C_1hn2 rectifiability and Q valued functions [Elektronische Ressource] / vorgelegt von Ulrich Menne

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• Mathematics

Informations

• Publié par: eberhard_karls_universitat_tubingen
• Publié le: 01 janvier 2008
• Nombre de lectures: 39
• Langue: Deutsch

Extrait

2C rectifiability andQ valued functions
Dissertation
zur Erlangung des Grades eines Doktors der Naturwissenschaften
der Fakultat fur Mathematik und Physik¨ ¨
der Eberhard-Karls-Universitat Tubingen¨ ¨
vorgelegt von
Ulrich Menne
aus Frankfurt am Main
2008Tag der mun¨ dlichen Qualifikation: 08.08.2008
Dekan: Prof. Dr. Nils Schopohl
1. Berichterstatter: Prof. Dr. Reiner Schatzle¨
2. Berich Prof. Dr. Tom IlmanenContents
Zusammenfassung in deutscher Sprache 1
Introduction 1
1 Approximation of integral varifolds 5
2 A Sobolev Poincar´e type inequality for integral varifolds 25
3 About the significance of the 1 tilt 34
A The Isoperimetric Inequality and its applications 36
B A differentiation theorem 42
C An example concerning tilt and height decays of integral vari-
folds 44
D Elementary properties of Q valued functions 47
References 52
iiZusammenfassung in deutscher Sprache
n+mIn dieser Arbeit werden integrale n Varifaltigkeiten in R betrachtet, wel-
pche eine Bedingung an die verallgemeinerte mittlere Krummung inL -Raumen¨ ¨
erfullen. Genauer wird der Zusammenhang von Großen, welche den klassischen¨ ¨
Tilt- und Height-Excess umfassen und verallgemeinern, untersucht, insbesonde-
2re im Hinblick auf die Frage moglic¨ herC -Rektifizierbarkeit solcher Varifaltigkei-
ten. Das Hauptresultat besagt, daß die Abweichung der integralen Varifaltigkeit
von einer eventuell mehrwertigen Ebene (Height-Excess) durch die Abweichung
der approximativen Tangentialraume¨ der integralen Varifaltigkeit von besagter
Ebene (Tilt-Excess) und die mittlere Krumm¨ ung kontrolliert werden kann.
Introduction
2 n+mThis work is concerned with C rectifiability of integral n varifolds in R ,
m,n ∈ N which are of locally bounded first variation. More precisely, given
2assumptions on the mean curvature, the relationship between C rectifiability
and decay of height or tilt quantities is examined.
First,somedefinitionswillberecalled. Supposethroughouttheintroduction
n+mthatm,nareasaboveandU isanonempty,opensubsetofR . Using[Sim83,
Theorem 11.8] as a definition, μ is a rectifiable [an integral] n varifold in U if
andonlyifμisaRadonmeasureonU andforμalmostallx∈U thereexistsan
napproximate tangent planeT μ∈G(n+m,n) with multiplicityθ (μ,x) ofμ atx
nx [andθ (μ,x)∈N],G(n+m,n) denoting the set ofn dimensional, unoriented
n+mplanes inR . The distributional first variation of mass of μ equals
R
1 n+m(δμ)(η) = div ηdμ whenever η∈C (U,R )μ c
where div η(x) is the trace of Dη(x) with respect to T μ. kδμk denotes theμ x
total variation measure associated to δμ and μ is said to be of locally bounded
first variation if and only if kδμk is a Radon measure. The tilt-excess and the
height-excess of μ are defined by
R
−n 2tiltex (x,%,T) :=% |T μ−T| dμ(ξ),μ ξB (x)%
R
−n−2 2heightex (x,%,T) :=% dist(ξ−x,T) dμ(ξ)μ B (x)%
n+mwhenever x ∈ R , 0 < % < ∞, B (x) ⊂ U, T ∈ G(n+m,n); here S ∈%
n+mG(n+m,n) is identified with the orthogonal projection ofR ontoS and|·|
n+m n+mdenotes the norm induced by the usual inner product on Hom(R ,R ).
From the above definition of a rectifiablen varifoldμ one obtains thatμ almost
all of U is covered by a countable collection of n dimensional submanifolds of
n+m 1
R of class C . This concept is extended to higher orders of differentiability
by adapting a definition of Anzellotti and Serapioni in [AS94] as follows: A
k,α krectifiable n varifold μ in U is called countably rectifiable of class C [C ],
k ∈ N, 0 < α ≤ 1, if and only if there exists a countable collection of n
n+m k,α kdimensional submanifolds ofR of class C [C ] covering μ almost all of U.
k,α kThroughout the introduction this will be abbreviated to C [C ] rectifiability.
k,1 k+1Note that C rectifiability and C rectifiability agree by [Fed69, 3.1.15].
Decays of tilt-excess or height-excess have been successfully used in [All72,
2Bra78, Sch04a, Sch04b]. The link toC rectifiability is provided in [Sch04b], see
1below. In order to explain some of these results, a mean curvature condition is
introduced. AnintegralnvarifoldinU issaidtosatisfy(H ), 1≤p≤∞, ifandp
p n+m~only if either p > 1 and for some H ∈ L (μ,R ), called the generalisedμ loc
mean curvature of μ,
R
1 n+m~(δμ)(η) =− H •ηdμ whenever η∈C (U,R ) (H )μ pc
or p = 1 and
μ is of locally bounded first variation; (H )1
n+mhere • denotes the usual inner product onR . Brakke has shown in [Bra78,
5.7] that
tiltex (x,%,T μ) =o (%), heightex (x,%,T μ) =o (%) as %↓ 0μ x x x xμ
for μ almost every x∈U provided μ satisfies (H ) and1
2−ε 2−εtiltex (x,%,T μ) =o (% ), heightex (x,%,T ) =o (% ) as %↓ 0μ x x x xμ
for every ε > 0 for μ almost every x ∈ U provided μ satisfies (H ). In case2
of codimension 1 and p > n Sch¨atzle has proved the following result yielding
optimal decay rates.
Theorem 5.1 in [Sch04a]. If m = 1, p > n, p ≥ 2, and μ is an integral n
varifold in U satisfying (H ), thenp
2 2tiltex (x,%,T μ) =O (% ), heightex (x,%,T μ) =O (% ) as %↓ 0μ x x x xμ
for μ almost all x∈U.
The importance of the improvement from 2−ε to 2 stems mainly from the
fact that the quadratic decay of tilt-excess can be used to compute the mean
~curvature vectorH in terms of the local geometry ofμ which had already beenμ
notedin[Sch01,Lemma6.3]. In[Sch04b]Sch¨atzleprovidestheabovementioned
2link to C rectifiability as follows:
Theorem 3.1 in [Sch04b]. If μ is an integral n varifold in U satisfying (H )2
then the following two statements are equivalent:
2(1) μ is C rectifiable.
(2) For μ almost every x∈U there holds
2 2
tiltex (x,%,T μ) =O (% ), heightex (x,%,T μ) =O (% ) as %↓ 0.μ x x x xμ
2The quadratic decay of heightex implies C rectifiability without the con-μ
dition (H ) as may be seen from the proof in [Sch04b]. However, (1) would not2
2nimply (2) ifμ were merely required to satisfy (H ) for somep with 1≤p< ,p n+2
an example will be provided in C.5. On the other hand, it is evident from the
Caccioppoli type inequality relating tiltex to heightex and mean curvature,μ μ
see e.g. [Bra78, 5.5], that quadratic decay of heightex implies quadratic decayμ
for tiltex under the condition (H ). This leads to the following question:μ 2
2Problem. Does quadratic decay of tiltex imply quadratic decay of heightexμ μ
under the condition (H )?2
More generally, suppose thatμ is an integraln varifold inU satisfying (H ),p
1≤p≤∞, and 0<α≤ 1, 1≤q<∞. Does
R 1/q−α−n/q qlimsupr |T μ−T μ| dμ(ξ) <∞ξ xB (x)rr↓0
for μ almost all x∈U imply
R 1/q−1−α−n/q q
limsupr dist(ξ−x,T μ) dμ(ξ) <∞xB (x)rr↓0
for μ almost all x∈U?
The answer to the second question will be shown in 2.8–2.10 to be in the
np
affirmative if and only if either p ≥ n or p < n and αq ≤ , yielding inn−p
particular a positive answer to the first question. The main task is to prove the
following theorem which in fact provides a quantitative estimate together with
qthe usual embedding in L spaces.
Theorem 2.8. Suppose Q ∈N, 0<α ≤ 1, 1 ≤p ≤n, and μ is an integral n
varifold in U satisfying (H ).p
Then the following two statements hold:
nq np1 1(1) If p<n, 1≤q <n, 1≤q ≤ min{ , · }, then for μ almost all1 2 n−q α n−p1
na∈U with θ (μ,a) =Q there holds
−α−1−n/q2 qlimsupr kdist(·−a,T μ)ka 2L (μxB (a))r
r↓0
−α−n/q1 q≤ Γ limsupr kT −T μkμ a 1(1) L (μxB (a))r
r↓0
where Γ is a positive, finite number depending only onm, n, Q, q , and(1) 1
q .2
n(2) If p =n, n<q≤∞, then for μ almost all a∈U with θ (μ,a) =Q there
holds
−α−1limsupr kdist(·−a,T μ)k ∞a L (μxB (a))r
r↓0
−α−n/q≤ Γ limsupr kT −T μk q(2) μ a L (μxB (a))r
r↓0
where Γ is a positive, finite number depending only on m, n, Q, and q.(2)
Here T denotes the function mapping x to T μ whenever the latter exists.μ x
The connection to higher order rectifiability is provided by the following simple
adaption of [Sch04b, Appendix A].
Lemma 3.1. Suppose 0 < α ≤ 1, μ is a rectifiable n varifold in U, and A
denotes the set of all x∈U such that T μ exists andx
R
−n−1−αlimsup% dist(ξ−x,T μ)dμ(ξ)<∞.xB (x)%%↓0
1,αThen μxA is C rectifiable.
3The analog of Theorem 2.8 in the case of weakly differentiable functions can
be proved simply by using the Sobolev Poincar´e inequality in conjunction with
an iteration procedure. In the present case, however, the curvature condition is
needed to exclude a behaviour like the one shown by the function f : R → R
defined by
∞X
−if(x) = (2 )χ −i−1 −i (x) whenever x∈R[2 ,2 [
i=0
1at 0; in fact an example of this behaviour occurring on a set of positive L
1/2measure is provided byf ◦g whereg is the distance function from a compact
1set C such that L (C)> 0 and for some 0<λ< 1
−3/2 1liminfr L ([x+λr,x+r[∼C)> 0 whenever x∈C.
r↓0
Therefore the strategy to prove Theorem 2.8 is to provide a special Sobolev
Poincar´e type inequality for integral varifolds involving curvature, see 2.4. In
the construction weakly differentiable functions are replaced by Lipschitzian Q
m ∼valuedfunctions, aQvaluedfunctionbeingafunctionwithvaluesinQ (R ) =Q
m Q(R ) ∼ where ∼ is induced by the action of the group of permutations of
m Q{1,...,Q} on (R ) .
Roughly speaking, the construction performed in a ballB (a)⊂U proceeds