Part II - Sets of finite perimeter

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Cambridge studies in advanced mathmatics 135
Sets of Finite
Perimeter and
Geometric
Variational
Problems
An Introduction to Geometric
Measure Theory
FRANCESCO MAGGI
CAMBRIDGE
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 135
Editorial Board
B. BOLLOB
´
AS, W. FULTON, A. KATOK, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
SETS OF FINITE PERIMETER AND GEOMETRIC
VARIATIONAL PROBLEMS
The marriage of analytic power to geometric intuition drives many of today’s
mathematical advances, yet books that build the connection from an elementary level
remain scarce. This engaging introduction to geometric measure theory bridges
analysis and geometry, taking readers from basic theory to some of the most celebrated
results in modern analysis.
The theory of sets of finite perimeter provides a simple and effective framework.
Topics covered include existence, regularity, analysis of singularities, characterization,
and symmetry results for minimizers in geometric variational problems, starting from
the basics about Hausdorff measures in Euclidean spaces, and ending with complete
proofs of the regularity of area-minimizing hypersurfaces up to singular sets of
codimension (at least) 8.
Explanatory pictures, detailed proofs, exercises, and remarks providing heuristic
motivation and summarizing difficult arguments make this graduate-level textbook
suitable for self-study and also a useful reference for researchers. Readers require only
undergraduate analysis and basic measure theory.
Francesco Maggi is an Associate Professor at the Universit
`
a degli Studi di Firenze,
Italy.
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
Editorial Board:
B. Bollob
´
as, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro
All the titles listed below can be obtained from good booksellers or from Cambridge University Press.
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132 J. V
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133 G. Malle & D. Testerman Linear algebraic groups and finite groups of Lie type
134 P. Li Geometric analysis
135 F. Maggi
Sets of finite perimeter and geometric variational problems
136
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137 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, I
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139 B. Helffer Spectral theory and its applications
Sets of Finite Perimeter and Geometric
Variational Problems
An Introduction to Geometric Measure Theory
FRANCESCO MAGGI
Universit
`
a degli Studi di Firenze, Italy
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town,
Singapore, S
˜
ao Paulo, Delhi, Mexico City
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9781107021037
C
Francesco Maggi 2012
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2012
Printed in the United Kingdom at the University Press, Cambridge
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Maggi, Francesco, 1978–
Sets of finite perimeter and geometric variational problems : an introduction to geometric
measure theory / Francesco Maggi, Universita degli Studi di Firenze, Italy.
pages cm – (Cambridge studies in advanced mathematics ; 135)
Includes bibliographical references and index.
ISBN 978-1-107-02103-7
1. Geometric measure theory. I. Title.
QA312.M278 2012
515
.42 – dc23 2012018822
ISBN 978-1-107-02103-7 Hardback
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party internet websites referred to
in this publication, and does not guarantee that any content on such
websites is, or will remain, accurate or appropriate.
To Chiara
Contents
Preface page xiii
Notation xvii
PART I RADON MEASURES ON R
n
1
1 Outer measures 4
1.1 Examples of outer measures 4
1.2 Measurable sets and σ-additivity 7
1.3 Measure Theory and integration 9
2 Borel and Radon measures 14
2.1 Borel measures and Carath
´
eodory’s criterion 14
2.2 Borel regular measures 16
2.3 Approximation theorems for Borel measures 17
2.4 Radon measures. Restriction, support, and push-forward 19
3 Hausdorff measures 24
3.1 Hausdorff measures and the notion of dimension 24
3.2 H
1
and the classical notion of length 27
3.3 H
n
= L
n
and the isodiametric inequality 28
4 Radon measures and continuous functions 31
4.1 Lusin’s theorem and density of continuous functions 31
4.2 Riesz’s theorem and vector-valued Radon measures 33
4.3 Weak-star convergence 41
4.4 Weak-star compactness criteria 47
4.5 Regularization of Radon measures 49
5Differentiation of Radon measures 51
5.1 Besicovitch’s covering theorem 52
viii Contents
5.2 Lebesgue–Besicovitch differentiation theorem 58
5.3 Lebesgue points 62
6 Two further applications of differentiation theory 64
6.1 Campanato’s criterion 64
6.2 Lower dimensional densities of a Radon measure 66
7 Lipschitz functions 68
7.1 Kirszbraun’s theorem 69
7.2 Weak gradients 72
7.3 Rademacher’s theorem 74
8 Area formula 76
8.1 Area formula for linear functions 77
8.2 The role of the singular set Jf = 080
8.3 Linearization of Lipschitz immersions 82
8.4 Proof of the area formula 84
8.5 Area formula with multiplicities 85
9 Gauss–Green theorem 89
9.1 Area of a graph of codimension one 89
9.2 Gauss–Green theorem on open sets with C
1
-boundary 90
9.3 Gauss–Green theorem on open sets with almost
C
1
-boundary 93
10 Rectifiable sets and blow-ups of Radon measures 96
10.1 Decomposing rectifiable sets by regular Lipschitz images 97
10.2 Approximate tangent spaces to rectifiable sets 99
10.3 Blow-ups of Radon measures and rectifiability 102
11 Tangential differentiability and the area formula 106
11.1 Area formula on surfaces 106
11.2 Area formula on rectifiable sets 108
11.3 Gauss–Green theorem on surfaces 110
Notes 114
PART II SETS OF FINITE PERIMETER 117
12 Sets of finite perimeter and the Direct Method 122
12.1 Lower semicontinuity of perimeter 125
12.2 Topological boundary and Gauss–Green measure 127
12.3 Regularization and basic set operations 128
12.4 Compactness from perimeter bounds 132
Contents ix
12.5 Existence of minimizers in geometric variational
problems 136
12.6 Perimeter bounds on volume 141
13 The coarea formula and the approximation theorem 145
13.1 The coarea formula 145
13.2 Approximation by open sets with smooth boundary 150
13.3 The Morse–Sard lemma 154
14 The Euclidean isoperimetric problem 157
14.1 Steiner inequality 158
14.2 Proof of the Euclidean isoperimetric inequality 165
15 Reduced boundary and De Giorgi’s structure theorem 167
15.1 Tangential properties of the reduced boundary 171
15.2 Structure of Gauss–Green measures 178
16 Federer’s theorem and comparison sets 183
16.1 Gauss–Green measures and set operations 184
16.2 Density estimates for perimeter minimizers 189
17 First and second variation of perimeter 195
17.1 Sets of finite perimeter and diffeomorphisms 196
17.2 Taylor’s expansion of the determinant close to the identity 198
17.3 First variation of perimeter and mean curvature 200
17.4 Stationary sets and monotonicity of density ratios 204
17.5 Volume-constrained perimeter minimizers 208
17.6 Second variation of perimeter 211
18 Slicing boundaries of sets of finite perimeter 215
18.1 The coarea formula revised 215
18.2 The coarea formula on H
n−1
-rectifiable sets 223
18.3 Slicing perimeters by hyperplanes 225
19 Equilibrium shapes of liquids and sessile drops 229
19.1 Existence of minimizers and Young’s law 230
19.2 The Schwartz inequality 237
19.3 A constrained relative isoperimetric problem 242
19.4 Liquid drops in the absence of gravity 247
19.5 A symmetrization principle 250
19.6 Sessile liquid drops 253
20 Anisotropic surface energies 258
20.1 Basic properties of anisotropic surface energies 258
20.2 The Wulff problem 262
x Contents
20.3 Reshetnyak’s theorems 269
Notes 272
PART III REGULARITY THEORY AND ANALYSIS
OF SINGULARITIES 275
21 (Λ, r
0
)-perimeter minimizers 278
21.1 Examples of (Λ, r
0
)-perimeter minimizers 278
21.2 (Λ, r
0
) and local perimeter minimality 280
21.3 The C
1,γ
-reguarity theorem 282
21.4 Density estimates for (Λ, r
0
)-perimeter minimizers 282
21.5 Compactness for sequences of (Λ, r
0
)-perimeter
minimizers 284
22 Excess and the height bound 290
22.1 Basic properties of the excess 291
22.2 The height bound 294
23 The Lipschitz approximation theorem 303
23.1 The Lipschitz graph criterion 303
23.2 The area functional and the minimal surfaces equation 305
23.3 The Lipschitz approximation theorem 308
24 The reverse Poincar
´
e inequality 320
24.1 Construction of comparison sets, part one 324
24.2 Construction of comparison sets, part two 329
24.3 Weak reverse Poincar
´
e inequality 332
24.4 Proof of the reverse Poincar
´
e inequality 334
25 Harmonic approximation and excess improvement 337
25.1 Two lemmas on harmonic functions 338
25.2 The “excess improvement by tilting” estimate 340
26 Iteration, partial regularity, and singular sets 345
26.1 The C
1,γ
-regularity theorem in the case Λ=0 345
26.2 The C
1,γ
-regularity theorem in the case Λ > 0 351
26.3 C
1,γ
-regularity of the reduced boundary, and the
characterization of the singular set 354
26.4 C
1
-convergence for sequences of (Λ, r
0
)-perimeter
minimizers 355
27 Higher regularity theorems 357
27.1 Elliptic equations for derivatives of Lipschitz minimizers 357
27.2 Some higher regularity theorems 359
Contents xi
28 Analysis of singularities 362
28.1 Existence of densities at singular points 364
28.2 Blow-ups at singularities and tangent minimal cones 366
28.3 Simons’ theorem 372
28.4 Federer’s dimension reduction argument 375
28.5 Dimensional estimates for singular sets 379
28.6 Examples of singular minimizing cones 382
28.7 A Bernstein-type theorem 385
Notes 386
PART IV MINIMIZING CLUSTERS 391
29 Existence of minimizing clusters 398
29.1 Definitions and basic remarks 398
29.2 Strategy of proof 402
29.3 Nucleation lemma 406
29.4 Truncation lemma 408
29.5 Infinitesimal volume exchanges 410
29.6 Volume-fixing variations 414
29.7 Proof of the existence of minimizing clusters 424
30 Regularity of minimizing clusters 431
30.1 Infiltration lemma 431
30.2 Density estimates 435
30.3 Regularity of planar clusters 437
Notes 444
References 445
Index 453
Preface
Everyone talks about rock these days;
the problem is they forget about the roll.
Keith Richards
The theory of sets of finite perimeter provides, in the broader framework of
Geometric Measure Theory (hereafter referred to as GMT), a particularly well-
suited framework for studying the existence, symmetry, regularity, and struc-
ture of singularities of minimizers in those geometric variational problems in
which surface area is minimized under a volume constraint. Isoperimetric-type
problems constitute one of the oldest and more attractive areas of the Calcu-
lus of Variations, with a long and beautiful history, and a large number of still
open problems and current research. The first aim of this book is to provide a
pedagogical introduction to this subject, ranging from the foundations of the
theory, to some of the most deep and beautiful results in the field, thus provid-
ing a complete background for research activity. We shall cover topics like the
Euclidean isoperimetric problem, the description of geometric properties of
equilibrium shapes for liquid drops and crystals, the regularity up to a singular
set of codimension at least 8 for area minimizing boundaries, and, probably for
the first time in book form, the theory of minimizing clusters developed (in a
more sophisticated framework) by Almgren in his AMS Memoir [Alm76].
Ideas and techniques from GMT are of crucial importance also in the study
of other variational problems (both of parametric and non-parametric charac-
ter), as well as of partial differential equations. The secondary aim of this book
is to provide a multi-leveled introduction to these tools and methods, by adopt-
ing an expository style which consists of both heuristic explanations and fully
detailed technical arguments. In my opinion, among the various parts of GMT,
xiv Preface
the theory of sets of finite perimeter is the best suited for this aim. Compared
to the theories of currents and varifolds, it uses a lighter notation and, virtually,
no preliminary notions from Algebraic or Differential Geometry. At the same
time, concerning, for example, key topics like partial regularity properties of
minimizers and the analysis of their singularities, the deeper structure of many
fundamental arguments can be fully appreciated in this simplified framework.
Of course this line of thought has not to be pushed too far. But it is my convic-
tion that a careful reader of this book will be able to enter other parts of GMT
with relative ease, or to apply the characteristic tools of GMT in the study of
problems arising in other areas of Mathematics.
The book is divided into four parts, which in turn are opened by rather de-
tailed synopses. Depending on their personal backgrounds, different readers
may like to use the book in different ways. As we shall explain in a moment, a
short “crash-course” is available for complete beginners.
Part I contains the basic theory of Radon measures, Hausdorff measures, and
rectifiable sets, and provides the background material for the rest of the book.
I am not a big fan of “preliminary chapters”, as they often miss a storyline,
and quickly become boring. I have thus tried to develop Part I as independent,
self-contained, and easily accessible reading. In any case, following the above
mentioned “crash-course” makes it possible to see some action taking place
without having to work through the entire set of preliminaries.
Part II opens with the basic theory of sets of finite perimeter, which is pre-
sented, essentially, as it appears in the original papers by De Giorgi [DG54,
DG55, DG58]. In particular, we avoid the use of functions of bounded vari-
ation, hoping to better stimulate the development of a geometric intuition of
the theory. We also present the original proof of De Giorgi’s structure theorem,
relying on Whitney’s extension theorem, and avoiding the notion of rectifiable
set. Later on, in the central portion of Part II, we make the theory of rectifiable
sets from Part I enter into the game. We thus provide another justification of
De Giorgi’s structure theorem, and develop some crucial cut-and-paste com-
petitors’ building techniques, first and second variation formulae, and slicing
formulae for boundaries. The methods and ideas introduced in this part are fi-
nally applied to study variational problems concerning confined liquid drops
and anisotropic surface energies.
Part III deals with the regularity theory for local perimeter minimizers, as
well as with the analysis of their singularities. In fact, we shall deal with the
more general notion of (Λ, r
0
)-perimeter minimizer, thus providing regular-
ity results for several Plateau-type problems and isoperimetric-type problems.
Finally, Part IV provides an introduction to the theory of minimizing clusters.
These last two parts are definitely more advanced, and contain the deeper ideas
Preface xv
and finer arguments presented in this book. Although their natural audience
will unavoidably be made of more expert readers, I have tried to keep in these
parts the same pedagogical point of view adopted elsewhere.
As I said, a “crash-course” on the theory of sets of finite perimeter, of about
130 pages, is available for beginners. The course starts with a revision of the
basic theory of Radon measures, temporarily excluding differentiation the-
ory (Chapters 1–4), plus some simple facts concerning weak gradients from
Section 7.2. The notion of distributional perimeter is then introduced and used
to prove the existence of minimizers in several variational problems, culminat-
ing with the solution of the Euclidean isoperimetric problem (Chapters 12–14).
Finally, the differentiation theory for Radon measures is developed (Chapter 5),
and then applied to clarify the geometric structure of sets of finite perimeter
through the study of reduced boundaries (Chapter 15).
Each part is closed by a set of notes and remarks, mainly, but not only,
of bibliographical character. The bibliographical remarks, in particular, are not
meant to provide a complete picture of the huge literature on the problems con-
sidered in this book, and are limited to some suggestions for further reading.
In a similar way, we now mention some monographs related to our subject.
Concerning Radon measures and rectifiable sets, further readings of excep-
tional value are Falconer [Fal86], Mattila [Mat95], and De Lellis [DL08].
For the classical approach to sets of finite perimeter in the context of func-
tions of bounded variation, we refer readers to Giusti [Giu84], Evans and
Gariepy [EG92], and Ambrosio, Fusco, and Pallara [AFP00].
The partial regularity theory of Part III does not follow De Giorgi’s origi-
nal approach [DG60], but it is rather modeled after the work of authors like
Almgren, Allard, Bombieri, Federer, Schoen, Simon, etc. in the study of area
minimizing currents and stationary varifolds. The resulting proofs only rely
on direct comparison arguments and on geometrically viewable constructions,
and should provide several useful reference points for studying more advanced
regularity theories. Accounts and extensions of De Giorgi’s original approach
can be found in the monographs by Giusti [Giu84] and Massari and Miranda
[MM84], as well as in Tamanini’s beautiful lecture notes [Tam84].
Readers willing to enter into other parts of GMT have several choices. The
introductory books by Almgren [Alm66] and Morgan [Mor09] provide initial
insight and motivation. Suggested readings are then Simon [Sim83], Krantz
and Parks [KP08], and Giaquinta, Modica, and Sou
ˇ
cek [GMS98a, GMS98b],
as well as, of course, the historical paper by Federer and Fleming [FF60]. Con-
cerning the regularity theory for minimizing currents, the paper by Duzaar and
Steffen [DS02] is a valuable source for both its clarity and its completeness.
Finally (and although, since its appearance, various crucial parts of the theory
xvi Preface
have found alternative, simpler justifications, and several major achievements
have been obtained), Federer’s legendary book [Fed69] remains the ultimate
reference for many topics in GMT.
I wish to acknowledge the support received from several friends and col-
leagues in the realization of this project. This book originates from the lecture
notes of a course that I held at the University of Duisburg-Essen in the Spring
of 2005, under the advice of Sergio Conti. The successful use of these unpub-
lished notes in undergraduate seminar courses by Peter Hornung and Stefan
M
¨
uller convinced me to start the revision and expansion of their content. The
work with Nicola Fusco and Aldo Pratelli on the stability of the Euclidean
isoperimetric inequality [FMP08] greatly influenced the point of view on sets
of finite perimeter adopted in this book, which has also been crucially shaped
(particularly in connection with the regularity theory of Part III) by several,
endless, mathematical discussions with Alessio Figalli. Alessio has also lec-
tured at the University of Texas at Austin on a draft of the first three parts, sup-
porting me with hundreds of comments. Another important contribution came
from Guido De Philippis, who read the entire book twice,givingmemuch
careful criticism and many useful suggestions. I was lucky to have the oppor-
tunity of discussing with Gian Paolo Leonardi various aspects of the theory of
minimizing clusters presented in Part IV. Comments and errata were provided
to me by Luigi Ambrosio (his lecture notes [Amb97] have been a major source
of inspiration), Marco Cicalese, Matteo Focardi, Nicola Fusco, Frank Morgan,
Matteo Novaga, Giovanni Pisante and Berardo Ruffini. Finally, I wish to thank
Giovanni Alberti, Almut Burchard, Eric Carlen, Camillo de Lellis, Michele
Miranda, Massimiliano Morini, and Emanuele Nunzio Spadaro for having ex-
pressed to me their encouragement and interest in this project.
I have the feeling that while I was busy trying to talk about the rock with-
out forgetting about the roll, some errors and misprints made their way to the
printed page. I will keep an errata list on my webpage.
This work was supported by the European Research Council through the
Advanced Grant n. 226234 and the Starting Grant n. 258685, and was com-
pleted during my visit to the Department of Mathematics and the Institute for
Computational Engineering and Sciences of the University of Texas at Austin.
My thanks to the people working therein for the kind hospitality they have
showntomeandmyfamily.
Francesco Maggi
Notation
Notation 1 We work in the n-dimensional Euclidean space R
n
, that is the n-
fold cartesian product of the space of real numbers R. Therefore x = (x
1
, ..., x
n
)
is the generic element of R
n
, and {e
i
}
n
i=1
is the canonical orthonormal basis
of R
n
. We associate with x ∈ R
n
\{0} the one-dimensional linear subspace
x
of R
n
,
x
= {tx : t ∈ R}, called the space spanned by x. We endow R
n
with
the Euclidean scalar product x · y =
n
i=1
x
i
y
i
. Given a linear subspace H of
R
n
, we denote by dim(H) its dimension. If dim(H) = k, then the orthogonal
space to H in R
n
is the (n − k)-dimensional linear space defined by
H
⊥
=
y ∈ R
n
:ifx ∈ H then y · x = 0
,
and we set x
⊥
=
x
⊥
for x 0. The Minkowski sum of E, F ⊂ R
n
is defined
as
E + F =
x + y : x ∈ E,y∈ F
,
with x + F = {x} + F if x ∈ R
n
.Ak-dimensional plane π in R
n
is a set of the
form π = x + H where x ∈ R
n
and H is a k-dimensional space in R
n
. When
k = 1 we simply say that π is a line in R
n
.GivenE ⊂ R
n
and λ>0weset
λ E =
λx : x ∈ E
.
Defining the Euclidean norm |x| = (
n
i=1
x
2
i
)
1/2
,theEuclidean open ball in
R
n
of center x and radius r > 0is
B(x, r) =
y ∈ R
n
: |y − x| < r
.
When x = 0wesetB(0, r) = B
r
and B
1
= B, so that B(x, r) = x + B
r
= x + rB.
We also set S
n−1
= ∂B = {x ∈ R
n
: |x| = 1} for the unit sphere in R
n
.Given
E, F ⊂ R
n
,thediameter of E and the distance between E and F are
diam(E) = sup
|x − y| : x,y ∈ E
,
dist(E, F) = inf
|x − y| : x ∈ E,y∈ F
.
xviii Notation
The interior, closure, and topological boundary (in the Euclidean topology)
of E ⊂ R
n
are denoted as usual as
˚
E, E, and ∂E respectively. We write E ⊂⊂ A
and say E is compactly contained in A if
E ⊂ A.
Notation 2 A family F of subsets of R
n
is disjoint if F
1
, F
2
∈F, F
1
F
2
implies F
1
∩ F
2
= ∅;itiscountable if there exists a surjective function
f : N →F;itisacovering of E ⊂ R
n
if E ⊂
F∈F
F.Apartition of E is a
disjoint covering of E which is composed of subsets of E.
Notation 3 (Linear functions) We denote by R
m
⊗ R
n
the vector space of
linear maps from R
n
to R
m
.IfT ∈ R
m
⊗ R
n
, then T (R
n
), the image of T ,is
a linear subspace of R
m
, and Ker T = {T = 0},thekernel of T , is a linear
subspace of R
n
. The dimension of T (R
n
) is called the rank of T , and T has
full rank if dim(T (R
n
)) = m.OnR
m
⊗ R
n
we define the operator norm,
T = sup
|Tx| : x ∈ R
n
, |x| < 1
, T ∈ R
m
⊗ R
n
.
We notice that T = Lip(T ), the Lipschitz constant of T on R
n
; see Chapter 7.
If T ∈ R
m
⊗ R
n
, then we define a linear map T
∗
∈ R
n
⊗ R
m
, called the adjoint
of T , through the identity
(Tx) ·y = x · (T
∗
y) , ∀x ∈ R
n
,y∈ R
m
.
Given v ∈ R
n
and w ∈ R
m
, we define a linear map w ⊗ v from R
n
to R
m
, setting
(w ⊗ v)x = (v · x)w, x ∈ R
n
.
When v 0 and w 0 we say that w ⊗v is a rank-one map, as we clearly have
(w ⊗ v)(R
n
) =
w
, Ker (w ⊗ v) = v
⊥
.
We also notice the useful relations
(w ⊗ v)
∗
= v ⊗ w, w ⊗ v = |v||w|.
Rank-one maps induce a canonical identification of R
m
⊗ R
n
with the space
R
m×n
of m × n matrices (a
i, j
)(1≤ i ≤ n,1≤ j ≤ m), having m rows and n
columns. Indeed, if V = {v
j
}
n
j=1
and W = {w
i
}
m
i=1
are orthonormal bases of R
n
and R
m
respectively, then, by definition of w
i
⊗ v
j
, we find that
T =
n
j=1
m
i=1
w
i
· (T v
j
)
w
i
⊗ v
j
.
Correspondingly, we associate T with the m ×n matrix (T
i, j
) with (i, j)th entry
given by T
i, j
= w
i
· (T v
j
). When n = m, this identification allows us to define
the notions of determinant and trace of a matrix for a linear map, by setting
det T = det(T
i, j
) , trace T = trace(T
i, j
) .
Notation xix
The functions det : R
n
⊗R
n
→ R and trace: R
n
⊗R
n
→ R are then independent
of the choice of V underlying the identification of R
m
⊗R
n
with the space R
m×n
,
and inherit their usual properties. For example, we have
det(TS) = det(T ) det(S ) , ∀T, S ∈ R
n
⊗ R
n
,
and det(Id
n
) = 1, where of course Id
n
x = x (x ∈ R
n
). If we denote by GL(n)
the set of invertible linear functions T ∈ R
n
⊗ R
n
, then
GL(n) =
T ∈ R
n
⊗ R
n
: det T 0
.
In particular, if n ≥ 2 then det(w ⊗ v) = 0 for every v, w ∈ R
n
. The trace defines
a linear function on R
n
⊗R
n
with trace(Id
n
) = n and, for every T, S ∈ R
n
⊗R
n
,
trace(T
∗
) = trace(T ) , trace(TS) = trace(ST) .
It is also useful to recall that for every v, w ∈ R
n
we have
trace(w ⊗ v) = v · w.
The trace operator can also be used to define a scalar product on R
m
⊗ R
n
:
T : S = trace(S
∗
T ) = trace(T
∗
S ) , T, S ∈ R
m
⊗ R
n
.
The norm corresponding to this scalar product (which does not coincide with
the operator norm) is defined as
|T | =
trace(T
∗
T ) , T ∈ R
m
⊗ R
n
.
Notation 4 (Standard product decomposition of R
n
into R
k
×R
n−k
) When we
need to decompose R
n
as the cartesian product R
k
× R
n−k
,1≤ k ≤ n − 1, we
denote by p : R
n
→ R
k
×{0} = R
k
and q : R
n
→{0}×R
n−k
= R
n−k
the horizontal
and vertical projections, so that x = (px , qx), x ∈ R
n
. We then introduce the
cylinder of center x ∈ R
n
and radius r > 0,
C(x, r ) =
y ∈ R
n
: |p(y − x)| < r , |q(y − x)| < r
,
and the k-dimensional ball of center z ∈ R
k
and radius r > 0,
D(z, r ) =
w ∈ R
k
: |z − w| < r
.
Moreover, we always abbreviate
C(0, r) = C
r
, C
1
= C , D(0, r) = D
r
, D
1
= D .
When k = n − 1, we alternatively set px = x
and qx = x
n
, so that x =
(x
, x
n
). Correspondingly we denote the gradient operator in R
n
and in R
n−1
,
respectively, by ∇ and ∇
= (∂
1
, ..., ∂
n−1
). If u : R
n
→ R has gradient ∇u(x) ∈
R
n
at x ∈ R
n
, then we set ∇u(x) = (∇
u(x),∂
n
u(x)).