HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL
CURRENTS
CAMILLO DE LELLIS AND EMANUELE NUNZIO SPADARO
Abstract. We give a new, simpler proof of the main approximation theorem for area
minimizing current contained in Almgren’s Big regularity paper, [2]. Our proof relies on
a new estimate concerning the higher integrability of the quantity called here the excess
density, which in turn we link to an analogous property of harmonic multiple valued
functions.
0. Introduction
In what follows, we consider integer rectifiable m-currents T supported in some open
cylinder Cr (y) = Br (y) × Rn ⊂ Rm × Rn and satisfying the following assumption:
π# T = Q !Br (y)"
and ∂T = 0,
(H)
where π : Rm × Rn → Rm is the orthogonal projection and m, n, Q are fixed positive
integers. For such currents, we denote by eT the non-negative excess measure and by
Ex(T, Cr (y)) the cylindrical excess, respectively defined by
!
"
eT (A) := M T (A × Rn ) − Q |A| for every Borel A ⊂ Br (y),
Ex(T, Cr (y)) :=
eT (Bs (x))
eT (Bs (x))
=
.
|Bs (x)|
ωm sm
The following theorem, proved by De Giorgi [5] in the case n = Q = 1, is due in its
generality to Almgren, who spends almost the entire third chapter of his Big regularity
paper [2] to accomplish it.
Theorem 0.1. There exist constants C, δ, ε0 > 0 with the following property. For every
mass-minimizing, integer rectifiable m-current T in the cylinder C4 which satisfies (H) and
E = Ex(T, C4 ) < ε0 , there exist a Q-valued function f ∈ Lip(B1 , AQ (Rn )) and a closed set
K ⊂ B1 such that
Lip(f ) ≤ CE δ ,
graph(f |K ) = T (K × R ) and |B1 \ K| ≤ CE
#
#
ˆ
# !
"
|Df |2 ##
1+δ
#M T C1 − Q ωm −
#
#≤CE .
2
B1
n
(0.1a)
1+δ
,
(0.1b)
(0.1c)
The most interesting aspects of Theorem 0.1 are the use of multiple-valued functions
(necessary when n > 1, as for the case of branched complex varieties) and the gain of
1
2
C. DE LELLIS AND E. N. SPADARO
a small power E δ in the three estimates (0.1). Observe that the usual approximation
theorems, which cover the case Q = 1 and stationary currents, are stated with δ = 0.
In this note, we provide a different, much simpler proof of Almgren’s theorem. In doing
so we establish a higher integrability estimate for the Lebesgue density dT of the excess
measure eT , called the excess density,
dT (x) := lim sup
s→0
eT (Bs (x))
.
ωm sm
Theorem 0.2. There exist constants p > 1 and C, ε0 > 0 such that, for every massminimizing, integer rectifiable m-current T satisfying (H) and E = Ex(T, C4 ) < ε0 , it
holds
ˆ
dp ≤ C E p .
(0.2)
{d≤1}∩B2
This estimate is not stated in [2], but it can be deduced from some of the arguments
therein. These arguments, which involve quite elaborate constructions and use several
intricate covering algorithms, are the most involved part of Almgren’s proof. In the case
Q = 1, we know a posteriori that T coincides with the graph of a C 1,α function over B2
(see [5], for instance). However, for Q ≥ 2 this conclusion does not hold and Theorem 0.2
has, therefore, an independent interest. The main contribution of the paper is to give a
much shorter and conceptually clearer derivation of (0.2).
However, we introduce several new ideas even in the derivation of Theorem 0.1 from
Theorem 0.2. In particular:
• we introduce a powerful maximal function truncation technique to approximate
general integer rectifiable currents with multiple valued functions;
• we give a simple compactness argument to conclude directly a first harmonic approximation of T ;
• we link Theorem 0.2 to a similar higher integrability property for the gradient of
Dir-minimizing multiple valued functions, observed here for the first time;
• we give a new proof of the existence of Almgren’s “almost projections” ρ# .
Remark 0.3. The careful reader will notice two important differences between the most
general approximation theorem of Almgren’s book and Theorem 0.1.
First of all, though the smallness hypothesis Ex(T, C4 ) < ε0 is the same, the estimates
corresponding to (0.1) are stated in [2] in terms of the “varifold excess”, a quantity smaller
than Ex. An additional argument, which we report in the appendix, shows that Ex and
the varifold excess are indeed comparable. This is obtained from a strenghtened version of
Theorem 0.1, included in the appendix, see Theorem A.1. Our proof is, to our knowledge
new and might have an independent interest.
Second, the most general result of Almgren is stated for currents in Riemannian manifolds. However, we believe that such generalization follows from standard modifications of
our arguments and we plan to address this issue elsewhere.
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
3
0.1. Graphical approximation of currents. Given a normal m-current T , following
[4] we can view the slice map x (→ )T, π, x* as a BV function taking values in the space
of 0-dimensional currents (endowed with the flat metric). Indeed, by a key estimate of
Jerrard and Soner (see [4] and [10]), the total variation of the slice map is controlled by
the mass of T and ∂T . In the same vein, following [6], Q-valued functions can be viewed
as Sobolev maps into the space of 0-dimensional currents.
Combining these points of view with standard truncation arguments, we develop a powerful and simple Lipschitz approximation technique, which gives a systematic tool to find
graphical approximations of integer rectifiable currents. For this purpose we introduce the
maximal function of the excess measure of a current T satisfying (H):
MT (x) :=
eT (Bs (x))
=
sup
Ex(T, Cs (x)).
ωm sm
Bs (x)⊂Br (y)
Bs (x)⊂Br (y)
sup
Our main approximation result is the following and relies on an improvement of the usual
Jerrard–Soner estimate.
Proposition 0.4 (Lipschitz approximation). There exist constants c, C > 0 with the
following property. Let T be an integer$rectifiable% m-current in C4s (x) satisfying (H)
and let η ∈ (0, c) be given. Set K := MT < η ∩ B3s (x). Then, there exists u ∈
1
Lip(B3s (x), AQ (Rn )) such that graph(u|K ) = T (K × Rn ), Lip(u) ≤ C η 2 and
"
C !
|B3s (x) \ K| ≤ eT {MT > η/2} .
(0.3)
η
In what follows, we will often choose η = E 2α (= Ex(T, C4s (x))2α ), for some α ∈
(0, (2m)−1 ). The map u given by Proposition 0.4 will then be called the E α -Lipschitz (or
briefly the Lipschitz ) approximation of T in C3s (x). We therefore conclude the following
estimates:
!
"
Lip(u) ≤ C E α , |B3s (x) \ K| ≤ C E −2 α eT {MT > E 2 α /2} ,
ˆ
!
"
(0.4)
|Du|2 ≤ eT {MT > E 2 α /2} .
B3s (x)\K
In particular, the function f in Theorem 0.1 is given by the E α -Lipschitz approximation
of T in C1 , for a suitable choice of α.
0.2. Harmonic approximation. The second step in the proof of Theorem 0.2 is a compactness argument which shows that, when T is mass-minimizing, the approximation f is
close to a Dir-minimizing function w, with an o(E) error.
Theorem 0.5 (o(E)-improvement). Let α ∈ (0, (2m)−1 ). For every η > 0, there exists
ε1 > 0 with the following property. Let T be a rectifiable, area-minimizing m-current in
C4s (x) satisfying (H). If E = Ex(T, C4s (x)) ≤ ε1 and f is the E α -Lipschitz approximation
of T in C3s (x), then
ˆ
B2s (x)\K
|Df |2 ≤ η eT (B4s (x)),
(0.5)
4
C. DE LELLIS AND E. N. SPADARO
and there exists a Dir-minimizing w ∈ W 1,2 (B2s (x), AQ (Rn )) such that
ˆ
ˆ
!
"2
2
G(f, w) +
|Df | − |Dw| ≤ η eT (B4s (x)).
B2s (x)
(0.6)
B2s (x)
This theorem is the multi-valued analog of De Giorgi’s harmonic approximation, which
is ultimately the heart of all the regularity theories for minimal surfaces. Our compactness
argument is, to our knowledge, new (even for n = 1) and particularly robust. Indeed, we
expect it to be useful in more general situations.
0.3. Higher integrability estimates and Theorem 0.2. The first key point in our
proof of Theorem 0.2 is that most of the energy of a Dir-minimizer lies where the gradient
is relatively small. The following quantitative version of this statement can be proved via
a classical reverse Hölder inequality (see [12] for a different proof and some improvements).
Curiously, though Chapter 3 of Almgren’s book has statements about the energy of Dirminimizing functions in various regions, Theorem 0.6 is stated nowhere and there is no
hint to reverse Hölder inequalities.
Theorem 0.6 (Higher integrability of Dir-minimizers). Let Ω& ⊂⊂ Ω ⊂⊂ Rm be open
domains. Then, there exist p > 2 and C > 0 such that
,Du,Lp (Ω! ) ≤ C ,Du,L2 (Ω)
for every Dir-minimizing u ∈ W 1,2 (Ω, AQ (Rn )).
(0.7)
Theorems 0.5 and 0.6 imply the following key estimate, which leads to Theorem 0.2 via
an elementary “covering and stopping radius” argument.
Proposition 0.7. For every κ > 0, there is ε2 > 0 with the following property. Let T be an
integer rectifiable, area-minimizing current in C4s (x) satisfying (H). If E = Ex(T, C4s (x)) ≤
ε2 , then
eT (A) ≤ κ Esm
for every Borel A ⊂ Bs (x) with |A| ≤ ε2 |B4s (x)|.
(0.8)
0.4. Almgren’s estimate and Theorem 0.1. Using now Theorem 0.2, we can prove
the following estimate, which is explicitely stated by Almgren and is the last ingredient,
in conjunction with Proposition 0.4, to derive Theorem 0.1.
Theorem 0.8. There exist constants σ, C > 0 such that, for every mass-minimizing,
integer rectifiable m-current T in C4 satisfying (H) and E = Ex(T, C4 ) < ε0 , it holds
!
"
eT (A) ≤ C E E σ + |A|σ
for every Borel A ⊂ B4/3 .
(0.9)
The proof of Theorem 0.8 is the only part where we follow essentially Almgren’s strategy.
The main point is to estimate the size of the set over which the graph of the Lipschitz
approximation f differs from T . As in many standard references, in the case Q = 1 this is
achieved comparing the mass of T with the mass of graph (f ∗ ρE ω ), where ρ is a smooth
convolution kernel and ω > 0 a suitably chosen constant.
However, for Q > 1, the space AQ (Rn ) is not linear and we cannot regularize f by
convolution. To bypass this problem, we follow Almgren and view AQ as a subset of a
large Euclidean space (via a biLipschitz embedding ξ). We can then convolve the map ξ ◦f
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
5
and project the convolution back on the set ξ(AQ ). But to do this efficiently in energetical
terms, we need an “almost” projection, denoted by ρ#µ , which is very little expensive in
terms of energy in the µ-neighborhood of ξ(AQ (Rn )) (µ is a parameter which will be tuned
accordingly). At this point Theorem 0.2 enters in a crucial way in estimating the size of
the set where the regularization of ξ ◦ f is far from ξ(AQ (Rn )).
We give here a clear and direct proof of the existence of ρ∗µ , which avoids some of
the technical complications of [2]. The ρ#µ are slightly different from Almgren’s almost
projections, but similar in spirit. Moreover, our argument is different and uses efficiently the
Kirzsbraun’s Theorem, yielding more explicit estimates in terms of the crucial parameters
involved.
0.5. Plan of the paper. The paper is organized as follows. Section 1 summarizes the
necessary tools from the theory of multiple valued functions. In particular, in this section
we prove a concentration-compactness lemma which is an essential step of Theorem 0.5.
Section 2 contains a summary of the Ambrosio-Kirchheim theory of currents with the
necessary improvement of the Jerrard–Soner estimate and gives the proof of Proposition
0.4 together with some results on the graphs of Q-valued functions. In Section 3 we
proceed with the proof of Theorem 0.5 and in Section 4 with that of the higher integrability
estimates in Theorem 0.6, Proposition 0.7 and Theorem 0.2. Section 5 is devoted to the
proof of Theorem 0.8 and the final derivation of the main Theorem 0.1. In Section 6 we
prove the existence of the projections ρ#µ . The Appendix contains three sections. Two of
them are devoted to give a strengthened version of Theorem 0.1, from which we derive a
comparison between Ex and the varifold excess. The third one gives a simple proof of the
structure properties of the graphs of Lipschitz multiple valued functions when viewed as
integer rectifiable currents.
1. Multiple valued functions
1.1. Q-valued functions. We follow here the notation and terminology of [6], which differ
slightly from those in Almgren’s big regularity paper [2]. Q-valued functions are maps from
a domain Ω ⊂ Rm taking values in the metric space AQ (Rn ) of unordered Q-tuple of vectors
in Rn . For the rigorous definitions of AQ , of the metric G, of Sobolev Q-valued maps and
of the Dirichlet energy, we refer to [6]. Here, we recall the following theorem (cp. to [6,
Theorem 2.1 and Corollary 2.2]).
Theorem 1.1 (The maps ξ and ρ). There exist N = N (Q, n) and an injective function
ξ : AQ (Rn ) → RN with the following three properties:
(i) Lip(ξ) ≤ 1;
(ii) Lip(ξ −1 |Q ) ≤ C(n, Q), where Q = ξ(AQ );
(iii) |Df | = |D(ξ ◦ f )| a.e. for every f ∈ W 1,2 (Ω, AQ ).
Moreover, there exists a Lipschitz projection ρ : RN → Q which is the identity on Q.
Many properties of classical Sobolev functions are inherited by Q-functions. In particular, we have the following results (cp. to [6, Section 2.2]).
6
C. DE LELLIS AND E. N. SPADARO
Lemma 1.2 (Lipschitz approximation of Sobolev maps). Let Ω ⊂ Rm be a Lipschitz
domain and f ∈ W 1,2 (Ω, AQ ). Then, for every ε > 0, there exists fε ∈ Lip(Ω, AQ ) such
that
ˆ
ˆ
!
"2
2
G(f, fε ) +
|Df | − |Dfε | ≤ ε.
Ω
Ω
If f |∂Ω ∈ W 1,2 (∂Ω, AQ ), then fε can be chosen to satisfy also
ˆ
ˆ
!
"2
2
G(f, fε ) +
|Df | − |Dfε | ≤ ε.
∂Ω
∂Ω
Lemma 1.3 (Interpolation lemma). There exists a constant C = C(m, n, Q) with the
following property. Assume f ∈ W 1,2 (Br , AQ (Rn )) and g ∈ W 1,2 (∂Br , AQ (Rn )) are given
maps such that f |∂Br ∈ W 1,2 (∂Br , AQ (Rn )). Then, for every ε ∈]0, r[ there is a function
h ∈ W 1,2 (Br , AQ (Rn )) such that h|∂Br = g and
ˆ
−1
Dir(h, Br ) ≤ Dir(f, Br ) + ε Dir(g, ∂Br ) + ε Dir(f, ∂Br ) + C ε
G(f, g)2 .
∂Br
Moreover, if f and g are Lipschitz, then h is as well Lipschitz with the estimate
&
'
−1
Lip(h) ≤ C Lip(f ) + Lip(g) + ε sup G(f, g) .
(1.1)
∂Br
1.2. Concentration compactness. The aim of this section is to show the following.
Lemma 1.4 (Concentration Compactness). Let (gl )l∈N be a sequence in W 1,2 (Ω, AQ ) with
(
supl Dir(gl , Ω) < +∞. Then, there are maps ζj ∈ W 1,2 (Ω, AQj ), with Q = Jj=1 Qj and
J ≥ 1, and points yjl ∈ Rn , with |yjl − yil | → +∞ for i 0= j, such that, up to a subsequence
(
(note relabeled), the Q-valued maps ωl = Jj=1 !τyjl ◦ ζj " satisfy
lim ,G(gl , ωl ),L2 (Ω) = 0 .
(1.2)
l→+∞
Moreover, if Ω& is an open subset of Ω and Jl a sequence of Borel sets with |Jl | → 0, then
)ˆ
*
ˆ
2
2
lim inf
|Dgl | −
|Dωl | ≥ 0,
(1.3)
l
and lim inf l
´
Ω! \Jl
Ω!
(|Dgl |2 − |Dωl |2 ) = 0 holds, if and only if lim inf l
´
(|Dgl | − |Dωl |)2 = 0.
Before coming to the proof, we recall two lemmas of [6].(Here, s(T ) and d(T ) are,
respectively, the separation and the diameter of a point T = i !Pi ":
$
%
s(T ) := min |Pi − Pj | : Pi 0= Pj
and d(T ) := max |Pi − Pj |.
i,j
Lemma 1.5. (Retractions [6, Lemma 3.7]). Let T ∈ AQ , r < s(T )/4. Then, there exists
a retraction ϑ : AQ → Br (T ) such that
(i) G(ϑ(S1 ), ϑ(S2 )) < G(S1 , S2 ) if S1 ∈
/ Br (T ),
(ii) ϑ(S) = S for every S ∈ Br (T ).
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
7
Q
Lemma 1.6. (Splitting [6, Lemma 3.8]). For every 0 < ε < 1, we set β(ε, Q) = (ε/3)3 .
Then, for every T ∈ AQ , there exists a point S ∈ AQ such that
β(ε, Q) d(T ) < s(S) and
G(S, T ) < ε s(S).
Proof of Lemma 1.4. First of all, by [6, Proposition 2.12], we can find ḡl ∈ AQ (Rn ) such
that
ˆ
ˆ
2
G(gl , ḡl ) ≤ c |Dgl |2 ≤ C,
where c and C are constants independent of l. We prove (1.2) by induction on Q and
distinguish two cases.
Case 1: lim inf l d(ḡl ) < ∞. After passing to a subsequence, we can then find yl ∈ Rn such
that the functions τyl ◦ gl are equi-bounded in the W 1,2 -distance. Hence, by the Sobolev
embedding [6, Proposition 2.11], there exists a Q-valued ζ such that τyl ◦ gl converges to ζ
in L2 . Note that, when Q = 1, we are always in this case.
Case 2: liml d(ḡl ) = +∞. By Lemma 1.6 there are points Sl ∈ AQ such that
s(Sl ) ≥ β1/8 d(ḡl ) and G(Sl , ḡl ) ≤ s(Sl )/8.
Set rl = s(Sl )/4 and let θl be the retraction into Brl (Sl ) provided by Lemma 1.5. Thus,
(
Sl = Ji=1 ki !Pli ", with mini*=j |Pli −Plj | = s(Sl ). In principle, the numbers I and ki depend
on l but, up to a subsequence, we can assume that they do not depend on l.
Clearly, the functions hl = θl ◦ gl satisfy Dir(hl , Ω) ≤ Dir(gl , Ω) and can be decomposed
as the superposition of ki -valued functions zli , with ki < Q,
J
+
# i$
hl =
zl ,
# $
with ,G(zli , ki Pli ),∞ ≤ rl .
i=1
The existence of ωl such that ,G(hl , ωl ),L2 → 0 follows, hence, by induction and, without loss of generality, we also can assume that liml |yil − yjl | = +∞ for i 0= j. Showing
,G(hl , gl ),L2 → 0, therefore, completes the proof of (1.2).
To this aim, recall first that {gl 0= hl } = {G (gl , Sl ) > rl } ⊆ {G (gl , ḡl ) > rl /2}. Thus,
ˆ
C
C
G (gl , ḡl ) ≤
.
| {gl 0= hl } | ≤ | {G (gl , ḡl ) > rl /2} | ≤ 2
rl {G(gl ,ḡl )> r2l }
(d(ḡl ))2
Since d(ḡl ) → +∞, we conclude |{gl 0= hl }| → 0. Next, since θl (ḡl ) = ḡl and Lip(θl ) = 1,
we have G(hl , ḡl ) ≤ G(gl , ḡl ). Therefore, by Sobolev embedding, for m ≥ 3 we infer
ˆ
ˆ
ˆ
ˆ
2
2
2
G(hl , gl ) =
G(hl , gl ) ≤ 2
G(hl , ḡl ) + 2
G(ḡl , gl )2
B2
{gl *=hl }
{hl *=gl }
{hl *=gl }
ˆ
2
≤4
G(ḡl , gl )2 ≤ ,G (gl , ḡl ),2L2∗ |{hl 0= gl }|1− 2∗
{hl *=gl }
≤
C
4
d(ḡl ) m−2
)ˆ
B2
2
|Dgl |
m+2
* m−2
.
8
C. DE LELLIS AND E. N. SPADARO
Recalling again that d(ḡl ) diverges, this shows ,G(hl , gl ),L2 → 0. The obvious modification
when m = 2 is left to the reader.
(
Now we come to the proof of (1.3). Arguing as in case 2, we find hl = i !zli " such that
,G(hl , gl ),L2 → 0, ,G(τ−yli ◦ zli , ζi ),L2 → 0 and |Dhl | ≤ |Dgl |. Therefore, we conclude that
∗
D(ξ ◦ τ−yil ◦ zli )0 D(ξ ◦ ζi ),
and hence
&
Dir(ζi , Ω ) =
ˆ
2
Ω!
|D(ξ ◦ ζi )| ≤ lim inf
l
ˆ
Ω! \J
l
|D(ξ ◦ τ−yil ◦
zli )|2
(1.4)
= lim inf
l
ˆ
Ω! \J
l
|Dzli |2 . (1.5)
(
Summing over i, we obtain (1.3). As for the final claim of the lemma, let ω = i !ζi " and
assume Dir(gl , Ω) → Dir(ω,
Ω). Set Jl := {gl 0= hl } and
! recall that" |Jl | → 0. Thus, by
´
2
(1.3), we conclude that Jl |Dgl | → 0 and hence, that |Dgl | − |Dhl | → 0 strongly in L2 .
On the other hand, we also infer
ˆ
ˆ
+ˆ
i 2
2
lim sup
|D(ξ ◦ τ−yil ◦ zl )| = lim sup |Dhl | ≤
|Dω|2 .
l
i
l
Ω
´
In conjunction with (1.5), this estimate leads to liml |D(ξ ◦ τ−yil ◦ zli )|2 = |D(ξ ◦ ζi )|2 ,
which, in turn, by (1.4), implies D(ξ ◦ τ−yil ◦ zli ) → D(ξ ◦ ζi ) strongly in L2 . Therefore,
|Dhl | → |Dω| in L2 , thus concluding the proof.
!
´
2. The Lipschitz approximation
2.1. The modified Jerrard–Soner estimate. For the sake of brevity, we do not introduce the machinery of metric space valued BV functions, developed by Ambrosio in [3],
which nevertheless remains the most elegant framework for this theory – cp. to [4]. We
adopt the definitions and the standard notation due to Federer, see [8] and [11]. An integer rectifiable
0-current S in Rn with finite mass is simply a finite sum of Dirac’s deltas:
(h
S = i=1 σi δxi , where h ∈ N, σi ∈ {−1, 1} for every i and the xi ’s are (not necessarily
distinct) points in Rn . The space of such measures, denoted by I0 (Rn ), is a Banach space
when endowed with the flat norm
$
%
F(S) := sup )S, ψ* : ψ ∈ C 1 (Rn ), ,ψ,∞ , ,Dψ,∞ ≤ 1 ,
(
where )S, ψ* = i σi ψ(xi ). Note that F(δx , δy ) = |x − y| if |x − y| ≤ 1.
Let T be an integer rectifiable m-dimensional normal current on C4 . The slicing map
x (→ )T, π, x* takes values in I0 (Rm+n ) and is characterized by (see Section 28 of [11])
ˆ
,
)T, π, x* , φ(x, ·) dx = )T, φ dx* for every φ ∈ Cc∞ (C4 ).
B4
Note that, in particular,
supp ()T,(
π, x*) ⊆ π −1 ({x}). Moreover, (H) implies that, if we
(
write )T, π, x* = i σi δ(x,yi ) , then i σi = Q.
Our estimates concerns the push-forwards of the slices )T, π, x* into the vertical direction,
!
"
Tx := q' )T, π, x* ∈ I0 (Rn ),
(2.1)
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
9
where q : Rm+n → Rn is the orthogonal projection on the last n components. Tx is
characterized through the identity
ˆ
)Tx , ψ* ϕ(x) dx = )T, ϕ(x) ψ(y) dx* for every ϕ ∈ Cc∞ (B4 ), ψ ∈ Cc∞ (Rn ).
B4
Proposition 2.1 (Modified BV estimate). Let T be an integer rectifiable current in C4
satisfying (H). For every ψ ∈ Cc∞ (Rn ), set Φψ (x) := )Tx , ψ*. If ,ψ,∞ , ,Dψ,∞ ≤ 1, then
Φψ ∈ BV (B4 ) and satisfies
!
"2
|DΦψ |(A) ≤ 2 eT (A) M (T (A × Rn ))
for every Borel A ⊂ B4 .
(2.2)
Note that (2.2) is a refined version of the usual Jerrard–Soner estimate, where the right
hand side would rather be (T (A × Rn ))2 (cp. to [4]). Note also that assumption (H) can
be dropped if in (2.2) eT is replaced by its total variation.
Proof. It is enough to prove (2.2) for every open set A ⊆ B4 . To this aim, recall that
&ˆ
'
∞
m
|DΦψ |(A) = sup
Φψ (x) divϕ(x) dx : ϕ ∈ Cc (A, R ), ,ϕ,∞ ≤ 1 .
(2.3)
A
For any vector field ϕ as in (2.3), (div ϕ(x)) dx = dα, where
+
α=
ϕj dx̂j and dx̂j = (−1)j−1 dx1 ∧ · · · ∧ dxj−1 ∧ dxj+1 ∧ · · · ∧ dxm .
j
Moreover, by the characterization of the slice map, we have
ˆ
ˆ
Φψ (x) divϕ(x) dx =
)Tx , ψ(y)* divϕ(x) dx = )T, ψ(y) div ϕ(x) dx*
A
B4
= )T, ψ dα* = )T, d(ψ α)* − )T, dψ ∧ α* = − )T, dψ ∧ α* ,
(2.4)
where in the last equality we used the hypothesis ∂T C4 = ∅.
Observe that the m-form dψ ∧ α has no dx component, since
dψ ∧ α =
m +
n
+
j=1 i=1
(−1)j−1
∂ψ
(y) ϕj (x) dy i ∧ dx̂j .
dy i
4 (see Section 25 of [11] for
Let 4e be the m-vector orienting Rm and write T4 = (T4 · 4e) 4e + S
4 · ,T , , dψ ∧ α*
the scalar product on m-vectors). We then conclude that )T, dψ ∧ α* = )S
and
ˆ
ˆ
ˆ
.
.
!
"2 /
!
"/
2
4
4
|S| d ,T , =
1 − T · 4e
d ,T , ≤ 2
1 − T4 · 4e d ,T , = 2 eT (A).
A×Rn
A×Rn
A×Rn
10
C. DE LELLIS AND E. N. SPADARO
Since |dψ ∧ α| ≤ ,Dψ,∞ ,ϕ,∞ ≤ 1, the Cauchy–Schwartz inequality yields
ˆ
ˆ
4
Φψ (x) div ϕ(x) dx ≤ | )T, dψ ∧ α* | = |)S · ,T , , dψ ∧ α*| ≤ |dψ ∧ α|
A
)ˆ
* 21
A×Rn
0
4 d ,T ,
|S|
M(T (A × Rn ))
n
A×R
0
√ 0
≤ 2 eT (A) M(T (A × Rn )).
≤
2
Taking the supremum over all such ϕ’s, we conclude (2.2).
4 d ,T ,
|S|
!
2.2. The Lipschitz approximation technique. For a vector measure ν in B4r , |ν| denotes its total variation and M (ν) its local maximal function:
M (ν)(x) :=
|ν|(Bs (x))
.
ωm sm
0<s<4 r−|x|
sup
We recall the following proposition (see for instance Section 6.6.2 of [7], up to the necessary
elementary modifications), a fundamental ingredient in the proof of Proposition 0.4.
Proposition 2.2. There is a dimensional constant C with the following property. If ν is
a vector measure in B4r , θ ∈]0, ∞[ and Jθ := {x ∈ B3r : M (ν) ≤ θ}, then
|Jθ | ≤
C
|ν|(B4r ).
θ
(2.5)
If in addition ν = Df for some f ∈ BV (B4r ), then
|f (x) − f (y)| ≤ C θ |x − y| for a.e. x, y ∈ Jθ .
(2.6)
Proof of Proposition 0.4. Since the statement is invariant under translations and dilations,
without loss of generality we assume x = 0 and s = 1. Consider
the slices Tx ∈ I0 (Rn )
´
of T (as defined in (2.1)). Recall that M(T A × Rn ) = A M(Tx ) for every open set A
(cp. to [11, Lemma 28.5]). Therefore,
M(T Cr (x))
≤ MT (x) + Q for almost every x.
r→0
ωm rm
M(Tx ) ≤ lim
Without loss of generality, we can assume c < 1. Hence, for almost every point in K,
being η < 1, we have that M(Tx ) < Q + 1. On the other hand, M(Tx ) ≥ Q for every x,
because π' T =(
Q !B4 ". Thus, Tx is the sum of Q positive Dirac’s delta for
( every x ∈ K,
that is, Tx =
δ
for
some
measurable
functions
g
.
We
set
g
:=
i
i gi (x)
i !gi ", so that
n
g : K → AQ (R ).
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
11
For every ψ ∈ Cc∞ (Rn ), by Proposition 2.1 we deduce that
)
*2
|DΦψ |(Br (x))
2 eT (Br (x)) M(T, Cr (x))
2
M (|DΦψ |)(x) = sup
≤ sup
|Br |
|Br |2
0<r≤4−|x|
0<r≤4−|x|
!
"
2eT (Br (x)) eT (Br (x)) + Q |Br |
≤ sup
|Br |2
0<r≤4−|x|
≤ 2 MT (x)2 + 2 Q MT (x) ≤ C MT (x).
Hence, by Proposition 2.2, this implies the existence of a constant C > 0 such that
#
#
#+
#
+
1
#
#
ψ(gi (x)) −
|Φψ (x) − Φψ (y)| = #
ψ(gi (y))# ≤ C η 2 |x − y| for a.e. x, y ∈ K.
#
#
i
i
Taking the supremum over all the ψ ∈ Cc∞ (Rn ) with ,ψ,∞ , ,Dψ,∞ ≤ 1, we deduce that
F(g(x) − g(y)) ≤ C η 1/2 |x − y|.
(2.7)
It is well-known that there is a constant C such that G(T1 , T2 ) ≤ C F(T1 − T2 ), for every
T1 , T2 ∈ AQ (Rn ) ⊂ I0 (Rn ), if F(T1 − T2 ) is small enough. Therefore, from (2.7), since
η < c and s = 1, for c small enough, we infer that g can be viewed as a Lipschitz map to
(AQ (Rn ), G). Recalling [6, Theorem 1.7], we can extend g to a map u : B3 → (AQ (Rn ), G)
with constant C η 1/2 . Clearly, u(x) = Tx for almost every point x ∈ K, which implies
graph(u|K ) = T (K × Rn ). Finally, (0.3) follows directly from Proposition 2.2.
!
2.3. Graphs of Q-valued functions. We conclude the section with a result
( on the graphs
¯
of Q-valued functions.
a Lipschitz f : Ω → AQ , we set f (x) = i !(x, fi (x))" and
$
( # Given
consider Df¯ = i Df¯i its differential (see [6, Section 1.3]). We set, moreover,
1
# #
!
"
#J f¯i # (x) = det Df¯i (x) · Df¯i T (x) ,
and, denoting by 4e the standard m-vector e1 ∧ . . . ∧ em in Rm ,
Df¯i (x)#4e
(e1 + ∂1 fi (x)) ∧ · · · ∧ (em + ∂m fi (x))
# #
T4fi (x) = ## ¯ ##
=
∈ Λm (Rm+n ).
¯
#
#
Jfi (x)
Jfi (x)
The current graph(f ) induced by the graph of f is, hence, defined by the following position:
ˆ +2
3# #
)graph(f ), ω* =
ω (x, fi (x)) , T4i (x) #J¯fi # (x) d Hm (x) ∀ ω ∈ D m (Rm+n ).
Ω
i
As one expects, we have the formula
ˆ +
ˆ +1
# #
!
"
k
¯
#
#
det Df¯i · Df¯iT d Hk .
M (graph(f )) =
Jfi d H =
Ω
Ω
i
(2.8)
i
Moreover ∂graph(f ) is supported in ∂Ω × R and is given by the current graph(f |∂Ω ). All
these facts are proved in Appendix C (see also [2, Section 1.5(6)]).
The following is a Taylor expansion for the mass of the graph of a Q-valued function.
n
12
C. DE LELLIS AND E. N. SPADARO
Proposition 2.3. There is a constant C > 1 such that, for every g ∈ Lip(Ω, AQ (Rn )) with
Lip(g) ≤ 1 and for every Borel set A ⊂ Ω, it holds
ˆ
ˆ
1 + C Lip(g)2
1 − C −1 Lip(g)2
2
|Dg| ≤ egraph(g) (A) ≤
|Dg|2 .
(2.9)
2
2
A
A
(
Proof. Note that det(Df¯i · Df¯iT )2 = 1 + |Dfi |2 + |α|≥2 (Miα )2 , where α is a multi-index
√
2
and Miα the corresponding minor of order |α| of Dfi . Since 1 + x2 ≤ 1 + x2 and Mfαi ≤
C |Df ||α| ≤ C |Df |2 Lip(f )|α|−2 ≤ C |Df |2 when |α| ≥ 2, we conclude
* 21
+ˆ )
+
M (graph(f |A )) =
1 + |Dfi |2 +
(Mfαi )2
i
A
≤ Q |A| +
ˆ
A
!1
2
|α|≥2
"
´
|Df |2 + C |Df |4 ≤ Q |A| + 12 (1 + C Lip(f )2 ) A |Df |2 .
√
2
4
On the other hand, exploiting the lower bound 1 + x2 − x4 ≤ 1 + x2 ,
+ˆ 0
+ˆ !
"
2
M (graph(f |A )) ≥
1 + |Dfi | ≥
1 + 21 |Dfi |2 − 41 |Dfi |4
i
≥
A
+ˆ )
i
i
1+
A
= Q |A| +
This concludes the proof.
1
2
!
1
|Dfi |2
2
A
1
4
2
2
− Lip(f ) |Dfi |
1 − 14 Lip(f )2
"´
A
*
|Df |2 .
!
3. The o(E)-improved approximation
In this section we prove Theorem 0.5. Both arguments for (0.5) and (0.6) are by contradiction and builds upon the construction of a suitable comparison current.
3.1. Proof of (0.5). Without loss of generality, assume x = 0 and s = 1. Arguing by
contradiction, there exist a constant c1 , a sequence of currents (Tl )l∈N and corresponding
Lipschitz approximations (fl )l∈N such that
ˆ
El := Ex(Tl , C4 ) → 0 and
|Dfl |2 ≥ c1 El .
B2 \Kl
El2´α /2}
Set Hl := {MTl ≤
expansion (2.9) gives
⊂ B3 . Since Tl and graph(fl ) coincide over Kl , the Taylor
2
|Df
l | ≤ C eTl (Kl \ Hl ). Together with (0.4), this leads to
Kl \Hl
ˆ
c1 El ≤
|Dfl |2 ≤ C eTl (B4 \ Hl ), ∀ s ∈ [2, 3],
Bs \Hl
which in turn, for 2 c2 = c1 /C, implies
eTl (Hl ∩ Bs ) ≤ eTl (Bs ) − 2 c2 El .
(3.1)
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
Since Lip(fl ) ≤ C Elα → 0, the Taylor expansion and (3.1) give, for l big enough,
ˆ
|Dfl |2
≤ eTl (Hl ∩ Bs ) ≤ eTl (Bs ) − c2 El , ∀ s ∈ [2, 3].
2
Hl ∩Bs
13
(3.2)
Our aim is to show that (3.2) contradicts the minimality of Tl . To this extent, we
construct a competitor current in different steps.
√
Step 1: splitting. Consider the maps gl := fl / El . Since supl Dir(gl , B3 ) < ∞ and
(
|B3 \ Hl | → 0, we can find maps ζj and ωl = Jj=1 !τyjl ◦ ζj " as in Lemma 1.4 such that
´
(a1 ) βl := B3 G(gl , ωl )2 → 0;
(b1 ) lim inf l (Dir(gl , Ω ∩ Hl ) − Dir(ωl , Ω)) ≥ 0 for every Ω ⊂ B3 .
(
Let ω := j !ζj " and note that |Dωl | = |Dω|.
Step 2: choice of a suitable radius. From the estimates in (0.4), one gets
!
"
!
"
!
"
M Tl − graph(fl ), C3 = M Tl , (B3 \ Kl ) × Rn + M graph(fl ), (B3 \ Kl ) × Rn
≤ Q |B3 \ Kl | + El + Q |B3 \ Kl | + C |B3 \ Kl | Lip(fl )
≤ El + C El1−2α ≤ C El1−2α .
(3.3)
With a slight abuse of notation, we write (Tl − graph(fl )) ∂Cr for )Tl − graph(fl ), ϕ, r*,
where ϕ(z, y) = |z| and introduce the real valued function ψl given by
ˆ
!
"
2α−1
−1
ψl (r) := El
M (Tl − graph(fl )) ∂Cr + Dir(gl , ∂Br ) + Dir(ω, ∂Br ) + βl
G(gl , ωl )2 .
∂Br
´3
From (a1 ), (b1 ) and (3.3), lim inf l 2 ψl (r) dr < ∞. By Fatou’s Lemma, there is r ∈ (2, 3)
and a subsequence, not relabeled, such that liml ψl (r) < ∞. Hence, it follows that:
´
(a2 ) ∂Br G(gl , ωl )2 → 0,
(b2 ) Dir(ω
some M < ∞,
! l , ∂Br ) + Dir(gl , ∂Br") ≤ M for
1−2α
(c2 ) M (Tl − graph(fl )) ∂Br ≤ C El
.
Step 3: Lipschitz approximation of ωl . We now apply Lemma 1.2 to the ζj ’s and find
Lipschitz maps ζ̄j with the following requirements:
(i) Dir(ζ̄j , Br ) ≤ Dir(ζj , Br ) + c2 /(2 Q),
(ii) ´Dir(ζ̄j , ∂Br ) ≤ Dir(ζj , ∂Br ) + 1/Q,
(iii) ∂Br G(ζ̄j , ω)2 ≤ c22 /(26 C Q (M + 1)), where C is the constant in the interpolation
Lemma 1.3.
(
The function 6l := !τyil ◦ ζ̄i " is, then, a Lipschitz approximation of ωl which, for (i)-(iii),
(b1 ), (b2 ) and (3.2), satisfies, for l big enough,
(a3 ) Dir(6l , Br ) ≤ Dir(ω, Br ) + c2 /2 ≤ 2 eTl (Br ) − c2 ,
(b3 ) ´Dir(6l , ∂Br ) ≤ Dir(ω, ∂Br ) + 1 ≤ M + 1,
(c3 ) ∂Br G(6l , ωl )2 ≤ c22 /(26 C (M + 1)).
14
C. DE LELLIS AND E. N. SPADARO
Step 4: patching graph(6l ) and Tl . Next, apply the interpolation Lemma 1.3 to 6l and
gl with ε = c2 /(24 (M + 1)). We then find maps ξl such that ξl |∂Br = gl |∂Br and, from (a2 ),
(a3 )-(c3 ), for l large enough,
ˆ
−1
Dir (ξl , Br ) ≤ Dir (6l , Br ) + ε Dir (6l , ∂Br ) + ε Dir(gl , ∂Br ) + C ε
G (6l , gl )2
≤
2 El−1 eTl (Br )
c2
c2 c2 c2
≤ 2 El−1 eTl (Br ) − .
− c2 + + +
8
8
4
2
∂Br
(3.4)
α−1/2
Moreover, from the last estimate in Lemma 1.3, if follows that Lip(ξl ) ≤ CEl
, since
+
α−1/2
α−1/2
Lip(gl ) ≤ C El
, Lip(6l ) ≤
Lip(ζ̄j ) ≤ C and ,G(6l , gl ),∞ ≤ C + C El
.
j
√
Set zl := El ξl and consider the current Zl := graph(ξl ). Since zl |∂Br = fl |∂Br , ∂Zl =
graph(fl ) ∂Br . Therefore, from (c2 ), M(∂(Tl Br − Zl )) ≤ CEl1−2α . From the isoperimetric inequality (see [11, Theorem 30.1]), there exists an integral current Rl such that
∂Rl = ∂(Tl Cr − Zl ) and M(Rl ) ≤ CE (1−2α)m/(m−1) .
Set finally Wl = Tl (C4 \ Cr ) + Zl + Rl . By construction, it holds obviously ∂Wl = ∂Tl .
Moreover, since α < 1/(2m), for l large enough, Wl contradicts to the minimality of Tl :
ˆ
(1−2 α)m
"
!
|Dzl |2
2α
M(Wl ) − M(Tl ) ≤ Q |Br | + 1 + C El
+ C El m−1 − Q|Br | − eTl (Br )
2
Br
)
*
(1−2 α)m
(3.4)!
"
c2 El
2α
+ C El m−1 − eTl (Br )
≤ 1 + C El
eTl (Br ) −
4
(1−2 α)m
m−1
≤ −c2 El + CEl1+2α + C El
< 0.
3.2. Proof of (0.6). The proof is again by contradiction. Let (Tl )l be a sequence with
vanishing El := Ex(Tl , C4 ) and contradicting (0.6), and perform again Steps 1 and 3.
Clearly, since (0.6) does not hold, up to extraction of a subsequence, we can assume that
´
´
(i) either liml B2 |Dgl |2 > |Dω|2 ,
(ii) or, for some j, ζ j is not Dir-minimizing in B2 .
Indeed, in case one between (i) and (ii) does not hold, it suffices to set w = ωl , because,
when each ζj is harmonic, inf x∈B2 G(τyil ◦ ζi (x), τyjl ◦ ζj (x)) → ∞ and, by the Maximum
principle [6, Proposition 3.5], ωl is harmonic for l large enough as well.
In case (i), since, for large l,
ˆ
ˆ
2
|Dωl | ≤
|Dgl |2 − 2 c2 ≤ El−1 eT (Br ) − c2 ,
Br
Br
for some positive constant c2 , we can arguing exactly as in the proof of (0.5).
In case (ii), we find a competitor for ζ j and, hence, new functions ω̂l such that ω̂l |∂Br =
ωl |∂Br and
ˆ
ˆ
ˆ
2
2
lim
|Dω̂l | ≤ lim
|Dωl | ≤ lim
|Dgl |2 − 2 c2 ≤ El−1 eT (Br ) − c2 .
l
Br
l
Br
l
Br
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
15
We then can argue as above with ω̂l in place of ωl , thus concluding the proof.
4. Higher integrability estimates
We come now to the proof of the various higher integrability estimates, culminating with
the proof of one of the two main result in the paper, Theorem 0.2.
4.1. Higher integrability of Dir-minimizers. In this section we prove Theorem 0.6.
The theorem is a corollary of Proposition 4.1 below and a Gehring’s type lemma due to
Giaquinta and Modica (see [9, Proposition 5.1]).
Proposition 4.1. Let 2 (m−1)
< s < 2. Then, there exist C = C(m, n, Q, s) such that, for
m
every u : Ω → AQ Dir-minimizing,
)ˆ
* 12
)ˆ
* 1s
$
%
2
s
−
|Du|
≤C −
|Du|
, ∀ x ∈ Ω, ∀ r < min 1, dist(x, ∂Ω)/2 .
Br (x)
B2r (x)
Proof. Let u : Ω → AQ (Rn ) be a Dir-minimizing map and let ϕ = ξ ◦ u : Ω → Q ⊂ RN .
Since the estimate is invariant under translations and rescalings, it is enough to prove it
for x = 0 and r = 1. We assume, therefore Ω = B2 . Let ϕ̄ ∈ RN be the average of ϕ on
B2 . By Fubini’s theorem, there exists ρ ∈ [1, 2] such that
ˆ
ˆ
s
s
(|ϕ − ϕ̄| + |Dϕ| ) ≤ C
(|ϕ − ϕ̄|s + |Dϕ|s ) ≤ C,Dϕ,sLs (B2 ) .
∂Bρ
B2
1
Consider ϕ|∂Bρ . Since 21 > 1s − 2 (m−1)
, we can use the embedding W 1,s (∂Bρ ) 8→ H 1/2 (∂Bρ )
(see, for example, [1]). Hence, we infer that
4
4
4ϕ|∂Bρ − ϕ̄4 1
≤ C ,Dϕ, s
,
(4.1)
L (B2 )
H 2 (∂Bρ )
where , · ,H 1/2 = , · ,L2 + | · |H 1/2 and | · |H 1/2 is the usual H 1/2 -seminorm. Let ϕ̂ be the
harmonic extension of ϕ|∂Bρ in Bρ . It is well known (one could, for example, use the result
in [1] on the half-space together with a partition of unity) that
ˆ
|Dϕ̂|2 ≤ C(m) |ϕ|2 12
.
(4.2)
H (∂Bρ )
Bρ
Therefore, using (4.1) and (4.2), we conclude ,Dϕ̂,L2 (Bρ ) ≤ C ,Dϕ,Ls (B2 ) . Now, since
ρ ◦ ϕ̂|∂Bρ = u|∂Bρ and ρ ◦ ϕ̂ takes values in Q, by the the minimality of u and the Lipschitz
properties of ξ, ξ −1 and ρ, we conclude
5ˆ
6 12
)ˆ
*1
)ˆ
*1
)ˆ
*1
B1
|Du|2
2
≤C
Bρ
|Dϕ̂|2
≤C
B2
|Dϕ|s
s
≤C
B2
|Du|s
s
.
!
16
C. DE LELLIS AND E. N. SPADARO
4.2. Weak Almgren’s estimate. Here we prove Proposition 0.7. Without loss of generality, we can assume s = 1 and x = 0. Let f be the E α -Lipschitz approximation in C3 ,
with α ∈ (0, 1/(2m)). Fix η = κ/4 and choose ε2 (κ) ≤ ε1 (η). Arguing as in Step 4 of
subsection 3.1, we find a radius r ∈ (2, 3) and a current R such that
∂R = (T − graph(f )) ∂Br
and M(R) ≤ CE (1−α)m/(m−1) .
Hence, by the minimality of T and using the Taylor expansion in Proposition 2.3, we have
M(T Cr ) ≤ M(graph(f ) Cr + R) ≤ M(graph(f ) Cr ) + C Ex(T, C4 )
ˆ
|Df |2
≤ Q |Br | +
+ C Ex(T, C4 )1+ν ,
2
Br
(1−2α) m
m−1
(4.3)
where ν is a fixed constant. On the other hand, using again the Taylor expansion for the
part of the current which coincides with the graph of f , we deduce as well that
!
"
!
"
M(T Cr ) ≥ M T ((Br \ K) × Rn ) + M T ((Br ∩ K) × Rn )
ˆ
!
"
|Df |2
n
≥ M T ((Br \ K) × R ) + Q |Br ∩ K| +
− C Ex(T, C4 )1+ν .
2
Br ∩K
(4.4)
Subtracting (4.4) from (4.3), by the choice of ε2 , we deduce from (0.5),
ˆ
|Df |2
κE
eT (Br \ K) ≤
+ CE 1+ν ≤
+ CE 1+ν .
2
2
Br \K
(4.5)
Let now A ⊂ B1 be such that |A| ≤ ε2 |B4 |. Combining (4.5) with the Taylor expansion
and Theorem 0.6, we finally get, for some constants C and q > 1 (independent of E) and
for ε2 (κ) sufficiently small,
ˆ
ˆ
|Df |2
|Dw|2 κ E
1+ν
eT (A) ≤ eT (A \ K) +
+CE
≤ eT (Br \ K) +
+
+ C E 1+ν
2
2
4
A
A
3κE
≤
+ C|A|1−1/q E + C E 1+ν ≤ κ E.
4
4.3. Proof of Theorem 0.2. The theorem is a consequence of the following estimate:
there exists √
constants γ ≥ 2m and β > 0 such that, for every c ∈ [1, (γ E)−1 ] and s ∈ [2, 4]
m
with s + 2/ c ≤ 4,
ˆ
ˆ
−β
d≤γ
d.
(4.6)
{γ c E≤d≤1}∩Bs
{ cγE ≤d≤1}∩Bs+ m√2
c
Iterate (4.6) to obtain
ˆ
{γ 2 k+1 E≤dT ≤1}∩B2
dT ≤ γ
−k β
ˆ
{γ E≤dT ≤1}∩B4
dT ≤ γ −k β 4m E,
(4.7)
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
for every k ≤ L := 6(logγ (λ/E)−1)/27 (note that, since γ ≥ 2m , it holds 2
Therefore, setting
Ak = {γ 2k−1 E ≤ dT < γ 2k+1 E} for k = 1, . . . , L,
(
k
17
γ −2k/m ≤ 2).
A0 = {dT < γ E} and AL+1 = {γ 2L+1 E ≤ dT ≤ 1},
for p < 1 + β/2, we conclude the theorem:
ˆ
ˆ
L+1
L+1 ˆ
+
+
p
p
(2k+1) (p−1) p−1
γ
E
dT ≤
dT =
B2
k=0
Ak ∩B2
(4.7)
Ak ∩B2
k=0
dT ≤ C
L+1
+
k=0
γ k (2p−β) E p ≤ C E p .
We now come to the proof of (4.6). Let NB be the constant in Besicovich’s covering
theorem and choose P ∈ N so large that NB < 2P −1 . Set γ = max{2m , 1/ε2 (2−2 m−P )} and
β = − logγ (NB /2P −1 ), where ε2 is the constant in Proposition 0.7.
Let c and s be any real numbers as above. First of all, we prove that, for a.e. x ∈
{γ c E ≤ dT ≤ 1} ∩ Bs , there exists rx such that
E(T, C4rx (x)) ≤ c E
E(T, Cρ (x)) ≥ c E
and
∀ρ ∈]0, 4 rx [.
m
Indeed, since dT (x) = limr→0 E(T, Cr (x)) ≥ γ c E ≥ 2 c E and
(4.8)
eT (Bρ (x))
4m E
4
≤
≤ c E for ρ ≥ √
,
m
m
m
ωm ρ
ρ
c
√
√
it suffices to choose 4rx = min{ρ ≤ 4/ m c : E(T, Cρ (x)) ≤ cE}. Note that rx ≤ 1/ m c.
Consider now the current T restricted to C4rx (x). We note that, for the choice of γ,
setting A = {γ c E ≤ dT },
!
"
E
Ex(T, C4rx (x)) ≤ c E ≤
≤ ε2 2−2m−P ,
γE
!
"
c E |B4rx (x)|
|A| ≤
≤ ε2 2−2m−P |B4rx (x)|.
cE γ
Hence, we can apply Proposition 0.7 to T C4rx (x) to get
ˆ
ˆ
dT ≤
dT ≤ eT (A) ≤ 2−2 m−P eT (B4rx (x))
E(T, Cρ (x)) =
Brx (x)∩{γ c E≤dT ≤1}
A
(4.8)
≤ 2−2 m−P (4 rx )m ωm Ex(T, C4rx (x)) ≤ 2−P eT (Brx (x)).
Thus,
eT (Brx (x)) =
ˆ
≤
ˆ
Brx (x)∩{dT >1}
dT +
A
(4.8), (4.9) !
≤
−P
2
dT +
ˆ
ˆ
Brx (x)∩{ cγE ≤dT ≤1}
Brx (x)∩{ cγE ≤dT ≤1}
+γ
−1
"
dT +
eT (Brx (x)) +
ˆ
dT +
cE
ωm rxm
γ
ˆ
Brx (x)∩{dT < cγE }
Brx (x)∩{ cγE ≤dT ≤1}
dT .
(4.9)
dT
(4.10)
18
C. DE LELLIS AND E. N. SPADARO
Therefore, recalling that γ ≥ 2m ≥ 4, from (4.9) and (4.10) we infer that
ˆ
ˆ
ˆ
2−P
−P +1
dT ≤
dT ≤ 2
dT .
1 − 2−P − γ −1 Brx (x)∩{ cγE ≤dT ≤1}
Brx (x)∩{γ c E≤dT ≤1}
Brx (x)∩{ cγE ≤dT ≤1}
Finally, by Besicovich’s covering theorem, we choose NB families of disjoint
balls Brx (x)
√
whose union covers {γ c E ≤ dT ≤ 1} ∩ Bs and, recalling that rx ≤ 2/ m c for every x, we
conclude
ˆ
ˆ
−P +1
d T ≤ NB 2
dT ,
{γ c E≤dT ≤1}∩Bs
{ cγE ≤dT ≤1}∩Bs+ m√2
c
which, for the above defined β, implies (4.6).
5. Almgren’s estimate and Theorem 0.1
5.1. Proof of Theorem 0.8. Let α ∈ (0, (2m)−1 ) and fix the E α -Lipschitz approximation
f . As pointed out in the introduction, the strategy here is to consider a suitable convolution
of the approximation f in order to find a competitor with energy over B3/2 \ K which is a
superlinear power of the excess. In order to do this, we need to regularize ξ ◦ f and to use
Almgren’s map ρ# whose properties are stated and proved in Proposition 6.1.
One of the main point in the proof is the following consequence of Theorem 0.2: since
|Df |2 ≤ C dT and dT ≤ E 2 α ≤ 1 in K, there exists q = 2 p > 2 such that
)ˆ
* 1q
1
|Df |q
≤ C E2.
(5.1)
K
Proposition 5.1. Let T be as in Theorem 0.1 and let f be its E α -Lipschitz approximation.
Then, there exist constants δ, C > 0 and a subset B ∈ [1, 2] with |B| > 1/2 such that, for
every s ∈ B, there exists a Q-valued function g ∈ Lip(Bs , AQ ) which satisfies g|∂Bs = f |∂Bs ,
Lip(g) ≤ C E α and
ˆ
ˆ
Bs
|Dg|2 ≤
Bs ∩K
|Df |2 + C E 1+δ .
(5.2)
Proof. We give an explicit construction of g & := ξ ◦ g starting from f & := ξ ◦ f and the
projection ρ#µ given in Proposition 6.1 with a constant µ > 0 to be fixed later: then,
composing with ξ −1 , we recover g. In order to simplify the notation, we simply write ρ#
in place of ρ#µ .
To this aim, let µ > 0 and ε > 0 be parameters and 1 < r1 < r2 < r3 < 2 be radii to
be fixed later. Let ϕ ∈ Cc∞ (B1 ) be a standard mollifier in RN and, for the sake of brevity,
let lin(h1 , h2 ) denote the linear interpolation in Br \ B̄s between two functions h1 |∂Br and
h2 |∂Bs . The function g & is defined as follows:
√
. !
. ! //
f
f
# √

√
E
lin
,
ρ
in Br3 \ Br2 ,


√
. E. ! / E . !
//
g & :=
E lin ρ# √fE , ρ# √fE ∗ ϕε
in Br2 \ Br1 ,
(5.3)

. !
/

√

 E ρ# √f ∗ ϕε
in Br1 .
E
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
19
Clearly g & |∂Br3 = f & |∂Br3 . We pass now to estimate its energy.
Step 1. Energy in Br3 \ Br2 . By the estimate on the linear interpolation, it follows directly
that
ˆ
ˆ
ˆ
& 2
& 2
|Dg | ≤ C
|Df | + C
|D(ρ# ◦ f & )|2 +
Br3 \Br2
Br3 \Br2
Br3 \Br2
# &
) & *#2
# f
#
#
# √ − ρ √f
#
# E
E #
CE
(r3 − r2 )2 Br3 \Br2
ˆ
−nQ+1
C E µ2
& 2
≤C
|Df | +
,
r3 − r2
Br3 \Br2
ˆ
+
where we used |ρ# (P ) − P | ≤ C µ2
−nQ
(5.4)
for all P ∈ Q.
Step 2. Energy in Br2 \ Br1 . Here, using the same interpolation inequality and the L2
estimate on convolution, we get
ˆ
ˆ
ˆ
C
& 2
& 2
|Dg | ≤ C
|Df | +
|f & − ϕε ∗ f & |2
2
(r
−
r
)
2
1
Br2 \Br1
Br2 \Br1
Br2 \Br1
ˆ
ˆ
ˆ
2
Cε
C ε2 E
& 2
& 2
≤C
|Df & |2 +
|Df
|
=
C
|Df
|
+
.
(5.5)
(r2 − r1 )2 B1
(r2 − r1 )2
Br2 \Br1
Br2 \Br1
#
Step 3. Energy in Br1 . For this estimate we use the;fine bounds on .
the projection
/ ρ . (see
<
!
Proposition 6.1). To this aim, consider the set Z := x ∈ Br1 : dist √fE ∗ ϕε , Q > µnQ .
Then, one can estimate
ˆ
ˆ
.
/ˆ
2
2
& 2
2−nQ
&
|Dg | ≤ 1 + C µ
|D (f ∗ ϕε )| + C
|D (f & ∗ ϕε )| =: I1 + I2 . (5.6)
Br1 \Z
Br1
Z
We consider I1 and I2 separately. For the first we have
.
/ˆ
2−nQ
I1 ≤ 1 + C µ
(|Df & | ∗ ϕε )2
≤
.
Br1
2
1+Cµ
/ˆ
−nQ
Br1
.
2−nQ
+2 1+Cµ
!
&
(|Df | χK ) ∗ ϕε
5
/ ˆ
Br1
!
"2
.
2−nQ
+ 1+Cµ
"2
(|Df & | χK ) ∗ ϕε )
6 12 5ˆ
/ˆ
Br1
Br1
!
!
"2
(|Df & | χB1 \K ) ∗ ϕε +
(|Df & | χB1 \K ) ∗ ϕε
"2
6 21
Next we notice that the following two estimates hold for the convolutions:
ˆ
ˆ
ˆ
ˆ
!
"2
2
&
&
& 2
(|Df | χK ) ∗ ϕε ≤
(|Df | χK ) ≤
|Df | +
|Df & |2
Br1
Br1 +ε
Br1 ∩K
Br1 +ε \Br1
.
(5.7)
(5.8)
20
C. DE LELLIS AND E. N. SPADARO
and, using Lip(f & ) ≤ C E α and |B1 \ K| ≤ C E 1−2α ,
ˆ
4
42
!
"2
(|Df & | χBr1 \K ) ∗ ϕε ) ≤ C E 2α 4χBr1 \K ∗ ϕε 4L2
Br1
4
42
C E 2−2 α
.
(5.9)
≤ C E 2α 4χBr1 \K 4L1 ,ϕε ,2L2 ≤
εN
Hence, putting (5.8) and (5.9) in (5.7), we get
)
*1
ˆ
.
/ˆ
2−2 α
2−2 α 2
1
C
E
C
E
−nQ
I1 ≤ 1 + C µ2
|Df & |2 + C
|Df & |2 +
+ C E2
εN
εN
Br1 ∩K
Br1 +ε \Br1
ˆ
ˆ
3
C E 2−2 α C E 2 −α
& 2
2−nQ
& 2
≤
|Df | + C µ
+
.
(5.10)
E+C
|Df | +
εN
εN/2
Br1 ∩K
Br1 +ε \Br1
For what concerns I2 , first we argue as for I1 , splitting in K and B1 \ K, to deduce that
ˆ
3
C E 2−2 α C E 2 −α
2
&
I2 ≤ C
((|Df | χK ) ∗ ϕε ) +
+
.
(5.11)
εN
εN/2
Z
Then, regarding the first addendum in
ˆ
2nQ
|Z| µ
≤
(5.11), we note that
# &
#
& #2
# f
# √ ∗ ϕε − √f # ≤ C ε2 .
# E
E#
Br1
(5.12)
Hence, using the higher integrability of |Df | in K, that is (5.1), we obtain
5ˆ
6 2q
)
* 2 (q−2)
ˆ
q
q−2
ε
2
q
((|Df & | χK ) ∗ ϕε ) ≤ |Z| q
((|Df & | χK ) ∗ ϕε )
≤CE
. (5.13)
µnQ
Z
Br1
Hence, putting all the estimates together, (5.6), (5.10), (5.11) and (5.13) give
ˆ
ˆ
ˆ
& 2
& 2
|Dg | ≤
|Df | + C
|Df & |2 +
Br1
Br1 ∩K
Br1 +ε \Br1
+CE
5
1
µ
2−nQ
E 1−2α E 2 −α
+ N + N/2 +
ε
ε
)
ε
µnQ
6
* 2 (q−2)
q
. (5.14)
Now we are ready to estimate the total energy of g & and conclude the proof of the
proposition. We start fixing r2 − r1 = r3 − r2 = λ. With this choice, summing (5.4), (5.5)
and (5.14),
ˆ
ˆ
ˆ
& 2
& 2
|Dg | ≤
|Df | + C
|Df & |2 +
Br3
Br3 ∩K
Br1 +3λ \Br1
+CE
5
µ2
−nQ+1
λ
1
ε2
E 2 −α
−nQ
+ 2 + µ2
+ N/2 +
λ
ε
)
ε
µnQ
6
* 2 (q−2)
q
.
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
21
We set ε = E a , µ = E b and λ = E c choosing
a=
1 − 2α
,
2N
b=
1 − 2α
4N nQ
and c =
1 − 2α
.
2nQ+2 N n Q
Now, for a choice of a constant C > 0 sufficiently large, there is a set B ⊂ [1, 2] with
|B| > 1/2 such that, for every r1 ∈ B, it holds
ˆ
ˆ
1−α
1+
& 2
|Df | ≤ C λ
|Df & |2 ≤ C E 2nQ+2 N n Q .
Br1 +3λ \Br1
Br1
Then, for a suitable δ = δ(α, n, N, Q) and for s = r3 , we conclude (5.2).
For what concerns the Lipschitz constant of g & , we notice that it is bounded by

&
&
&
α
in Br1 ,

Lip(g ) ≤ C Lip(f ∗ ϕε ) ≤! C !Lip(f ) ≤ C E
.f −f ∗ϕε .L∞
ε
&
&
&
α
Lip(g ) ≤ C Lip(f ) + C
≤ C(1 + λ ) Lip(f ) ≤ C E
in Br2 \ Br1 ,
λ

−nQ
2

Lip(g & ) ≤ C Lip(f & ) + C E 1/2 µ λ ≤ C E α + C E 1/2 ≤ C E α
in Br3 \ Br2 .
!
Now we are ready for the conclusion of the proof of Theorem 0.8. Consider the set
B ⊂ [1, 2] given in Proposition 5.1 and, as done in subsection 4.2, choose r ∈ B and a
integer rectifiable current R such that
!
"
∂R = T − graph(f ) ∂Br and M(R) ≤ CE (1−2α)m/(m−1) .
Since g|∂Bs = f |∂Bs , we use graph(g) + R as competitor for the current T . In this way we
obtain, for a suitable σ,
ˆ
ˆ
(5.2)
|Dg|2
|Df |2
1+α
M (T Cs ) ≤ Q |Bs | +
+CE
≤ Q |Bs | +
+ C E 1+σ . (5.15)
2
2
Bs
Bs ∩K
On the other hand, again using Taylor’s expansion (2.9),
M (T Cs ) = M (T (Bs \ K) × Rn ) + M (graph(f |Bs ∩K ))
ˆ
|Df |2
n
≥ M (T (Bs \ K) × R ) + Q |K ∩ Bs | +
− C E 1+σ .
2
K∩Bs
(5.16)
Hence, from (5.15) and (5.16), we get eT (Bs \ K) ≤ C E 1+σ .
This is enough to conclude the proof. Indeed, for A ⊂ B1 , using the higher integrability
of |Df | in K, possibly changing σ, we get
ˆ
|Df |2
eT (A) ≤ eT (A ∩ K) + eT (A \ K) ≤
+ C E 1+σ
2
A∩K
)ˆ
* p2
. q−2
/
q−2
≤ C |A ∩ K| q
|Df |q
+ C E 1+σ ≤ C E |A| q + E σ .
A∩K
22
C. DE LELLIS AND E. N. SPADARO
5.2. Proof of Theorem 0.1. Choose α < min{(2m)−1 , (2(1 + σ))−1 σ}, where σ is the
constant in Theorem 0.8 and let f be the E α -Lipschitz approximation of T C4/3 .
Clearly (0.1a) follows directly from (0.4) for δ < α. Set A = {MT > E 2 α /2} ⊂ B4/3 .
Applying (0.9) to A, since by (2.5) |A| ≤ CE 1−2α , we get (0.1b), for some positive δ,
|B1 \ K| ≤ C E −2 α eT (A) ≤ C E 1+σ−2 α + CE 1+σ−2(1+σ)α ≤ C E 1+δ .
On the other hand, (0.1c) is consequence of (0.9) and (2.9). Indeed, if we set Γ = graph(f ):
#
#
#
#
ˆ
ˆ
2#
2#
#
# !
"
|Df
|
|Df
|
#M T C1 − Q ωm −
# ≤ eT (B1 \ K) + eΓ (B1 \ K) + #eΓ (B1 ) −
#
#
#
2 #
2 #
B1
B1
ˆ
(0.9), (2.9)
1+σ
2
≤ CE
+ C |B1 \ K| + C Lip(f )
|Df |2
B1
! 1+σ
"
1+2 α
1+δ
≤C E
+E
=CE .
6. Almgren’s projections ρ#µ
In this section we complete the proof of Theorem 0.1 showing the existence of the almost
projections ρ#µ . Compared to the original maps introduced by Almgren, our ρ# ’s have the
advantage of depending on a single parameter. Our proof is different from Almgren’s
and relies heavily on the classical Theorem of Kirszbraun on the Lipschitz extensions of
Rd –valued maps. A feature of our proof is that it gives more explicit estimates.
Proposition 6.1. For every µ > 0, there exists ρ#µ : RN (Q,n) → Q = ξ(AQ (Rn )) such that:
(i) the following estimate holds for every u ∈ W 1,2 (Ω, RN ),
ˆ
ˆ
.
/ˆ
#
2
2−nQ
2
|D(ρµ ◦ u)| ≤ 1 + C µ
|Du| + C
{dist(u,Q)≤µnQ }
{dist(u,Q)>µnQ }
|Du|2 ,
(6.1)
with C = C(Q, n);
−nQ
(ii) for all P ∈ Q, it holds |ρ#µ (P ) − P | ≤ C µ2 .
From now on, in order to simplify the notation we drop the subscript µ. We divide the
proof into two parts: in the first one we give a detailed description of the set Q; then, we
describe rather explicitly the map ρ#µ .
6.1. Linear simplicial structure of Q. In this subsection we prove that the set Q can
be decomposed in a families of sets {Fi }nQ
i=0 , here called i-dimensional faces of Q, with the
following properties:
(p1) Q = ∪nQ
i=0 ∪F ∈Fi F ;
(p2) F := ∪Fi is made of finitely many disjoint sets;
(p3) each face F ∈ Fi is a convex open i-dimensional cone, where open means that for
every x ∈ F there exists an i-dimensional disk D with x ∈ D ⊂ F ;
(p4) for each F ∈ Fi , F̄ \ F ⊂ ∪j<i ∪G∈Fj G.
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
23
In particular, the family of the 0-dimensional faces F0 contains an unique element, the
origin {0}; the family of 1-dimensional faces F1 consists of finitely many lines of the form
lv = {λ v : λ ∈]0, +∞[} with v ∈ SN −1 ; F2 consists of finitely many 2-dimensional cones
delimited by two half lines lv1 , lv2 ∈ F1 ; and so on.
To prove this statement, first of all we recall the construction of ξ. After selecting a
suitable finite collection of non zero vectors {el }hl=1 , we define the linear map L : Rn Q → RN
given by
!
"
L(P1 , . . . , PQ ) := P1 · e1 , . . . , PQ · e1 , P1 · e2 , . . . , PQ · e2 , . . . , P1 · eh , . . . , PQ · eh .
=
>?
@ =
>?
@
=
>?
@
w1
w2
wh
Then, we consider the map O : RN → RN which maps (w1 . . . , wh ) into the vector
(v 1 , . . . , v h ) where each v i is obtained from wi ordering its components in increasing order.
Note that the composition O◦L : (Rn )Q → RN is now invariant under the action of the symmetric group PQ . ξ is simply the induced map on AQ = RnQ /PQ and Q = ξ(AQ ) = O(V )
where V := L(Rn Q ).
Consider the following equivalence relation ∼ on V :
A
wji = wki ⇔ zji = zki
(w1 , . . . , wh ) ∼ (z 1 , . . . , z h )
if
∀ i, j, k ,
(6.2)
wji > wki ⇔ zji > zki
i
i
where wi = (w1i , . . . , wQ
) and z i = (z1i , . . . , zQ
) (that is two points are equivalent if the map
O rearranges their components with the same permutation). We let E denote the set of
corresponding equivalence classes in V and C := {L−1 (E) : E ∈ E}. The following fact is
an obvious consequence of definition (6.2):
L(P ) ∼ L(S) implies L(Pπ(1) , . . . , Pπ(Q) ) ∼ L(Sπ(1) , . . . , Sπ(Q) ) ∀ π ∈ PQ .
Thus, π(C) ∈ C for every C ∈ C and every π ∈ PQ . Since ξ is injective and is induced by
O◦L, it follows that, for every pair E1 , E2 ∈ E, either O(E1 ) = O(E2 ) or O(E1 )∩O(E2 ) = ∅.
Therefore, the family F := {O(E) : E ∈ E} is a partition of Q.
Clearly, each E ∈ E is a convex cone. Let i be its dimension. Then, there exists a
i-dimensional disk D ⊂ E. Denote by x its center and let y be any other point of E. Then,
by (6.2), the point z = (1 + ε) y − ε x belongs as well to E for any ε > 0 sufficiently small.
The convex envelope of D ∪ {z}, which is contained in E, contains in turn an i-dimensional
disk centered in y. This establishes that E is an open convex cone. Since O|E is a linear
injective map, F = O(E) is an open convex cone of dimension i. Therefore, F satisfies
(p1)-(p3).
Next notice that, having fixed w ∈ E, a point z belongs to Ē \ E if and only if
• wji ≥ wki implies zji ≥ zki for every i, j and k;
• there exists r, s and t such that wsr > wtr and zsr = ztr .
Thus, if d is the dimension of E, Ē \ E ⊂ ∪j<d ∪G∈Ej G, where Ed is the family of ddimensional classes. Therefore,
O(Ē \ E) ⊂ ∪j<d ∪H∈Fj H,
(6.3)
24
C. DE LELLIS AND E. N. SPADARO
from which (recalling F = O(E)) we infer that
O(Ē \ E) ∩ F = O(Ē \ E) ∩ O(E) = ∅.
(6.4)
Now, since O(Ē \ E) ⊂ O(Ē) ⊂ O(E) = F̄ , from (6.4) we deduce O(Ē \ E) ⊂ F̄ \ F .
On the other hand, it is simple to show that F̄ ⊂ O(Ē). Hence, F̄ \ F ⊂ O(Ē) \ F =
O(Ē) \ O(E) ⊂ O(Ē \ E). This shows that F̄ \ F = O(Ē \ E), which together with (6.3)
proves (p4).
6.2. Construction of ρ#µ . The construction is divided into three steps:
(1) first we specify ρ#µ in Q;
(2) then we find an extension on a µnQ -neighborhood of Q, QµnQ ;
(3) finally we extend the ρ#µ to all RN .
6.2.1. Construction on Q. The construction of ρ#µ on Q is made through a recursive procedure whose main building block is the following lemma.
Lemma 6.2. Let b > 2 and D ∈ N. There exists a constant C such that the following
holds for every τ ∈]0, 1[. Let V d ⊂ RD be a d-dimensional cone and let v : ∂Bb ∩ V d → RD
satisfy Lip(v) ≤ 1 + τ and |v(x) − x| ≤ τ . Then, there exists an extension w of v,
w : Bb ∩ V d → RD , such that
√
Lip(w) ≤ 1 + C τ, |w(x) − x| ≤ C τ and w(x) = 0 ∀ x ∈ Bτ ∩ Vd .
Proof. First extend v to Bτ ∩ Vd by setting it identically 0 there. Note that such a function
is still Lipschitz continuous with constant 1+C τ . Indeed, for x ∈ ∂Bb ∩V d and y ∈ Bτ ∩V d ,
we have that
|v(x) − v(y)| = |v(x)| ≤ |x| + τ = b + τ ≤ (1 + C τ )(b − τ ) ≤ (1 + C τ ) |x − y|.
Let w be an extension of v to Bb ∩ V d with the same Lipschitz constant, whose existence
is guaranteed by the classical Kirzbraun’s
Theorem, see [8, Theorem 2.10.43]. We claim
√
that w satisfies |w(x) − x| ≤ 1 + C τ , thus concluding the lemma.
To this aim, consider x ∈ Bb \ Bτ and set y = b x/|x| ∈ ∂Bb . Consider, moreover,
the line r passing through 0 and w(y), let π be the orthogonal projection onto r and set
z = π(w(x)). Note that, if |x| ≤ C τ , then obviously |w(x) − x| ≤ |x| + |w(x)| ≤ C τ .
Thus, without loss of generality, we can assume that |x| ≥ C τ for some constant τ . In
this case, the conclusion is clearly a consequence of the following estimates:
√
|z − w(x)| ≤ C τ ,
(6.5)
|x − z| ≤ C τ.
(6.6)
|z − w(y)| ≤ (1 + Cτ )|x − y| ≤ b − |x| + C τ
(6.7)
To prove (6.5), note that Lip(π ◦ w) ≤ 1 + Cτ and, hence,
|z| = |π ◦ w(x) − π ◦ w(0)| ≤ (1 + Cτ )|x| ≤ |x| + C τ.
Then, by the triangle inequality,
|z| ≥ |w(y)| − |w(y) − z| ≥ b (1 − τ ) − b + |x| − Cτ ≥ |x| − Cτ.
(6.8)
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
25
Since |x| ≥ C τ , the left hand side of (6.8) can be supposed nonnegative and we obtain
(6.5),
|z − w(x)|2 = |w(x)|2 − |z|2 ≤ (1 + Cτ )2 |x|2 − (|x| − Cτ )2 ≤ Cτ.
For what concerns (6.6), note that
#
#
#
#
#x − |x| w(y)# ≤ |x| |y − w(y)| ≤ |x| C τ ≤ C τ.
#
#
b
b
(6.9)
On the other hand, since by (6.7) |z −w(y)| ≤ b−|x|+Cτ ≤ b−τ ≤ |w(y)| and w(y)·z ≥ 0,
we have also
#
# #
#
#
#
#
# #
#
#
#
|x|
|x|
|x|
#z −
# = #|z| −
# ≤ ||z| − |x|| + #|x| −
# ≤ Cτ,
w(y)
|w(y)|
|w(y)|
#
#
# #
#
#
b
b
b
!
which together with (6.9) gives (6.6).
Now we pass to the construction of the map ρ#µ . To fix notation, let Sk denote the kskeleton of Q, that is the union of all the k-faces, Sk = ∪F ∈Fk F . For every k = 1 . . . , nQ−1
and F ∈ Fk , let F̂a,b denote the set
$
%
F̂a,b = x ∈ Q : dist(x, F ) ≤ a , dist(x, Sk−1 ) ≥ b ,
where a, b > 0 are given constants. In the case of maximal dimension F ∈ FnQ , for every
a > 0, F̂a denotes the set
$
%
F̂a = x ∈ F : dist(x, SnQ−1 ) ≥ a .
Next we choose constants 1 = cnQ−1 < cnQ−2 < . . . < c0 such that, for every 1 ≤ k ≤
nQ − 1, each family {F̂2ck ,ck−1 }F ∈Fk is made by pairwise disjoint sets. Note that this is
possible: indeed, since the number of faces is finite, given ck one can always find a ck−1
such that the F̂2ck ,ck−1 ’s are pairwise disjoint for F ∈ Fk .
Moreover, it is immediate to verify that
nQ−1
B B
k=1 F ∈Fk
F̂2ck ,ck−1 ∪
B
F ∈FnQ
F̂cnQ−1 ∪ B2c0 = Q.
To see this, let Ak = ∪F ∈Fk F̂2ck ,ck−1 and AnQ = ∪F ∈FnQ F̂cnQ−1 : if x ∈
/ ∪nQ
k=1 Ak , then
dist(x, Sk−1 ) ≤ ck−1 for every k = 1, . . . , nQ, that means in particular that x belongs to
B2c0 .
Now we are ready to define the map ρ#µ inductively on the Ak ’s. On AnQ we consider
the map fnQ = Id . Then, we define the map fnQ−1 on AnQ ∪ AnQ−1 starting from fnQ and,
in general, we define inductively the map fk on ∪nQ
l=k Al knowing fk+1 .
nQ
Each map fk+1 : ∪l=k+1 Al → Q has the following two properties:
(ak+1 ) Lip(fk+1 ) ≤ 1 + C µ2
−nQ+k+1
and |fk+1 (x) − x| ≤ C µ2
−nQ+k+1
;
26
C. DE LELLIS AND E. N. SPADARO
(bk+1 ) for every k-dimensional face G ∈ Fk , setting coordinates in G2ck ck−1 in such a way
that G ∩ G2ck ,ck−1 ⊂ Rk × {0} ⊂ RN , fk+1 factorizes as
fk+1 (y, z) = (y, hk+1 (z)) ∈ R × R
k
N −k
∀ (y, z) ∈ G2ck ck−1 ∩
nQ
B
l=k+1
Al .
The constants involved depend on k but not on the parameter µ.
Note that, fnQ satisfies (anQ ) and (bnQ ) trivially, because it is the identity map. Given
fk+1 we next show how to construct fk . For every k-dimensional
face G ∈" Fk , setting
!
coordinates as in (bk+1 ), we note that the set Vy := G2ck ck−1 ∩ {y} × RN −k ∩ B2ck (y, 0)
is the intersection of a cone with the ball B2ck (y, 0). Moreover, hk+1 (z) is defined on
Vy ∩(B2ck (y, 0)\Bck (y, 0)). Hence, according to Lemma 6.2, we can consider an extension wk
−nq+k
of hk+1 |{|z|=2ck } on Vy ∩ B2ck (again not depending on y) satisfying Lip(wk ) ≤ 1 + C µ2
,
2−nq+k
|z − wk (z)| ≤ C µ
and wk (z) ≡ 0 in a neighborhood of 0 in Vy .
Therefore, the function fk defined by
A
(y, wk (z)) for x = (y, z) ∈ G2ck ,ck−1 ⊂ Ak ,
C
fk (x) =
(6.10)
fk+1 (x)
for x ∈ nQ
l=k+1 Al \ Ak ,
satisfies the following properties:
(ak ) First of all the estimate
|fk (x) − x| ≤ C µ2
−nQ+k
(6.11)
comes from Lemma 6.2. Again from Lemma 6.2, we conclude Lip(fk ) ≤ 1 +
−nQ+k+1
C µ2
on every G2ck ,ck−1 . Now, every pair of points x, y contained, respectively,
into two different G2ck ,ck−1 and H2ck ,ck−1 are distant apart at least one. Therefore,
.
/
(6.11)
−nQ+k
−nQ+k
|fk (x) − fk (y)| ≤ |x − y| + C µ2
≤ 1 + Cµ2
|x − y| .
−nQ+k
This gives the global estimate Lip(fk ) ≤ 1 + C µ2
.
(bk ) For every (k − 1)-dimensional face H ∈ Fk−1 , setting coordinates in H2ck−1 ,ck−2 in
such a way that H ∩ H2ck−1 ,ck−2 ⊂ Rk−1 × {0} ⊂ RN −k+1 , fk factorizes as
&
&
&
&
fk (y , z ) = (y , hk (z )) ∈ R
k−1
×R
N −k+1
&
&
∀ (y , z ) ∈ H2ck−1 ,ck−2 ∪
nQ
B
l=k
Al .
Indeed, when H ⊂ ∂G, with G ∈ Fk+1 and z & = (z1& , z) where (y, z) is the coordinate
system selected in (bk+1 ) for G, then
hk (z & ) = (z1& , wk (z)) .
After nQ steps, we get a function f0 = ρ#0 : Q → Q which satisfies
Lip(ρ#0 ) ≤ 1 + C µ2
−nQ
and |ρ#0 (x) − x| ≤ C µ2
−nQ
.
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
27
Moreover, since the extensions wk coincide with the projection in balls BCµ2−nQ+k−1 around
the origin, hence, in particular on balls Bµ , it is easy to see that, for every face F ∈ Fk ,
the map ρ#0 coincides with the projection on F for x ∈ Fµ,2ck−1 , that is
ρ#0 (x) = πF (x) ∀ x ∈ Fµ,2ck−1 .
(6.12)
6.2.2. Extension to QµnQ . Now we need extend the map ρ#0 : Q → Q to a neighborhood
of Q preserving the same Lipschitz constant.
We start noticing that, since the number of all the faces is finite, when µ is small
enough, there exists a constant C = C(N ) with the following property. If x, y are two
points contained, respectively, in Fµi+1 \ ∪j<i ∪G∈Fj Gµj+1 and Hµi+1 \ ∪j<i ∪G∈Fj Gµj+1 ,
where F 0= H ∈ Fi , then
dist(x, y) ≥ C µi .
(6.13)
The extension ρ#1 is defined inductively. We start this time from a neighborhood of the
0-skeleton of Q. i.e. the ball Bµ (0). The extension g0 has the constant value 0 on Bµ (0)
(note that this is compatible with the ρ#0 by (6.12)).
Now we come to the inductive step. Suppose we have an extension gk of ρ#0 , defined on
the union of the µl+1 -neighborhoods of the l-skeletons Sl , for l running from 0 to k, that
is, on the set
k B
B
Λk := Q ∪ Bµ ∪
Fµl+1 .
l=1 F ∈Fl
2−nQ
Assume that Lip(gk ) ≤ 1 + C µ
. Then, we define the extension of gk to Λk+1 in the
following way. For every face F ∈ Fk+1 , we set
A
gk in (Sk )µk+1 ∩ Fµk+2 ,
gk+1 :=
(6.14)
πF in {x ∈ RN : |πF (x)| ≥ 2 ck } ∩ Fµk+2 .
Consider now a connected component C of Λk+1 \ Λk . As defined above, gk+1 maps a
portion of C̄ into the closure K of a single face of Q. Since K is a convex closed set, we
can use Kriszbraun’s Theorem to extend gk+1 to C̄ keeping the same Lipschitz constant of
−nQ
gk , which is 1 + C µ2 .
Next, notice that if x belongs the intersection of the boundaries of two connected components C1 and C2 , then it belongs to Λk . Thus, the map gk+1 is continuous. We next
bound the global Lipschitz constant of gk+1 . Indeed consider points x ∈ Fµk+2 \ Λk and
y ∈ Fµ& k+2 \ Λk , with F, F & ∈ Fk+1 . Since by (6.13) |x − y| ≥ C µk , we easily see that
|gk+1 (x) − gk+1 (y)| ≤ 2 µk+1 + |gk (πF (x)) − gk (πF ! (y))|
≤ 2 µk+1 + (1 + C µ2
−nQ
)|πF (x) − πF ! (y)|
"
−nQ !
−nQ
≤ 2 µk+1 + (1 + C µ2 ) |x − y| + 2 µk+1 ≤ (1 + C µ2 ) |x − y|.
Therefore, we can conclude again that Lip(gk+1 ) ≤ 1 + C µ2
step.
−nQ
, finishing the inductive
28
C. DE LELLIS AND E. N. SPADARO
After making the step above nQ times we arrive to a map gnQ which extends ρ#0 and is
defined in a µnQ -neighborhood of Q. We denote this map by ρ#1 .
6.2.3. Extension to RN . Finally, we extend ρ#1 to RN with a fixed Lipschitz constant.
This step is immediate recalling the Lipschitz extension theorem for Q-valued functions.
Indeed, taken ξ −1 ◦ ρ#1 : SµnQ → AQ , we find a Lipschitz extension h : RN → AQ of it with
Lip(h) ≤ C. Clearly, the map ρ#µ := ξ ◦ h fulfills all the requirements of Proposition 6.1.
Appendix A. A variant of Theorem 0.1
Theorem A.1. There are constants C, α, ε1 > 0 such that the following holds. Assume T
satifes the assumptions of Theorem 0.1 with E4 := Ex(T, C4 ) < ε1 and set Er := Ex(T, Cr ).
Then there exist a radius s ∈]1, 2[, a set K ⊂ Bs and a map f : Bs → AQ (Rn ) such that:
Lip(f ) ≤ CEsα ,
(A.1a)
graph(f |K ) = T (K × R ) and |Bs \ K| ≤
#
#
ˆ
# !
"
|Df |2 ##
#M T Cs − Q ωm sm −
≤ C Es1+α .
#
2 #
n
CEs1+α ,
(A.1b)
(A.1c)
Bs
The theorem will be derived from the following lemma, which in turn follows from
Theorem 0.1 through a standard covering argument.
Lemma A.2. There are constants C, β, ε2 > 0 such that the following holds. Assume T is
an area-minimizing, integer recitifiable current in Cρ , satisfying (H) and E := Ex(T, Cρ ) <
ε2 . Set r = ρ(1 − 4 E β ). Then there exist a set K ⊂ Br and a map f : Br → AQ (Rn ) such
that:
Lip(f ) ≤ CE β ,
graph(f |K ) = T (K × Rn ) and |Br \ K| ≤ CE 1+β rm ,
#
#
ˆ
2#
# !
"
|Df
|
#M T Cr − Q ωm rm −
# ≤ C E 1+β rm .
#
2 #
(A.2a)
(A.2b)
(A.2c)
Br
Proof. Without loss of generality we prove the lemma for ρ = 1. Fix β > 0 and ε2 > 0
and assume T as in the statement. We choose a family of balls B i = BE β (ξi ) satisfying
the following conditions:
(i) the numer N of such balls is bounded by CE −mβ ;
(ii) B4E β (ξi ) ⊂ B1 and {BE β /2 (ξi )} covers Br = B1−4E β ;
(iii) each B i intersects at most M balls B j .
The constants C and M are dimensional and do not depend on E, β and ε2 . Moreover,
observe that
Ex(T, C4E β (ξi )) ≤ 4−m E −mβ Ex(T, C1 ) ≤ C E 1−mβ .
Fix now ε2 such that ε1−mβ
≤ ε0 , with ε0 the constant in Theorem 0.1. Applying (the
2
obvious scaled version of) Theorem 0.1, for each B i we obtain a set Ki ⊂ Bi and a map
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
29
fi : Bi → AQ (Rn ) such that
Lip(fi ) ≤ CE (1−mβ)δ ,
(A.3)
E ,
(A.4)
graph(fi |Ki ) = T (Ki × R ) and |B \ Ki | ≤ CE
#
#
ˆ
2
# !
"
|Df | ##
(1−mβ)(1+δ) mβ
#M T CE β (ξi ) − Q ωm E mβ −
E .
(A.5)
#
#≤CE
2
i
B
D
Set next I(i) := {j : B j ∩ B i 0= ∅} and Ji := Ki ∩ j∈I(i) Kj . By (iii) and (A.4), we have
n
i
(1−mβ)(1+δ)
mβ
|B i \ Ji | ≤ CE (1−mβ)(1+δ)+mβ .
C
(A.6)
Define K := Ji . Since fi |Ji ∩Jj = fj |Jj ∩Ji , there is a function f : K → AQ (Rn ) such that
f |Ji = fi . Choose β so small that (1 − mβ)(1 + δ) ≥ 1 + β. Then, (A.2b) holds because of
(i) and (A.6).
We claim next that f satisfies the Lipschitz bound (A.2a). First take x, y ∈ K such
that |x − y| ≤ E β /2. Then, by (ii), x ∈ BE β /2 (ξi ) for some i and hence x, y ∈ B i . By the
definition of K, x ∈ Jj ⊂ Kj for some j. On the other hand, B j ∩ B i 0= ∅ and thus, by the
definition of Jj , we necessarily have x ∈ Ki . For the same reason we conclude y ∈ Ki . It
follows from (A.3) and the choice of β ≤ (1 − mβ) δ that
|f (x) − f (y)| = |fi (x) − fi (y)| ≤ CE β |x − y|.
Next, assume that x, y ∈ K and |x − y| ≥ E β /2. On the segment σ = [x, y], fix N ≤
8E −β |x − y| points ζi with ζ0 = x, ζN = y and |ζi+1 − ζi | ≤ E β /4. We can choose ζi so
that, for each i ∈ {1, N − 1}, B̂ i := BE β /8 (ζi ) ⊂ Br . Obviously, if β and ε2 are chosen
small enough, (A.2b) implies that B̂ i ∩ K 0= ∅ and we can select zi ∈ B̂ i ∩ K 0= ∅. But
then |zi+1 − zi | ≤ E β /2 and hence |f (zi+1 ) − f (zi )| ≤ CE 2β . Setting zN = ζN = y and
z0 = ζ0 = x, we conclude the estimate
|f (x) − f (y)| ≤
N
+
i=0
|f (i + 1) − f (i)| ≤ CN E 2β ≤ CE β |x − y| .
Thus, f can be extended to Br with the Lipschitz bound (A.2a). Finally, a simple argument
using (A.2a), (A.2b), (A.5) and (i) gives (A.2c) and concludes the proof.
!
Proof of Theorem A.1. Let β be the constant of Lemma A.2 and choose α ≤ β/(2 + β).
Set r0 := 2 and E0 := Ex(T, Cr0 ), r1 := 2(1 − 4E0β ) and E1 := Ex(T, Cr1 ). Obviously, if ε1 is
sufficiently small, we can apply Lemma A.2 to T in Cr0 . We also assume of having chosen
1+β/2
ε1 so small that 2(1 − 4E0β ) > 1. Now, if E1 ≥ E0
, then f satisfies the conclusion of
β
the theorem. Otherwise we set r2 = r1 (1 − 4E1 ) and E2 := Ex(T, Cr2 ). We continue this
process and stop only if
(a) either rN < 1;
1+β/2
(b) or EN ≥ EN −1 .
30
C. DE LELLIS AND E. N. SPADARO
First of all, notice that, if ε1 is chosen sufficiently small, (a) cannot occur. Indeed, we have
(1+β/2)i
1+iβ/2
Ei ≤ E0
≤ ε1
and thus
β 2 /2
+ β+iβ 2 /2
+ β
ε1
ri +
Ei ≥ −8
ε1
≥ −8 εβ1
.
log =
log(1 − 4Eiβ ) ≥ −8
β 2 /2
2
1 − ε1
(A.7)
Clearly, for ε1 sufficiently small, the right and side of (A.7) is larger than log(2/3), which
gives ri ≥ 4/3.
Thus, the process can stop only if (b) occurs and in this case we can apply Lemma A.2
to T in CrN −1 and conclude the theorem for the radius s = rN . If the process does not
stop, we conclude that Ex(T, CrN ) → 0. If s := limN rN , we then conclude that s > 1 and
that Ex(T, Cs ) = 0. But then, because of (H),(this implies that there are Q points qi ∈ Rn
(not (
necessarily distinct) such that T Cs = i !Bs × {qi }". Thus, if we set K = Bs and
f ≡ i !qi ", the conclusion of the theorem holds trivially.
!
Appendix B. The varifold excess
As pointed out in Remark 0.3, though the approximation theorems of Almgren have
(essentially) the same hypotheses of Theorem 0.1, the main estimates are stated in terms
of the “varifold excess” of T in the cylinder C4 . More precisely, consider the representation
of the rectifiable current T as T4 ,T ,. As it is well-known, T4 (x) is a simple vector of the
form v1 ∧ . . . ∧ vm with )vi , vj * = δij . Let τx be the m-plane spanned by v1 , . . . , vm and
let πx : Rm+n → τx be the orthogonal projection onto τx . Finally, for any linear map
L : Rm+n → Rm , denote by ,L, the operator norm of L. Then, the varifold excess is
defined by
ˆ
VEx(T, Cr (x0 )) =
(B.1)
|T4 (x) − 4em |2 d,T ,(x) .
(B.2)
Cr (x0 )
whereas
Ex(T, Cr (x0 )) =
,πx − π,2 d,T ,(x) ,
ˆ
Cr (x0 )
The two quantities differ. If on the one hand VEx ≤ CEx for trivial reasons (indeed,
,πx − π, ≤ C,T4 (x) − 4em , for every x), VEx might, for general currents, be much smaller
than Ex. However, Almgren’s statements can be easily recovered from Theorem 0.1 thanks
to the following proposition.
Proposition B.1. There are constants ε3 , C > 0 with the following properties. Assume T
is as in Theorem A.1 and consider the radius s given by its conclusion. If Ex(T, C2 ) ≤ ε3 ,
then Ex(T, Cr ) ≤ CVEx(T, Cr ).
Proof. Note that there are constants c0 , C1 such that |T4 (x) − 4em | ≤ C1 and |T4 (x) − 4em | ≤
C1 ,πx − π, if |T4 (x) − 4em | < c0 . Let now D := {x ∈ Cr : |T4 (x) − 4em | > c0 }. We can then
write
Ex(T, Cr ) ≤ C1 VEx(T, Cr ) + 2 M(T D).
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
31
On the other hand, from the bounds (A.1), it follows immediately that M(T D) ≤
CEx(T, Cr )1+α . If ε3 is chosen sufficiently small, we conclude
2−1 Ex(T, Cr ) ≤ Ex(T, Cr ) − CEx(T, Cr )1+α ≤ C1 VEx(T, Cr ) .
!
Appendix C. Push-forward of currents under Q-functions
In this appendix we give the rigorous definition of the integer rectifiable current associated to the graph of a Q-valued function and prove that the boundary of such currents is
given by the graph of the trace of the function.
(
Given a Q-valued function f : Rm → AQ (Rn ), we set f¯ = i !(x, fi (x))", f¯ : Rm →
AQ (Rm+n ). Let R ∈ Dk (Rm ) be a rectifiable current associated to a k-rectifiable set M
with multiplicity θ. In the notation of [11], R = τ (M, θ, ξ), where ξ is a simple Borel
k-vector field orienting M . If f is a proper Lipschitz Q-valued function, we can define the
push-forward of T under f as follows.
Definition C.1. Given R = τ (M, θ, ξ) ∈ Dk (Rm ) and f ∈ Lip(Rm , AQ (Rn )) as above, we
denote by Tf,R the current in Rm+n defined by
ˆ
+,
)Tf,R , ω* =
θ
ω ◦ f¯i , DM f¯i# ξ d Hk ∀ ω ∈ D k (Rm+n ),
(C.1)
M
i
$
( #
where i DM f¯i (x) is the differential of f¯ restricted to M .
Remark C.2. Note that, by Rademacher’s theorem [6, Theorem 1.13] the derivative of a
Lipschitz Q-function is defined a.e. on smooth C 1 and, hence, also on rectifiable sets.
As a simple consequence of the Lipschitz decomposition in [6, Proposition 1.6], there
exist {Ej }j∈N closed subsets of Ω, positive integers kj,l , Lj ∈ N and Lipschitz functions
fj,l : Ej → Rn , for l = 1, . . . , Lj , such that
k
H (M \ ∪j Ej ) = 0 and f |Ej =
(
Lj
+
l=1
kj,l !fj,l " .
(C.2)
From the definition, Tf,R = j,l kj,l f¯j,l# (R Ej ) is a sum of rectifiable currents defined by
the push-forward under single-valued Lipschitz functions. Therefore, it follows that Tf,R is
!
"
rectifiable and coincides with τ f¯(M ), θf , T4f , where
DM f¯j,l# ξ(x)
θf (x, fj,l (x)) = kj,l θ(x) and T4f (x, fj,l (x)) = M ¯
∀ x ∈ Ej .
|D fj,l# ξ(x)|
By the standard area formula, using the above decomposition of Tf,R , we get an explicit
expression for the mass of Tf,R :
ˆ
+1
!
"
M (Tf,R ) =
|θ|
det DM f¯i · (DM f¯i )T d Hk .
(C.3)
M
i
32
C. DE LELLIS AND E. N. SPADARO
C.1. Boundaries of Lipschitz Q-valued graphs. With a slight abuse of notation, when
R = !Ω" ∈ Dm (Rm ) is given by the integration over a Lipschitz domain Ω ⊂ Rm of the
standard m-vector 4e = e1 ∧ · · · ∧ em , we write simply Tf,Ω for Tf,R . We do the same for
Tf,∂Ω , understanding that ∂Ω is oriented as the boundary of !Ω". We give here a proof of
the following theorem.
Theorem C.3. For every Ω Lipschitz domain and f ∈ Lip(Ω, AQ ), ∂ Tf,Ω = Tf,∂Ω .
This theorem is of course contained also in Almgren’s monograph [2]. However, our
proof is different and considerably shorter. The main building block is the following small
variant of [6, Homotopy Lemma 1.8].
Lemma C.4. There exists a constant cQ with the following property. For every closed cube
C ⊂ Rm centered at x0 and u ∈ Lip(C, AQ ), there exists h ∈ Lip(C, AQ ) with the following
properties:
(i) h|∂C = u|∂C , Lip(h) ≤ cQ Lip(u) and ,G(u, h),L∞ ≤ cQ Lip(u) diam(C);
(
(
(ii) u = Jj=1 !uj ", h = Jj=1 !hj ", for some J ≥ 1 and Lipschitz (multi-valued) maps
uj , hj ; each Thj ,C is a cone over Tuj ,∂C :
Thj ,C = !(x0 , aj )" ×
× Tuj ,∂C ,
for some aj ∈ Rn .
Proof. The proof is essentially contained in [6, Lemma 1.8]. Indeed, (i) follows in a straightforward way from the conclusions there. Concerning (ii), we follow the inductive argument
in the proof of [6, Lemma 1.8]. By the obvious invariance of the problem under translation
and dilation, it is enough to prove (
the following. If we consider the cone-like extension
of a vector-valued map u, h(x) = i !,x,ui (x/,x,)", where ,x, = supi |xi | is the uniform norm, then Th,C1 =
× Tu,∂C1 , with C1 = [−1, 1]m . This follows easily from the
(!0" ×
decomposition Tu,∂C1 = j,l kj,l ūj,l# (R Ej ) described in the previous subsection. Indeed,
setting
Fj = {tx : x ∈ Ej , 0 ≤ t ≤ 1},
clearly h decomposes in Fj as u in Ej and h̄j,l# (R Fj ) = !0" ×
× ūj,l# (R Ej ).
!
Proof of Theorem C.3. Without loss of generality, we can can reduce to the case where
Ω is the unit cube [0, 1]m . First of all, assume that Ω is biLipschitz equivalent to a
single cube and let φ : Ω → [0, 1]m be the corresponding homeomorphism. Set g =
f ◦ φ−1 . Define φ̃ : Ω × Rn → [0, 1]m × Rn . Following [11, Remark 27.2 (3)] and using the
characterization Tf,Ω = τ (f (Ω), θf , T4f ), it is simple to verify that φ̃# Tf,Ω = Tg,[0,1]m and
analogously φ̃# Tf,∂Ω = Tg,∂[0,1]m . So, since the boundary and the push-forward commute,
the case of Ω biLipschiz equivalent to [0, 1]m is reduced to the the case Ω = [0, 1]m .
Next, using a grid-type decomposition, any Ω, can be decomposed into finitely many
disjoint Ωi ⊂ Ω, all homeomorphic to a cube via a biLipschitz map, with the property that
∪Ωi = Ω. The conclusion for Ω follows then from the corresponding conclusion for each Ωi
and the obvious cancellations for the overlapping portions of their boundaries.
Assuming therefore Ω = [0, 1]m , the proof is by induction on m. For m = 1, by the
Lipschitz selection principle (cp. to [6, Proposition 1.2]) there exist single-valued Lipschitz
HIGHER INTEGRABILITY AND APPROXIMATION OF MINIMAL CURRENTS
functions fi such that f =
∂Tf,Ω
33
(
!fi ". Hence, it is immediate to verify that
+
+!
"
δfi (1) − δfi (0) = Tf |∂Ω .
=
∂Tfi ,Ω =
i
i
i
For the inductive argument, consider the dyadic decompositions of scale 2−l of Ω,
B
Ω=
Qk,l , with Qk,l = 2−l (k + [0, 1]m ) .
k∈{0,...,2l −1}m
In each Qk,l , let hk,l be the cone-like extension given by Lemma C.4 and
+
Tl :=
Thk,l ,Qk,l = Thl ,
k
with hl the Q-function which coincides with hk,l in Qk,l . Note that the hl ’s are equi-Lipschitz
and converge uniformly to f by Lemma C.4 (i).
By inductive hypothesis, since each face F of ∂Qk,l is a (m−1)-dimensional cube, ∂Tf,F =
Tf,∂F . Taking into account the orientation of ∂F for each face, it follows immediately that
∂Tf,∂Qk,l = 0.
(C.4)
Moreover, by Lemma C.4, each Thk,l ,Qk,l is a sum of cones. Therefore, using (C.4) and
∂(!0" ×
× T ) = T − !0" ×
× ∂T (see [11, Section 26]), ∂(Tl Qk,l ) = ∂Thk,l ,Qk,l = Tf,∂Qk,l .
Considering the different orientations of the boundary faces of adjacent cubes, it follows
that all the contributions cancel except those at the boundary of Ω, thus giving ∂Tl = Tf,∂Ω .
The integer m-rectifiable currents Tl , hence, have all the same boundary, which is integer
rectifiable and has bounded mass. Moreover, the mass of Tl can be easily bounded using
the formula (C.3) and the fact that sup Lip(hl ) < ∞. By the compactness theorem for
integral currents (see [11, Theorem 27.3]), there exists an integral current S which is the
weak limit for a subsequence of the Tl (not relabeled). Clearly, ∂S = liml→∞ ∂Tl = Tf,∂Ω .
We claim that Tf,Ω = S, thus concluding the proof.
To show the claim, notice that, since hl → f in L∞ , then supp (S) ⊆ graph(f ). So,
we need only to show that the multiplicity of the currents S and Tf,Ω coincide almost
everywhere. Consider a point x ∈ Ej , for some Ej in (C.2). From the Lipschitz continuity
of f and hl , in a neighborhood U of x, hl and S can be decomposed in the same way as f ,
hl |U =
Lj
+
p=1
!hl,p "
and S (U × R ) =
n
Lj
+
Sp ,
p=1
where the hl,p ’s are kj,p -valued and the Sp are integer rectifiable m-currents with disjoint
supports. By definition, the density of Tf,Ω in (x, fj,p (x)) is kj,p . On the other hand, since
π' Sp = lim π' Thl,p ,U = kj,l !U "
l
and supp (Sp ) ∩ ({x} × Rn ) = (x, fj,p (x)),
it follows that the density of Sp (and hence of S) in (x, fj,p (x)) equals kj,p . Since |Ω\∪j Ej | =
0, this implies S = Tf,Ω .
!
34
C. DE LELLIS AND E. N. SPADARO
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Zürich university
E-mail address: [email protected] and [email protected]