2 Proof for curves

The energy spaces of the tangent point energies

Simon Blatt

August 31, 2011

In this small note, we will give a necessary and sucient condition un- der which the tangent point energies introduced by Heiko von der Mosel and Pawel Strzelecki in [SvdM11,SvdM10] are bounded. We show that an admissible submanifold has boundedEq-energy if and only if it is injective and locally agrees with the graph of functions that belongs to Sobolev- Slobodeckij space W^2−mq,q. The known Morrey embedding theorems of Heiko von der Mosel and Pawel Strzelecki can then be interpreted as stan- dard Morrey embedding theorem for these spaces. Especially, this show that the Hölder exponents for the embeddings in [SvdM11] are sharp.

1 Introduction

Very recently Heiko von der Mosel and Pawel Strzelecki started the investigation of so called tangent point energies for curves and surfaces and showed that they posses regularizing and self repulsive eects for a broad class of sets of arbitrary dimension and codimension which they called admissible sets. One of the main results in this work was that boundedness of these energies for an admissible set implies that this set is locally a graph of aC^1,1−2mq function. For curves they could give an example that shows that this Hölder exponent is optimal. For object of higher dimensions this was not known.

The tangent point energy of an m-rectiable set Σ ⊂ Rⁿ is given by the double integral

Eq(Σ) :=

Z

Σ

Z

Σ

1 Rtp(x, y)

q

dH^m(x)dH^m(y) (1.1)

∗Departement Mathematik, ETH Zürich, Rämistrasse 101, CH-8004 Zürich, Switzerland, si- mon.blatt@math.ethz.ch

where H^m denotes them-dimensional Hausdor measure and Rtp(x, y) := |x−y|²

dist(x−y, T_xΣ) for allx6=y∈Σsuch that the tangent spaceTxΣonΣexists.

In this small note, we want to classify all admissible sets with nite tangent point energy under the assumption that the exponentqis bigger than the critical exponent 2m. As in the case of O'Hara's knot energies (cf. [Bla10]), it turns out that this classication can be given with the help of Sobolev Slobodeckij spaces. For an open subsetΩ⊂Rⁿ ands∈(0,1), p∈[1,∞)these are dened using the seminorms

|f|s,p:=

Z

Ω

Z

Ω

|f(x)−f(y)|p

|x−y|^n+sp dxdy

!¹_p .

Fork∈N0, the Sobolev-Slobodeckij spaceW^k+s,p(R^m,R)is the space of all functions f ∈W^k,p(R^m,R)with|∂^αf|s,p<∞for all multiindicesα∈N^m0 with length |α|=k. TheW^k+1,p-norm is then given by

kfk_Wk,p:=kfk_Wk,p+ X

α∈Nn 0

|α|=k

|∂^αf|s,p.

As usual, the spaceW^k+s,p(R^m,Rⁿ)is dened component wise. More information on these spaces can be found for example in [Ada75,Tar07] The main result of this article is the following

Theorem 1.1. Let Σ be a compact m-dimensional embedded C¹ submanifold and q >2m. Then Eq(Γ)<∞if and only if Σis an embeddedW^2−mq,q submanifold.

Here an embedded W^s,p submanifold for s−_m^p >1 is a submanifold which locally agrees with the graph of aW^s,p function.

Combining this with Theorem 1.4 in [SvdM11] we get the following corollary Corollary 1.2. Let Σbe an admissible set andq >2m. ThenE_q(Σ)<∞if and only if Σis an embeddedW^2−mq,q manifold.

Especially, together with the sharpness of the Morrey embedding theorem this proves that the exponent1−_2q^m in Theorem 1.4 of [SvdM11] is optimal. In Section2we rst give a proof of Theorem 1.1 for curves to illustrate the main idea for this simple toy problem. Afterwards, with the help of some further techniques we prove the full statement in Section 3.

2 Proof for curves

Of course it is enough to show the theorem for curves of length 1. LetΓ∈C¹(R/Z,Rⁿ), be a regular embedded curve parametrized by arc length andΣ = Γ(R/Z). Then

Eq(Γ) :=Eq(Γ(R/Z)) = Z

R/Z 1/2

Z

−1/2

(Rtp(Γ(x),Γ(x+w))^−qdxdw

We chose aδ >0 such that

|Γ⁰(x)−Γ⁰(y)| ≤√

2 (2.1)

for allx, y∈R/Zwith|x−y| ≤δ. If we denote byΠ_Γ(x)the orthogonal projection of Rⁿ onto the normal space ofΓ(R/Z)at the pointΓ(x), i.e.

Π_Γ(x)(v) :=v− hv,Γ⁰(x)iΓ⁰(x) we get using Equation (2.1) that there is aC <∞such that

C⁻¹|Γ⁰(x)−Γ⁰(y)| ≤ kΠ_Γ(x)−Π_Γ(x)k ≤C|Γ⁰(x)−Γ⁰(y)|

for allx, y∈R/Zwith|x−y| ≤δ. From

dist(Γ(x)−Γ(y), TxΓ) =

Π_Γ(x)(Γ(x)−Γ(y)) , one then sees that

Eq(Γ) = Z

(R/Z) 1/2

Z

−1/2

|Π_Γ(x)(Γ(x)−Γ(x+w))|^q

|Γ(x)−Γ(x+w)|^2q dwdx

= 1 2

Z

(R/Z) 1/2

Z

−1/2

|Π_Γ(x)(Γ(x)−Γ(x+w))|^q+|Π_Γ(x+w)(Γ(x+w)−Γ(x))|^q

|Γ(x)−Γ(x+w)|^2q dwdx

≥C⁻¹ Z

(R/Z) δ

Z

−δ

|ΠΓ(x)−ΠΓ(x+w)|^q

|Γ(x)−Γ(x+w)|^qdwdx

≥C⁻¹ Z

(R/Z) δ

Z

−δ

|Γ⁰(x)−Γ⁰(x+w)|^q

|w|^q dwdx.

Hence,

Z

R/Z 1/2

Z

−1/2

|Γ⁰(x+w)−Γ⁰(x)|^q

|w|^q dxdw≤C(E_q(R/Z) +δ^q−1).

This implies that Γ∈W^2−1q,q(R/Z)ifEq(Γ)is nite.

To get the other implication, rst note that every embedded curveΓ∈C¹(R/Z,Rⁿ) parametrized by arc length satises a bi-Lipschitz estimate, i.e. there is a constant C=C(Γ)<∞such that

|w| ≤C|Γ(x+w)−Γ(x)|

for all|w| ≤ ¹₂. Hence,

E_q(Σ) = Z

R/Z 1/2

Z

−1/2

|Π_Γ(x)(Γ(x+w)−Γ(x))|^q

|Γ(x+w)−Γ(x)|^q

dwdx

≤C Z

R/Z 1/2

Z

−1/2

|ΠΓ(x)(R1

0 Γ⁰(x+τ w)dτ|

|w|

q

dwdx

Jensen's inequality

≤ C

Z

R/Z 1/2

Z

−1/2 1

Z

0

|Γ⁰(s)−Γ⁰(s+τ w)|^q

|w|^q

dwdsdτ

As

|ΠΓ(x)Γ⁰(y)|=|ΠΓ(x)(Γ⁰(y)−Γ⁰(x))| ≤ |Γ⁰(y)−Γ⁰(x)|

this implies

Eq(Σ)≤C Z

R/Z 1

Z

0 1−2

Z

−1/2

|Γ⁰(s)−Γ⁰(s+τ w)|^q

|w|^q

dwdsdτ

Substitutingw˜=τ w, the right hand side can be written as

1

Z

0

τ^p−1 Z

R/Z τ /2

Z

−τ /2

|Γ⁰(s)−Γ⁰(s+ ˜w)|^p

|w|˜ ^p

dwdsdτ˜

=1 p

Z

R/Z 1/2

Z

−1/2

|Γ⁰(s)−Γ⁰(s+ ˜w)|^p

|w|˜^p

dwds <˜ ∞.

ThusE_q(Σ)is nite ifΓ∈W^2−1/q,q.

3 Proof for submanifolds

In the case of submanifolds, surprisingly a similar idea as in the case of curves works though the technical details are more involved.

Formvectorsv₁, . . . , v_m∈Rⁿ, letA_{vi}be then×mmatrix whose columns consist of v_i and Π_{vi} be the orthogonal projection onto the vector space spanned by v_i, i = 1, . . . , m. If the v₁, . . . , v_m are linearly independent, the normal equations (cf.

[Sto99, pp. 235-237] ) lead to the representation Π_{vi}=A_{vi}

A_{vi}A^∗_{v

i}

−1

A^∗_{v

i}

and thus the map

(v₁, . . . , v_n)→Π_{vi}

is locally Lipschitz with respect to the Euclidean norm on(Rⁿ)^mon the set{(vi, . . . , v_m)∈ (Rⁿ)^m:v_i are linearly independent}.

Now, letf ∈C¹(R^m,R^n−m)be such that kfkL^∞ ≤1,

g(x) := (f(x), x), and let e1, . . . , em be a basis of R^m. Then there is aC ≤ ∞ such that for any orthogonal projection P onto am-dimensional subspace and anyx∈R we have

kP−P_{g(x+ei)−g(x)}k ≤C sup

i=1,...,m

|(P−id_R^m)(g(x+ei)−g(x))|. (3.1) Indeed, forε0:=ε0(e1, . . . , em)>0 small enough,

|P(g(x+e_i)−g(x))−(g(x+e_i)−g(x))| ≤ε₀

implies that the vectorsP(g(x+e_i)−g(x))are linearly independent as there projections onto the rst mcoordinates are linearly independent. Furthermore,

|P(g(x+ei)−g(x))| ≤ε0+ 2 sup

i=1,...,m

|ei|

due to the Lipschitz estimate onf. Hence the local Lipschitz continuity proven above implies that

kP−Π_{{g(x+ei)−g(x)}}k=kΠ_{{P(g(x+ei)−g(x))}}−Π_{{g(x+ei)−g(x)}}k

≤C sup

i=1,...,m

|(P−id_R^m)(g(x+ei)−g(x))|

if

sup

i=1,...,m

|(P−id_R^m)(g(x+ei)−g(x))≤ε0. Since we always have kP−Π_{g(x+ei)−g(x)}k ≤2, we deduce

kP − Π_{{g(x+ei)−g(x)}}k ≤ (C + 2/ε0) sup

i=1,...,m

|(P − id_Rⁿ)(g(x + ei) − g(x))|

which proves Equation (3.1)

To prove the theorem, lete˜_i,i= 1, . . . , mbe an orthonormal basis of R^m and let

ej :=

(˜e1, ifj= 1

1

2(˜e₁+ ˜e_j)ifj= 2, . . . , m.

Sincee_i,i= 1, . . . , mis a basis of R^m, we get kP−Π_{g(x+rei)−g(x)}k ≤C sup

i=1,...,m

|(P−id_Rm)(g(x+re_i)−g(x))|.

and scaling leads to kP−Πg(x+rei)−g(x)

r

k ≤C sup

i=1,...,m

(P−id_Rm)

g(x+e_i)−g(x r

≤C(

(P−id)

g(x+r˜e1)−g(x) r

+

m

X

i=2

(P−id)

g(x+^r₂(˜e1+ ˜ei)−g(x) r

for allr≥0.

Since both sides of this inequality are invariant under orthogonal transformation of the ˜ei, the constant in it does not depend on the choice of the orthonormal basis

˜

e1, . . . ,e˜m. Exchanginge˜1 with−˜e1and xwithx+ ˜e1 and observing that this leaves the vector space spanned by g(x+rei)−g(x) invariant, we furthermore get that for all orthogonal projections QofRⁿ onto anm-dimensional subspace we have

kQ−Πg(x+rei)−g(x) r

k

≤C

(Q−id)

g(x)−g(x+r˜e₁) r

+

m

X

i=2

(Q−id)

g(x+^r₂(˜e₁+ ˜e_i)−g(x+r˜e₁) r

and hence kP−Qk

≤C

(P−id)

g(x+r˜e1)−g(x) r

+

m

X

i=2

(P−id)

g(x+^r₂(˜e1+ ˜ei)−g(x) r

+C

(Q−id)

g(x)−g(x+r˜e₁) r

+

m

X

i=2

(Q−id)

g(x+^r₂(˜e₁+ ˜e_i)−g(x+r˜e₁) r

(3.2) for all orthogonal projections P, QofRⁿ ontom-dimensional subspaces.

Now letΣbe an m-dimensional C¹ submanifold with niteEq energy andx0∈Σ. After some rotation and translation we can assume thatx0= 0andTx₀Σ =R^m× {0}

and that there is an r0>0 and anf ∈C¹(R^m,R^n−m)withf(0) = 0 such that

kf⁰kL^∞ ≤1 (3.3)

and

g(B_r0(0))⊂Σ.

LetΠ_g(x) denote the orthogonal projection ofRⁿ onto the tangent space ong(R^m) at g(x). For x∈ B_r0/2(0), r ≤ r0/2 and any orthonormal basis e1, . . . em ofR^m we have by the above discussion

kΠ_g(x)+ Π_g(x+re1)k

≤C

(Π_g(x)−id)

g(x)−g(x+r˜e1) r

+C

m

X

i=2

(Π_g(x)−id)

g(x+^r₂(˜e1+ ˜ei)−g(x+r˜e1) r

+C

(Πg(x+re₁)−id)

g(x+r˜e₁)−g(x) r

+C

m

X

i=2

(Π_g(x+re1)−id)

g(x+^r₂(˜e₁+ ˜e_i)−g(x) r

≤CRtp(g(x), g(x+re1))⁻¹ r

|g(x)−g(x+r˜e₁)|² +C

m

X

i=2

Rtp(g(x+r

2(e1+ei), g(x+e1))⁻¹ r

|g(x+^r₂(˜e1+ ˜ei)−g(x+r˜e1)|² +CR_tp(g(x+re₁), g(x))⁻¹ r

|g(x+re1)−g(x)|² +C

m

X

i=2

Rtp(g(x+r

2(e1+ei), g(x+e1))⁻¹ r

|g(x+^r₂(˜e₁+ ˜e_i)−g(x+r˜e₁)|² Using that due to the Lipschitz bound on f we have that|g(x)−g(y)| ≤2|x−y|, we deduce

kΠ_g(x)−Π_g(x+re1)k^q

r^q ≤CR_tp(g(x), g(x+re₁))^−q +C

m

X

i=2

R_tp(g(x+r

2(e₁+e_i), g(x+e₁))^−q +CRtp(g(x+re1), g(x))^−q

+C

m

X

i=2

Rtp(g(x+r

2(e1+ei), g(x+e1))^−q.

To get rid of the dependence on a special orthonormal basis, we consider the space of orthonormal matricesO(m)as am⁰ :=^m(m−1)₂ dimensional submanifold ofR⁽m×m) and the fact that for a function f ∈L¹(S^m−1,R)we have

1 H^m0(O(m))

Z

A∈O(m)

f(Ai)dH^m0(A) = 1 H^m−1(S^m−1)

Z

S^m−1

f(ω)dH^m(ω).

This leads to Z

B_r0/2(0;R^m)

Z

B_r0/2(0;R^m)

kΠx−Πx+wk^q

|w|^q dH^m(w)dH^m(x)

= Z

B_r

0/2(0;R^m) r0/2

Z

0

r^m−1 Z

S^m−1

kΠx−Πx+rek^q

|w|^q dH^m−1(e)drdH^m(x)

=C(m) Z

B_r

0/2(0;R^m) r₀/2

Z

0

r^m−1 Z

O(m)

kΠx−Πx+rA₁k^q

|w|^q dH^m0(A)dH^m(w)

≤C· Z

B_r0/2(0;R^m) r₀/2

Z

0

r^m−1 Z

O(m)

Rtp(g(x), g(x+rA1))^−q

+

m

X

i=2

R_tp g

x+r

2(A₁+A_i)

, g(x+A₁)^−q +R_tp(g(x+rA₁), g(x))^−q

+

m

X

i=2

Rtp

g

x+r

2(A1+Ai)

, g(x+A1)−q!

dH^m0(A)drdH^m(x)

≤C Z

B_r0/2(0;R^m) r₀/2

Z

0

Z

S^m−1

Rtp g(x), g(x+re)−q

+ (m−1)Rtp

g x+ 2^−1/2re

, g(x)−q

+Rtp g(x+re), g(x)−q

+ (m−1)R_tp

g x+ 2^−1/2re

, g(x)−q!

dH^m−1(e)drdH^m(x)

≤CEq(Σ).

Together with the estimate

kf⁰(x)−f⁰(y)k ≤CkΠ_g(x)−Π_g(y)k this proves hatΣis a W^2−1q,q submanifold ifEq(Σ) is nite.

Do get the other implication, we use that ifΣis a compactW^2−1/q,qmanifold there is anr₀>0such that for allx₀inΣthere is a functionf ∈W^2−1/q,q(R^m,R^n−m)with f(0) = 0 such that

kf⁰k_L^∞ ≤1

and anA∈SO(n)such that the functiong(x) := (f(x), x)satises A(g(B_r0/2(0;R^m))) +x0⊂Σ.

For xedx₀∈Σwe then use

kΠ_g(x)−Π_g(y)k ≤ kf⁰(x)−f⁰(y)k, and

|g(x)−g(y)| ≥ |x−y|

to get Z

B_r0/2(0)

Z

B_r0/2(0)

|Π_g(x)(g(x+w)−g(x)|^q

|g(x+w)−g(x)|^2q dH^m(w)dH^m(x)

≤C Z

B_r0/2(0)

Z

B_r0/2(0) 1

Z

0

|Π_g(x)g⁰(x+τ w|^q

|w|^q dτ dH^m(w)dH^m(x)

≤C Z

B_r0/2(0)

Z

B_r0/2(0) 1

Z

0

|Π_g(x)(g⁰(x+τ w)−g⁰(x)|^q

|w|^q dτ dH^m(w)dH^m(x)

≤C Z

B_r0/2(0)

Z

B_r0/2(0) 1

Z

0

|(g⁰(x+τ w)−g⁰(x)|^q

|w|^q dτ dH^m(w)dH^m(x).

Substitutingw˜=τ wthis expression can be estimated as in the case of curves by

C(r₀) Z

B_r0/2(0)

Z

B_r0/2(0)

|(g⁰(x+w)−g⁰(x)|^q

|w|^q dH^m(w)dH^m(x)≤C(r₀)kfk

W²⁻

1 q,q

(Br(0))

LetdΣdenote the geodesic distanceΣandBr(x; Σ)denote the geodesic ball of radius r in Σ. Then the above calculation combined with the fact that the gradient of f is bounded leads to

Z

B_r0/2(x0,Σ)

Z

y∈Σ dΣ (x,y)≤r0

2

1

Rtp(x, y)^qdH^m(y)H^m(x)

≤ Z

B_r0/2(0)

Z

B_r0/2(0)

|Πg(x)(g(x+w)−g(x)|^q

|g(x+w)−g(x)|^2q dH^m(w)dH^m(x)

<∞ for allx₀∈Σ.

SinceΣis compact an easy covering argument then gives x

y,x∈Σ dΣ (x,y)≤r0/4

1

Rtp(x, y)^qdH(y)H^m(x)<∞ SinceΓ is an embeddedC¹ manifold we furthermore have that

µ:= inf

kx−yk

dΣ(x, y) :x, y∈Σ, d_Σ(x, y)> r₀/4

>0 and hence we nally get

Eq(Σ) = Z

Σ

Z

x∈Σ,dΣ(x,y)≤r0/4

1

R_tp(x, y)^qdH^m(y)H^m(x)

+ Z

Σ

Z

x∈Σ,d_Σ(x,y)≥r₀/4

1

Rtp(x, y)^qdH^m(y)H^m(x)

≤ Z

Σ

Z

x∈Σ,dΣ(x,y)≤r0/4

1

Rtp(x, y)^qdH^m(y)H^m(x) + 1

µ^qH^m(Σ)<∞.

This nishes the proof of the theorem.

References

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[Sto99] Josef Stoer. Numerische Mathematik. 1: Eine Einführung - unter Berück- sichtigung von Vorlesungen von F. L. Bauer. 8., neu bearb. u. erw. Au.

Berlin: Springer, 1999.

[SvdM10] Pawel Strzelecki and Heiko von der Mosel. Tangent-point self-avoidance energies for curves. June 2010.

[SvdM11] Pawel Strzelecki and Heiko von der Mosel. Tangent-point repulsive poten- tials for a class of non-smoothm-dimensional sets inrⁿ. part i: Smoothing and self-avoidance eects. February 2011.

[Tar07] Luc Tartar. An introduction to Sobolev spaces and interpolation spaces, volume 3 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin, 2007.