁿ→Randτ ∈Sⁿ⁻¹. Forh∈R\ {0}, we denote by τhu:Rⁿ →Rthe incremental ratio ofuin the directionτ:

τhu(x) := u(x +hτ)−u(x)

h .

By the invariance of the Lebesgue measure under translations, if u ∈L¹_loc(Rⁿ),v :Rⁿ→RboundedLⁿ-measurable with compact support andh∈R\ {0}:

Z

u(x +hτ)v(x)dLⁿ(x) = Z

u(x)v(x −hτ)dLⁿ(x),

hence Z

τ_hu(x)v(x)dLⁿ(x) =− Z

u(x)τ−hv(x)dLⁿ(x). (4)

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Notation

Letu:Rⁿ→Randτ ∈Sⁿ⁻¹. Forh∈R\ {0}, we denote by τhu:Rⁿ →Rthe incremental ratio ofuin the directionτ:

τhu(x) := u(x +hτ)−u(x)

h .

By the invariance of the Lebesgue measure under translations, if u ∈L¹_loc(Rⁿ),v :Rⁿ→RboundedLⁿ-measurable with compact support andh∈R\ {0}:

Z

u(x +hτ)v(x)dLⁿ(x) = Z

u(x)v(x −hτ)dLⁿ(x),

hence Z

τ_hu(x)v(x)dLⁿ(x) =− Z

u(x)τ−hv(x)dLⁿ(x). (4)

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Proposition (5.9 )

Let f :Rⁿ→Rbe a Lipschitz function. Then f ∈W^1,∞_loc (Rⁿ).

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Fréchet Differentiability

LetU⊂Rⁿopen andf :U→R. Recall thatf is differentiable at x₀∈U in the sense of Fréchet if there existsA∈L(Rⁿ,R)such that

h→0lim

f(x₀+h)−f(x₀)−A·h

khk =0.

If that is the case,f has first order partial derivatives atx₀,Asatisfying the above condition is unique and coincides withh∇f(x₀),·i:Rⁿ→R; Ais calledFréchet derivativeoff atx₀and denoted by Df(x₀).

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Fréchet Differentiability

LetU⊂Rⁿopen andf :U→R. Recall thatf is differentiable at x₀∈U in the sense of Fréchet if there existsA∈L(Rⁿ,R)such that

h→0lim

f(x₀+h)−f(x₀)−A·h

khk =0.

If that is the case,f has first order partial derivatives atx₀,Asatisfying the above condition is unique and coincides withh∇f(x₀),·i:Rⁿ→R; Ais calledFréchet derivativeoff atx₀and denoted by Df(x₀).

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Exercise (characterization of Fréchet differentiability; 5.10)

Let U ⊂Rⁿopen and f :U→R. Then f is differentiable at x₀∈U iffthere exists A∈L(Rⁿ,R)such that

lim

t→0⁺

f(x₀+tv)−f(x₀)

t =A·v

uniformly in v ∈Sⁿ⁻¹. If so A=Df(x₀).

Exercise (weak gradients under scaling and translations; 5.11)

Let x ∈Rⁿ, h>0, T :Rⁿ→Rⁿgiven byτ 7→x+hτ and u∈W^1,1_loc(Rⁿ).

Then u◦T ∈W^1,1_loc(Rⁿ)and∇^w(u◦T)(τ) =h∇^wu(x +hτ).

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Exercise (characterization of Fréchet differentiability; 5.10)

Let U ⊂Rⁿopen and f :U→R. Then f is differentiable at x₀∈U iffthere exists A∈L(Rⁿ,R)such that

lim

t→0⁺

f(x₀+tv)−f(x₀)

t =A·v

uniformly in v ∈Sⁿ⁻¹. If so A=Df(x₀).

Exercise (weak gradients under scaling and translations; 5.11)

Let x ∈Rⁿ, h>0, T :Rⁿ→Rⁿgiven byτ 7→x+hτ and u∈W^1,1_loc(Rⁿ).

Then u◦T ∈W^1,1_loc(Rⁿ)and∇^w(u◦T)(τ) =h∇^wu(x +hτ).

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Theorem (Rademacher’s theorem; 5.12)

Let f :Rⁿ→Rbe Lipschitz. Then f is differentiable in the sense of FréchetLⁿ-a.e. and∇f =∇^wf Lⁿ-a.e.

Exercise (5.13)

Let f :Rⁿ→Rbe Lipschitz. The set D_f of points where f is differentiable in the sense of Fréchet is Borel measurable and Df :D_f →L(Rⁿ,R)is Borelian.

Corollary (5.14)

IfΩ⊂Rⁿ open and f : Ω→Ris locally Lipschitz, then f isLⁿ|_Ω-a.e.

differentiable in the sense of Fréchet.

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Exercise (5.13)

Corollary (5.14)

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Exercise (5.13)

Corollary (5.14)

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Corollary (5.16)

IfΩ⊂Rⁿ open and f : Ω→R^mis locally Lipschitz, then f isLⁿ|_Ω-a.e.

Corollary (5.17)

1 Let f :Rⁿ→R^mbe locally Lipschitz and Z_f :={x ∈Rⁿ|f(x) =0}.

ThenDf(x) =0forLⁿ-a.e. x ∈Z_f.

2 Let f,g :Rⁿ→Rⁿbe locally Lipschitz and Y :={x ∈Rⁿ |g f(x)

=x}. ThenDg f(x)

◦Df(x) =id_Rⁿ for Lⁿ-a.e. x ∈Y .

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Corollary (5.16)

IfΩ⊂Rⁿ open and f : Ω→R^mis locally Lipschitz, then f isLⁿ|_Ω-a.e.

Corollary (5.17)

1 Let f :Rⁿ→R^mbe locally Lipschitz and Z_f :={x ∈Rⁿ|f(x) =0}.

ThenDf(x) =0forLⁿ-a.e. x ∈Z_f.

2 Let f,g :Rⁿ→Rⁿbe locally Lipschitz and Y :={x ∈Rⁿ |g f(x)

=x}. ThenDg f(x)

◦Df(x) =id_Rⁿ for Lⁿ-a.e. x ∈Y .

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Definition (5.18)

Let V and W be finite-dimensional Hilbert spaces.

1 A linear mapO :V→W is called anorthogonal injectionif

∀x,y ∈V,hO·x,O·yi=hx,yi. Notation:O(V,W).

2 LetT :V→W be a linear map. We denote byT^∗ theadjointofT with respect to the inner products on V and W. If V=W and T =T^∗, we callT self-adjointorsymmetric(notation: Sym(V)).

3 We say that a linear mapT :V→V ispositiveif it is symmetric and∀x ∈V,hT ·x,xi ≥0.

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Theorem (5.20)

LetVandWbe finite-dimensional Hilbert spaces and L:V→Wbe a linear map.

1 Ifdim V≤dim W, there exists a positive S∈Sym(V)and O∈O(V,W)such that

L=O◦S.

Moreover, in the above decomposition, S ∈Sym(V)positive is unique, and so is O∈O(V,W)if L is injective.

2 Ifdim V≥dim W, there exists a positive S∈Sym(W)and O∈O(W,V)such that

L=S◦O^∗.

Moreover, in the above decomposition, S ∈Sym(W)positive is unique, and so is O∈O(W,V)if L is surjective.

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Theorem (5.20)

LetVandWbe finite-dimensional Hilbert spaces and L:V→Wbe a linear map.

1 Ifdim V≤dim W, there exists a positive S∈Sym(V)and O∈O(V,W)such that

L=O◦S.

Moreover, in the above decomposition, S ∈Sym(V)positive is unique, and so is O∈O(V,W)if L is injective.

2 Ifdim V≥dim W, there exists a positive S∈Sym(W)and O∈O(W,V)such that

L=S◦O^∗.

Moreover, in the above decomposition, S ∈Sym(W)positive is unique, and so is O∈O(W,V)if L is surjective.

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Definition (Jacobian of a linear map; 5.21)

Let V and W be finite-dimensional Hilbert spaces andL∈L(V,W), with polar decompositionO◦Sif dim V≤dim W orS◦O^∗if dim V>dim W.

We define theJacobianofLby:

JLK:=|detS|.

Remark (5.22)

1 Note thatJLKis well-defined, by the uniqueness ofSin the polar decomposition.

2 It is clear that

JLK=JL^∗K= (√

detL^∗L if dim V≤dim W

√

detLL^∗ if dim V≥dim W.

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Definition (Jacobian of a linear map; 5.21)

Let V and W be finite-dimensional Hilbert spaces andL∈L(V,W), with polar decompositionO◦Sif dim V≤dim W orS◦O^∗if dim V>dim W.

We define theJacobianofLby:

JLK:=|detS|.

Remark (5.22)

1 Note thatJLKis well-defined, by the uniqueness ofSin the polar decomposition.

2 It is clear that

JLK=JL^∗K= (√

detL^∗L if dim V≤dim W

√

detLL^∗ if dim V≥dim W.

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Theorem (5.23)

LetVandWbe finite-dimensional Hilbert spaces with n=dim V≤dim W=m. If L∈L(V,W), then

JLK=

s X

B∈µ(m,n)

(detB)²,

whereµ(m,n)is the set of n×n minors in some matrix representation of L with respect to orthonormal bases onVandW.

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Definition (5.25)

Letf :Rⁿ→R^mbe Lipschitz.

We define, for each pointx wheref is differentiable, theJacobianoff atx,

Jf(x) :=JDf(x)K.

Note that Jf is a Borelian function defined on the complement of a Borel subset ofRⁿofLⁿ-null measure.

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Notation

For a Lipschitz mapf :Rⁿ →R^m, we will use henceforth the following notation:

D_f :={x ∈Rⁿ| ∃Df(x)};

J_f⁺:={x ∈D_f |Jf(x)>0};

J_f⁰:={x ∈D_f |Jf(x) =0}.

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Lemma (5.27)

If L:Rⁿ→R^m is linear and n≤m, then∀A⊂Rⁿ, Hⁿ L(A)

=JLKLⁿ(A). (5)

Exercise (5.28)

Let T ∈L(Rⁿ,R^m), n≤m.

1 If R ∈L(Rⁿ), thenJT ◦RK=JTKJRK.

2

JTK≤ kTkⁿ. If T is 1-1, thenkT⁻¹k⁻ⁿ≤JTK≤ kTkⁿ.

3 If m≤k and R∈L(R^m,R^k), thenJR◦TK≤ kRkⁿJTK. If R is 1-1, thenkR⁻¹k⁻ⁿJTK≤JR◦TK≤ kRkⁿJTK.

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Lemma (5.27)

=JLKLⁿ(A). (5)

Exercise (5.28)

2

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Lemma (5.27)

=JLKLⁿ(A). (5)

Exercise (5.28)

2

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Lemma (5.27)

=JLKLⁿ(A). (5)

Exercise (5.28)

2

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Lemma (5.29)

Let f :Rⁿ→R^mbe Lipschitz, with n≤m, and A⊂RⁿLⁿ-measurable.

Then:

1 f(A)isHⁿ-measurable.

2 The function N(f|_A) :R^m→[0,∞]given by y 7→H⁰(A∩f⁻¹{y})is Hⁿ-measurable.

3 R

R^mH⁰(A∩f⁻¹{y})dHⁿ(y)≤(Lipf)ⁿLⁿ(A).

Definition (multiplicity function;5.30)

With the notation from the previous lemma,

N(f|_A) :y 7→H⁰(A∩f⁻¹{y})is called themultiplicity functionoff|_A.

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Remark (5.31)

IfX is a complete, separable metric space,Y a Hausdorff topological space,µa Borel measure onY andf :X →Y continuous, then

∀A∈BX,f(A)isµ-measurable.

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Definition (5.32)

Letf :Rⁿ→R^mbe a Lipschitz map withn≤mandt >1. We say that (E,S)is at-linearizationforf ifE ∈BRⁿ andS ∈Sym(n)∩GL(Rⁿ) satisfy:

i) ∀x ∈E,f is differentiable atx and Jf(x)>0;

ii) ∀x,y ∈E,t⁻¹kS·x−S·yk ≤ kf(x)−f(y)k ≤tkS·x −S·yk;

iii) ∀x ∈E,∀v ∈Rⁿ,t⁻¹kS·vk ≤ kDf(x)·vk ≤tkS·vk.

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Proposition (5.33)

Let E ∈B_Rⁿ such that condition i) in the previous definition holds and S ∈Sym(n)∩GL(Rⁿ). Then(E,S)is a t-linearization for f ifff|_E is 1-1 with Lipschitz inverse and satisfies:

ii’) Lipf|_E◦S⁻¹≤t andLipS◦(f|_E)⁻¹≤t;

iii’) ∀x ∈E ,kDf(x)◦S⁻¹k ≤t andkS◦Df(x)⁻¹k ≤t.

Corollary (5.34)

Let f :Rⁿ→R^mbe a Lipschitz map with n≤m, t >1and(E,S)a t-linearization for f . Then∀x ∈E ,

t⁻ⁿ|detS| ≤Jf(x)≤tⁿ|detS|. (6)

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Proposition (5.33)

Let E ∈B_Rⁿ such that condition i) in the previous definition holds and S ∈Sym(n)∩GL(Rⁿ). Then(E,S)is a t-linearization for f ifff|_E is 1-1 with Lipschitz inverse and satisfies:

ii’) Lipf|_E◦S⁻¹≤t andLipS◦(f|_E)⁻¹≤t;

iii’) ∀x ∈E ,kDf(x)◦S⁻¹k ≤t andkS◦Df(x)⁻¹k ≤t.

Corollary (5.34)

Let f :Rⁿ→R^mbe a Lipschitz map with n≤m, t >1and(E,S)a t-linearization for f . Then∀x ∈E ,

t⁻ⁿ|detS| ≤Jf(x)≤tⁿ|detS|. (6)

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Theorem (5.35)

Let f :Rⁿ→R^mbe a Lipschitz map with n≤m, t >1and

J_f⁺={x ∈Rⁿ| ∃Df(x)and Jf(x)>0}. Then there exists a countable disjoint family(E_k)k∈NinB_Rⁿ such that J_f⁺=∪__k∈_NE_k and,∀k ∈N, there exists S_k ∈Sym(n)∩GL(Rⁿ)such that(E_k,S_k)is a

t-linearization for f .

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Theorem (5.36)

Let f :Rⁿ→R^mbe Lipschitz, n≤m. Then, for all A∈σ(Lⁿ), Z

A

JfdLⁿ = Z

R^m

H⁰(A∩f⁻¹{y})dHⁿ(y).

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Corollary (5.37)

If f :Rⁿ→R^m is Lipschitz, n≤m, then forHⁿ-a.e. y ∈R^m, f⁻¹{y}is countable.

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Corollary (5.38)

Let f :Rⁿ→R^mbe Lipschitz, n≤m. Then for all g :Rⁿ →R Lⁿ-measurable with g≥0or g summable,

Z

Rⁿ

gJfdLⁿ= Z

R^m

X

x∈f⁻¹{y}

g(x)

dHⁿ(y).

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Corollary (5.39)

Let f :Rⁿ→R^mbe Lipschitz 1-1, n≤m.

1 ∀A∈σ(Lⁿ),Hⁿ f(A)

=R

AJfdLⁿ. In particular, we have

f_#(Lⁿ

x

^Jf^{) =}^Hⁿ

x

^Im^f ⁽⁷⁾

(equality as Borel regular outer measures onR^m).

2 If g:Rⁿ→RisLⁿ-measurable with g≥0or g ∈L¹(Lⁿ), then R

Imf g◦f⁻¹dHⁿ=R

RⁿgJfdLⁿ. In particular, if g :Imf →[0,∞]is Borelian, then

Z

Imf

gdHⁿ= Z

g◦fJfdLⁿ. (8)

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Corollary (5.39)

Let f :Rⁿ→R^mbe Lipschitz 1-1, n≤m.

1 ∀A∈σ(Lⁿ),Hⁿ f(A)

=R

AJfdLⁿ. In particular, we have

f_#(Lⁿ

x

^Jf^{) =}^Hⁿ

x

^Im^f ⁽⁷⁾

(equality as Borel regular outer measures onR^m).

2 If g:Rⁿ→RisLⁿ-measurable with g≥0or g ∈L¹(Lⁿ), then R

Imf g◦f⁻¹dHⁿ=R

RⁿgJfdLⁿ. In particular, if g :Imf →[0,∞]is Borelian, then

Z

Imf

gdHⁿ= Z

g◦fJfdLⁿ. (8)

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Example (5.40)

1 (length of a curve) Let−∞<a<b<∞andγ : [a,b]→R^mbe Lipschitz 1-1. We may extendγ to a Lipschitz function onR, which we still denote byγ. Then

Z b a

kγ⁰(t)kdt=H¹ γ([a,b]) .

2 (area of a graph) Letg :Rⁿ→Rbe Lipschitz andf :Rⁿ →Rⁿ⁺¹ be given byf(x) := x,g(x)

.

For eachU⊂Rⁿ open, the “surface area” of the graph ofg over U,Γ = Γ(g;U) :={ x,g(x)

|x ∈U}=f(U), is given by:

Hⁿ(Γ) = Z

U

JfdLⁿ= Z

U

q

1+k∇g(x)k²dx.

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Example (5.40)

1 (length of a curve) Let−∞<a<b<∞andγ : [a,b]→R^mbe Lipschitz 1-1. We may extendγ to a Lipschitz function onR, which we still denote byγ. Then

Z b a

kγ⁰(t)kdt=H¹ γ([a,b]) .

2 (area of a graph) Letg :Rⁿ→Rbe Lipschitz andf :Rⁿ →Rⁿ⁺¹ be given byf(x) := x,g(x)

.

For eachU⊂Rⁿ open, the “surface area” of the graph ofg over U,Γ = Γ(g;U) :={ x,g(x)

|x ∈U}=f(U), is given by:

Hⁿ(Γ) = Z

U

JfdLⁿ= Z

U

q

1+k∇g(x)k²dx.

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Exercise (5.41)

The area formula and its corollaries remain valid for locally Lipschitz maps defined on open subsets ofRⁿ.

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R

Exercise (5.42)

For any smooth embedded k -Riemannian submanifoldM⊂Rⁿ, the measure induced by the Riemannian metric onM(i.e. the Lebesgue measure ofM) coincides with the traceH^k|_M. Conclude that

H-dimM =k and, ifMis closed (i.e. topologically closed),H^k

x

^M^is

a Radon measure onRⁿ.

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Theorem (5.48)

Let f :Rⁿ→R^mbe Lipschitz, n≥m. Then, for eachLⁿ-measurable A⊂Rⁿ,

Z

A

JfdLⁿ= Z

R^m

H^n−m(A∩f⁻¹{y})dL^m(y). (9)

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Remark (5.49)

1 Iff :Rⁿ≡R^m×R^n−m →R^m is the projection on the first factor, we have Jf ≡1 and the coarea formula reduces to Fubini-Tonelli’s theorem.

2 Ifn=m, the coarea formula coincides with the area formula 22.

3 If we take the Borel set

A:= (Rⁿ\D_f)∪J_f⁰={x ∈Rⁿ|@Df(x)or Jf(x) =0}in the coarea formula, we conclude thatH^n−m(A∩f⁻¹{y}) =0 for L^m-a.e. y ∈R^m.

That may be interpreted as a measure theoretic version of Morse-Sard’s theorem:L^m-a.e. y ∈R^mis a measure theoretic

“regular value” off, in the sense that, up toH^n−mnull sets,f⁻¹{y} lies in the setJ_f⁺of points where Df has maximal rank.

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Remark (5.49)

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Remark (5.49)

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Remark (5.49)

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Corollary (curvilinear Fubini-Tonelli’s theorem; 5.50)

Let f :Rⁿ→R^mbe Lipschitz, n≥m. Then for all g :Rⁿ →R Lⁿ-measurable with g≥0or g summable,

Z

Rⁿ

gJfdLⁿ= Z

R^m

Z

f⁻¹{y}

g(x)dH^n−m(x)

dL^m(y), (10) meaning that the iterated integrals in second member make sense and the equality holds.

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Exercise (Coarea Formula for locally Lipschitz maps; 5.51)

The coarea formula and its corollary remain valid for locally Lipschitz maps defined on open subsets ofRⁿ.

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Proposition (polar coordinates; 5.52)

If g :Rⁿ →RisLⁿ-measurable and g≥0or g∈L¹(Lⁿ), then Z

Rⁿ

gdLⁿ= Z ∞

0

Z

∂B(0,r)

gdHⁿ⁻¹

dr. (11)

Proposition (5.53)

LetΩ⊂Rⁿopen and f : Ω→Rbe locally Lipschitz. Then Z

Ω

k∇fkdLⁿ= Z ∞

−∞

Hⁿ⁻¹({f =t})dt.

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Proposition (polar coordinates; 5.52)

If g :Rⁿ →RisLⁿ-measurable and g≥0or g∈L¹(Lⁿ), then Z

Rⁿ

gdLⁿ= Z ∞

0

Z

∂B(0,r)

gdHⁿ⁻¹

dr. (11)

Proposition (5.53)

LetΩ⊂Rⁿopen and f : Ω→Rbe locally Lipschitz. Then Z

Ω

k∇fkdLⁿ= Z ∞

−∞

Hⁿ⁻¹({f =t})dt.

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Theorem (6.45)

Let n ≥2, f :Rⁿ⁻¹→RLipschitz andΩ :=epi_S f (hence∂Ω =grf ).

Then

i) Hⁿ⁻¹

x

^∂Ωis a Radon measure onRⁿ;

ii) there exists a Borel measurable unit vector fieldν :∂Ω→Rⁿ, unique up toHⁿ⁻¹

x

∂Ω-null sets, such that, for allϕ∈C¹_c(Rⁿ),

Z

Ω

∇ϕdLⁿ= Z

∂Ω

ϕνdHⁿ⁻¹, (12)

or, equivalently, such that, for allϕ∈C¹_c(Rⁿ,Rⁿ), Z

Ω

divϕdLⁿ= Z

∂Ω

ϕ·νdHⁿ⁻¹. (13)

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With the notation from the previous theorem,ν is calledouter unit normalto∂Ω.

Remark

Up toHⁿ

x

∂Ω-null sets, on each point pointx = x⁰,f(x⁰) in

∂Ω =grf whose abscissax⁰is a differentiability point off,

ν(x) = ∇f(x⁰),−1

p1+k∇f(x⁰)k². (14) In particular, iff is C¹,νcoincides with the usual outer unit normal from Differential Geometry.

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With the notation from the previous theorem,ν is calledouter unit normalto∂Ω.

Remark

Up toHⁿ

x

∂Ω-null sets, on each point pointx = x⁰,f(x⁰) in

∂Ω =grf whose abscissax⁰is a differentiability point off, ν(x) = ∇f(x⁰),−1

p1+k∇f(x⁰)k². (14) In particular, iff is C¹,νcoincides with the usual outer unit normal from Differential Geometry.