NOTE ON LUSIN (N ) CONDITION AND THE DISTRIBUTIONAL
DETERMINANT
LUIGI D’ONOFRIO, STANISLAV HENCL, JAN MALÝ AND ROBERTA SCHIATTARELLA
Abstract. Let Ω ⊂ Rn be an open set. We show that for a continuous mapping
f ∈ W 1,n−1 (Ω, Rn ) with Jf ∈ L1 (Ω) the validity of the Lusin (N ) condition implies
that the distributional Jacobian equals to the pointwise Jacobian.
1. Introduction
In the pioneering works [1] and [2] J. Ball studied a class of mappings that could be
used to model nonlinear elasticity. The theory was later extended for example in the
works of V. Šverák [26], I. Fonseca and W. Gangbo [6] and S. Müller, S. J. Spector
and Q. Tang [22]. The whole theory is nowadays very rich and we recommend the
monographs [15] and [11] for an overview of the field, discussion of interdisciplinary
links and further references.
Let Ω ⊂ Rn be a domain. In the theory one assumes that we have mapping f : Ω →
Rn (that corresponds to the deformation of a body in space) and we would like to find
weakest conditions that guarantee various properties like continuity, differentiability
a.e., invertibility, regularity of the inverse, null sets are mapped to null sets, mapping
does not change orientation and so on. In the theory one usually assumes that
f ∈ W 1,n (Ω, Rn ) (or that f belongs to some function space very close to W 1,n like
the closure of the grand-Sobolev space W 1,n) (see [16] for the definition) and using
this shows that the Jacobian is equal to the distributional Jacobian, i.e. that (1.1)
below holds for every ϕ ∈ Cc∞ (Ω) (see [9]). This fact is the main ingredient for the
later proof of various positive properties mentioned above (see e.g. [15] and [11] for
details).
The main aim of this note is to show that Sobolev regularity f ∈ W 1,n can be
essentially weakened if we a priori know that f is continuous and satisfies the Lusin
(N ) condition, i.e. for every N ⊂ Ω with |N | = 0 we have |f (N )| = 0. In fact
we show that if f ∈ W 1,n−1 is a continuous mapping which satisfies the Lusin (N )
condition, then the distributional Jacobian equals to the Jacobian and as corollary
we obtain various positive properties which were known before only with the stronger
assumption f ∈ W 1,n .
2000 Mathematics Subject Classification. 26B10, 30C65, 46E35.
The second author was supported by the ERC CZ grant LL1203 of the Czech Ministry of Education. The third author was supported by the grant GA ČR P201/15-08218S of the Czech Grant
Agency. The first author has been partially supported by Legge 5/2007 Regione Campania and the
first and the fourth author have been supported by the Gruppo Nazionale per l’ Analisi Matematica, la Probabilita’ e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica
(INdAM) .
1
2
LUIGI D’ONOFRIO, STANISLAV HENCL, JAN MALÝ AND ROBERTA SCHIATTARELLA
Theorem 1.1. Let Ω ⊂ Rn , n ≥ 2, be a domain and let f ∈ W 1,n−1 (Ω, Rn ) be
a continuous mapping with Jf ∈ L1 (Ω). Suppose that f satisfies the Lusin (N )
condition. Then the distributional Jacobian equals to the pointwise Jacobian, i.e.
Z
Z
(1.1)
ϕ(x)Jf (x) dx = − f1 (x)J(ϕ(x), f2 (x), . . . , fn (x)) dx
Ω
Ω
for every ϕ ∈ Cc∞ (Ω).
The version of this theorem was shown before in [10] for mappings that satisfy
the INV condition, Jf > 0 a.e. and that are essentially one to one a.e. We show
that this result holds even without those assumption but moreover we require that
our mapping is continuous as is natural in models that do not allow cavitation. The
proof follows the ideas of [3] and [10] (see also [19]).
Inspired by [17] and [11] we can show the following corollary (see Preliminaries for
the definition of sense-preserving mapping). Recall that f is said to be a mapping
of finite distortion if Jf ∈ L1 (Ω), Jf (x) ≥ 0 a.e. and the derivative Df (x) vanishes
a.e. in the zero set of the Jacobian Jf (x). The optimal (outer) distortion function is
defined as
( |Df (x)|n
if Jf (x) 6= 0
Jf (x)
K(x) :=
1
if Jf (x) = 0.
Corollary 1.2. Let Ω ⊂ Rn , n ≥ 2, be a domain and let f ∈ W 1,n−1 (Ω, Rn ) be a
continuous mapping of finite distortion which satisfies the Lusin (N ) condition. Then
a) f is sense preserving.
b) f is p-weakly monotone for all 1 ≤ p < n such that f ∈ W 1,p .
c) f is differentiable a.e. if f ∈ W 1,p (Ω, Rn ), p > n − 1 for n ≥ 3 or p ≥ 1 for
n = 2.
d) f is either constant or both open and discrete, i.e. maps open sets to open sets
and preimage of each point is a discrete set (=does not have an accumulation
point), if K ∈ Lp , p > n − 1 for n ≥ 3 or p ≥ 1 for n = 2.
In [3, Lemma 4.3] Conti and De Lellis showed a version of the area formula using
distributional Jacobian for mappings that satisfy the INV condition, Jf > 0 a.e. and
that are essentially one to one a.e. Its analogy with the degree holds for any sense
preserving continuous mapping even without the assumption that the mapping is
one-to-one.
Theorem 1.3. Let Ω ⊂ Rn , n ≥ 2, be a domain and let f ∈ W 1,n−1 (Ω, Rn ) be a
nonconstant, open and discrete continuous mapping. Then the distributional Jacobian
is a Radon measure and for all balls B ⊂⊂ Ω with Ln (f (∂B)) = 0 we have
Z
(1.2)
|Jf (B)| =
| deg(f, B, z)| dz .
Rn
Moreover, if the measure
of the branching set satisfies Ln (f (Bf )) = 0, then the last
R
integral is equal to Rn N (f, B, z) dz.
LUSIN CONDITION AND THE DISTRIBUTIONAL DETERMINANT
3
Let us briefly mention the idea of the proof of Theorem 1.1. Assume for simplicity
that n = 2 and that f is a homeomorphism. Then we can find a sequence of smooth
homeomorphisms fk that converge to f uniformly and in W 1,1 (see [12] and [14]). As
f satisfies the
R Lusin (N ) condition we know by area formula (see e.g. [11, Theorem
A.35]) that
R A Jf = |f (A)| for every measurable set A and similarly for smooth fk
we have A Jfk = |fk (A)|. Since
R fk → f Runiformly we have |fk (A)| → |f (A)| and
therefore it is easy to see that Ω ϕJfk → Ω ϕJf . Formula (1.1) holds for fk as they
are smooth and since fk → f uniformly and in W 1,1 it is easy to see that we can pass
to the limit also for the righthand side of (1.1) and we obtain the desired equality
for f . Unfortunately the approximation of homeomorphism by diffeomorphisms is
not known in higher dimension (see [13]) and therefore we have to choose a different
approach even for homeomorphisms. We have to use an explicit approximation of f
by mollification and take advantage of its special structure for careful computation
with the terms involved. This idea and the use of the degree formula goes back to
the works of Müller [20].
We would like to know if the assumptions of Corollary 1.2 are sharp:
Problem 1: Let f ∈ W 1,p , p < n − 1, be a continuous mapping of finite distortion
that satisfies the Lusin condition (N ). Does it follow that f is sense-preserving?
Problem 2: Let f ∈ W 1,p , p < n − 1, be a continuous mapping of finite distortion
that satisfies the Lusin condition (N ). Does it follow that f is p-weakly monotone?
Without the (N ) condition such an example exists even for p < n (see [17, Example
4]).
2. Preliminaries
If A is a real n × n matrix, we denote the cofactor matrix of A by cof A, the cof A
is the transpose of the adjugate of A. The inner product of vectors x, y ∈ Rn will be
denoted by hx, yi. A point x ∈ Rn is denoted in coordinates as (x1 , x2 , . . . , xn ).
The coordinates functions of a map f : Ω ⊂ Rn → Rn are denoted by f (x) =
(f1 (x), . . . , fn (x)).
Topological degree. Given a smooth map f from Ω ⊂ Rn into Rn we can define
the topological degree as
X
deg(f, Ω, y0 ) =
sgn(Jf (x))
{x∈Ω:f (x)=y0 }
if y0 ∈
/ f (∂Ω), Jf (x) 6= 0 for each x ∈ f −1 (y0 ). This definition can be extended to
arbitrary continuous mappings and each point see e.g. [6].
We will use the fact that the topological degree is stable under homotopy. That is
for every continuous mapping H : Ω × [0, 1] → Rn and p ∈ Rn such that p ∈
/ H(∂Ω, t)
for all t ∈ [0, 1] we have
deg(H(·, 0), Ω, p) = deg(H(·, 1), Ω, p).
A continuous mapping f : Ω → Rn is called sense-preserving if
deg(f, Ω0 , y0 ) > 0
for all domains Ω0 ⊂⊂ Ω and all y0 ∈ f (Ω0 ) \ f (∂Ω0 ). Similarly we define sensereversing mappings by opposite inequality.
4
LUIGI D’ONOFRIO, STANISLAV HENCL, JAN MALÝ AND ROBERTA SCHIATTARELLA
Let us recall that each homeomorphism on a domain is either sense-preserving or
sense-reversing see [23, II.2.4., Theorem 3]. The branch set Bf is defined as a set of
points where f fails to be locally injective - see [25, Chapter I.4] for more details.
Degree formula. We need the following proposition from [10, Proposition 2.10].
Proposition 2.1. Let B be a ball B ⊂ Rn , n ≥ 2 and let f ∈ W 1,n−1 (∂B, Rn ) ∩
L∞ (∂B, Rn ). Then there exists a unique L1 (Rn ) function, denoted by deg∗ (f, B, ·),
such that
Z
Z
n−1
hg(f (x)) cof Df (x), νidH (x) =
deg∗ (f, B, y) div g(y)dy
Rn
∂B
for all g ∈ C 1 (Rn , Rn ), where ν denotes the unit exterior normal to B. Moreover, if
f is continuous, then deg∗ (f, B, ·) coincides with the topological degree defined above
a.e. in Rn \ f (∂B).
1,1
Change of variables. Let f ∈ Wloc
(Ω, Rn ) and E ⊂ Ω be a measurable set. The
multiplicity function N (f, E, y) of f is defined as the number of preimages of y under
f in E. We say that the area formula holds for f on E if
Z
Z
(2.1)
η(f (x)) |Jf (x)| dx =
η(y) N (f, E, y) dy
Rn
E
for any nonnegative Borel measurable function on Rn . It is well known that there
exists a set Ω0 ⊂ Ω of full measure such that the area formula holds for f on Ω0 . Also,
the area formula holds on each set on which the Lusin condition (N ) is satisfied. This
follows from [5, Theorem 3.1.4, 3.1.8, 3.2.5], namely, it can be found there that Ω
can be covered up to a set of measure zero by countably many sets the restriction to
which of f is Lipschitz continuous.
The following change of variables formula using degree will be essential for us.
A similar statement holds for W 1,1 -mappings under the additional assumption that
f has a regular approximate differential (according to the terminology in [8]) a.e., see
[6, Theorem 5.27]. Any W 1,p -function has a regular approximate differential a.e. if
p > n−1 (see [8], [24] for the definition and result), but not necessarily in the limiting
case p = n − 1.
Theorem 2.2. Let Ω ⊂ Rn be an open, bounded set and let f ∈ W 1,n−1 (Ω, Rn ) ∩
C(Ω, Rn ) satisfy the Lusin (N ) property and Jf ∈ L1 (Ω). Then for every bounded,
open set G ⊂⊂ Ω with Ln (∂G) = 0 and v ∈ L∞ (Rn ) we have v(y) deg(f, G, y) ∈
L1 (Rn ) and
Z
Z
v(f (x))Jf (x) dx =
v(y) deg(f, G, y) dy .
Rn
G
Proof. Consider first the case that v = 1 in Rn . Consider a point x0 ∈ G which is an
(n−1)-Lebesgue point for Df and a Lebesgue point for Jf with Jf (x0 ) 6= 0. Further,
consider a sequence (rk )k of radii, rk & 0. We may assume that B(x0 , 2rk ) ⊂ G.
Since x0 is an (n−1)-Lebesgue point for Df , the rescalled functions
gk (ξ) = (Df (x0 ))−1
f (x0 + rk ξ) − f (x0 )
rk
LUSIN CONDITION AND THE DISTRIBUTIONAL DETERMINANT
5
converge in W 1,n−1 (B(0, 1)) to the limit function g which is nothing else than the
identity. Passing to a subsequence we may assume that
∞
X
kgk − gkW 1,n−1 (B(0,1)) < ∞.
k=1
1
,1
2
It follows that there is a radius t ∈
∞
X
such that
kgk − gkW 1,n−1 (∂B(0,t)) < ∞.
k=1
Write B = B(0, t). By Proposition 2.1,
Z
Z
(2.2)
deg(gk , B, ζ) div ϕ(ζ) dζ =
Rn
hϕ(gk (ξ)) cof Dgk (ξ), ν(ξ)i dHn−1 (ξ)
∂B
and
Z
Z
(2.3)
deg(g, B, ζ) div ϕ(ζ) dζ =
Rn
hϕ(g(ξ)) cof Dg(ξ), ν(ξ)i dHn−1 (ξ)
∂B
Cc∞ (Rn ).
for any test function ϕ ∈
The equality (2.2) together with the estimate
Z
Z
Z
| deg(gk , B, ζ)| dζ ≤
N (gk , B, ζ) dζ ≤
|Jgk (ξ)| dξ ≤ C
Rn
Rn
B
(which follows from (2.1)) yield a bound for the BV -norm of deg(gk , B, ·), independent
of k (for similar estimates see also [21, Lemma 3.1] and [7, Theorem 7]). Now, we
shall show that the right hand part of (2.2) converges to the right hand part of
(2.3). Indeed, we may assume that gk → g a.e., then by the Lebesgue dominated
convergence theorem
Z
h(ϕ(gk (ξ)) − ϕ(g(ξ))) cof Dg(ξ), ν(ξ)i dHn−1 (ξ) → 0.
∂B
Also
Z
n−1
∂B
(ξ)
hϕ(gk (ξ))(cof Dgk (ξ) − cof Dg(ξ)), ν(ξ)i dH
Z
≤ kϕk∞
| cof Dgk (ξ) − cof Dg(ξ)| dHn−1 (ξ) → 0.
∂B
Hence the convergence in the left hand part holds as well, namely
Z
Z
(2.4)
deg(gk , B, ζ) div ϕ(ζ) dζ →
deg(g, B, ζ) div ϕ(ζ) dζ.
Rn
Rn
Cc∞ (Rn ).
for each test function ϕ ∈
Together with the BV -bound, it follows that
deg(gk , B, ·) → deg(g, B, ·) = χB weak* in BV (Rn ). By the compact embedding,
deg(gk , B, ·) → χB in L1loc (Rn ), in particular in L1 (B). It follows that Ln (B\gk (B)) →
0. By the isoperimetric inequality (see the proof of [21, Lemma 3.1] and a routine
approximation argument)
Z
Z
n
n
n−1 n−1
L (gk (B)) ≤
| deg(gk , B, ζ)| dζ ≤ cn
| cof Dgk | dH
Rn
∂B
Z
n
n−1 n−1
→ cn
| cof Dg| dH
= Ln (B).
∂B
6
LUIGI D’ONOFRIO, STANISLAV HENCL, JAN MALÝ AND ROBERTA SCHIATTARELLA
where we have used the W 1,n−1 -convergence gk → g. Hence also Ln (gk (B) \ B) → 0.
n
By the BV -embedding theorem, the sequence k deg(gk , B, ·)k n−1
is bounded and thus
Z
Z
n−1
n1
n
m
| deg(gk , B, ζ)| dζ ≤
| deg(gk , B, ζ)| n−1 dζ
Ln (gk (B) \ B)
→ 0.
Rn \B
Rn \B
We finally have proved that
Z
Z
deg(gk , B, ζ) dζ →
Rn
deg(g, B, ζ) dζ
Rn
which cannot be obtained directly from (2.4) as ϕ(ζ) = ζ in not a valid test function
and we have no L∞ bound for gk .
An elementary change of variables gives
Z
Z
−n
rk
deg(f, B(x0 , trk ), z) dz = Jf (x0 )
deg(gk , B, ζ) dζ
Rn
Rn
Z
deg(g, B, ζ) dζ = Jf (x0 ) Ln (B)
→ Jf (x0 )
Rn
Z
=
Jf (x0 ) dξ.
B
But x0 is also a Lebesgue point for Jf , whence
Z
Z
Z
−n
Jf (x0 ) dξ.
Jf (x0 + rk ξ) dξ →
rk
Jf (x) dx =
B
B
B(x0 ,trk )
Thus, given ε > 0, with almost every y ∈ G with Jf (y) 6= 0 we can associate a set
Ry ⊂ (0, ∞) such that inf Ry = 0, B(y, sup Ry ) ⊂ G and
Z
Z
deg(f,
B(f
(y),
r),
z)
dz
−
J
(x)
dx
r ∈ Ry .
≤ εLn (B(y, r)),
f
Rn
B(y,r)
This gives a fine Vitali covering of almost all of G0 := {y ∈ G : Jf (y) 6= 0}. By the
Vitali covering theorem, there exists an a disjointed system (B(yi , ri ))i of balls in G
which covers almost all of G0 such that ri ∈ Ryi for each index i. Write
[
G00 =
B(yi , ri ).
i
Summing over i, we obtain
Z
Z
00
deg(f, G , z) dz −
(2.5)
G00
Rn
Jf (x) dx < εLn (G)
Obviously
Z
Z
Jf (x) dx =
G00
Jf (x) dx.
G
Now, we use the (N )-property to infer that the area formula holds for f . It follows
that deg(f, G, ·) = deg(f, G00 , ·) a.e. and thus
Z
Z
00
deg(f, G , z) dz =
deg(f, G, z) dz.
Rn
Rn
LUSIN CONDITION AND THE DISTRIBUTIONAL DETERMINANT
Letting ε → 0 in (2.5) we obtain
Z
Z
Jf (x) dx =
7
deg(f, G, z) dz,
Rn
G
which completes the proof of the case v = 1. Now, let v ∈ Cc (Rn ) be nonnegative
and set
Gs = {x ∈ G : v(f (x)) > s}.
Then Ln (∂Gs ) = 0 for almost every s > 0 and
(
0
if z ∈
/ f (∂Gs ) and v(z) ≤ s,
deg(f, Gs , z) =
deg(f, G, z) if z ∈
/ f (∂Gs ) and v(z) > s.
By the above part of the proof,
Z ∞ Z
Z ∞ Z
Z
v(f (x))Jf (x) dx =
Jf (x) dx ds =
deg(f, Gs , z) dz ds =
0
G
Rn
Z
Z0 ∞ ZGs
deg(f, G, z) dz ds =
v(z) deg(f, G, z) dz.
=
Rn
{v(z)>s}
0
Finally, the case of an arbitrary v ∈ L∞ (Rn ) can be obtained by decomposition into
the positive and negative part and a routine approximation.
3. Lusin (N ) condition implies the equality of Jacobians
Proof of Theorem 1.1. Set g(x) = (x1 , 0, · · · , 0). We recall the definition of distributional Jacobian, namely, for any ϕ ∈ Cc∞ (Ω)
Z D
Z
E
g(f (x)) cof Df (x), Dϕ(x) dx.
Jf (ϕ) = − f1 (x)J(ϕ(x), f2 (x), . . . , fn (x)) dx =
Ω
Ω
Cc∞ (R)
Let ψ ∈
(0, 1) and
be an even function supported in [−1, 1] such that ψ ≥ 0, ψ 0 ≤ 0 on
Z
ψ(|x|) dx = 1.
Rn
For each t > 0 we denote by φt the usual convolution kernel, that is
|x| φt (x) = ψt (|x|) = t−n ψ
.
t
Then by the Lebesgue dominated Theorem we have
Z D
E
Z
Jf (ϕ) = − lim
g(f (x)) cof Df (x),
ϕ(z)Dφt (x − z) dz
dx.
t→0+
Ω
B(x,t)
x
It is easy to see that Dφt (x) = ψt 0 (|x|)ν, where ν = |x|
is the normal vector. By
Fubini theorem and change to polar coordinates we get
Z
Z
Z t
0
Jf (ϕ) = − lim
ϕ(z)
ψt (r)
hg(f (x)) cof Df (x), νB(z,r) idHn−1 (x) dr dz.
t→0+
supp ϕ
0
∂B(z,r)
8
LUIGI D’ONOFRIO, STANISLAV HENCL, JAN MALÝ AND ROBERTA SCHIATTARELLA
By assumption Lusin condition (N ) holds for f and hence Ln (f (∂B)) = 0. Since f is
continuous, Proposition 2.1 is valid with the topological degree and the last integral
is equal to
Z
Z
Z t
0
− lim
ϕ(z)
ψt (r)
deg(f, B(z, r), y) div g(y) dy dr dz.
t→0+
supp ϕ
Rn
0
With the help of Lusin (N ) condition we can now use the degree formula (Theorem
2.2) to get
Z
Z
Z t
0
Jf (ϕ) = − lim
ϕ(z)
ψt (r)
div g(f (x))Jf (x) dx dr dz.
t→0+
Now we define
Z
G(r) =
supp ϕ
0
B(z,r)
Z rZ
div g(f (x))Jf (x)dx =
0
B(z,r)
div g(f (x))Jf (x) dHn−1 (x) dρ.
∂B(z,ρ)
Clearly
0
Z
div g(f (x))Jf (x) dHn−1 (x), a.e. r ∈ (0, t).
G (r) =
∂B(z,r)
Integrating by parts and using that G(0) = 0 and ψ(1) = 0 we get
Z
Z
Jf (ϕ) = lim
ϕ(z)
φt (z − x) div g(f (x))Jf (x) dx dz.
t→0+
supp ϕ
B(z,t)
By the definition of g we know that div g = 1. Since the convolution approximation
of Jf ∈ L1 converge to Jf in L1 we can pass to the limit and get
Z
Z
Jf (ϕ) =
ϕ(x) div g(f (x))Jf (x) dx =
ϕ(x)Jf (x) dx.
Ω
Ω
Proof of Corollary 1.2. a) Using the Lusin (N ) condition we obtain that the Jacobian
equals to the distributional Jacobian by Theorem 1.1. The result now follows by [17,
Theorem 2.4].
b) By [11, Theorem 2.17] we know that mapping is p-weakly monotone; in fact,
the assumption there require higher regularity of f but in its proof we only need to
know that Jf = Jf which is known by Theorem 1.1.
c) The claim follows from [11, Theorem 2.24] and part b).
d) This follows from [17, Theorem 3.1]. To verify its assumption we only need to
check the assumption (2.1) of that paper. This is the well-known divergence formula
and its proof is given under our assumption in the proof of Theorem 1.1 (just choose
g ∈ C 1 arbitrarily).
4. Change of variables formula
Proof of Theorem 1.3. By [25, paragraph after Theorem 4.6] we know that each open
and discrete mapping is either sense-preserving or sense-reversing. Without loss of
generality we can assume that f is sense-preserving as the reflection only changes the
LUSIN CONDITION AND THE DISTRIBUTIONAL DETERMINANT
9
sign of Jf and deg. Using the same test function as in the proof of Theorem 1.1 we
recall that with our choice div g = 1 we get the following equality
Z
Z
Z t
0
Jf (ϕ) = − lim
ϕ(z)
ψt (r)
deg(f, B(z, r), y) dy dr dz.
t→0+
supp ϕ
0
Rn
Then as f is sense-preserving (deg ≥ 0) and ψt0 ≤ 0 we get that the distributional
Jacobian is non-negative and each non-negative distribution is a Radon measure.
By [4, Theorem 4.7] we know that the absolutely continuous part of distributional
Jacobian is equal to the pointwise Jacobian.
As the distributional Jacobian is a non-negative Radon measure we can write:
Z
Jf (B(y, r)) = lim
Φδ (|x − y|) dJf (x)
δ→0+
where
Φδ (s) =


1
(r−s)
 δ
0
if s ≤ r − δ
if r − δ ≤ s ≤ r
if s ≥ r.
Using the definition of distributional Jacobian for ϕ = Φδ (|x − y|) , we have
Z
Z
(4.1)
Φδ (|x − y|) dJf (x) = Jf (Φδ (|x − y|)) = hf1 (x) cof Df (x), DΦδ (x − y)i dx.
Ω
Ω
In fact distributional Jacobian is defined only for Cc∞ functions and we have only
Φδ ∈ W 1,∞ . However, we can approximate it by ϕk ∈ Cc∞ such that ϕk → Φδ
uniformly and Dϕk → DΦδ weak* in L∞ . Identity as (4.1) holds for ϕk and we
can clearly pass to the limit using ϕk → Φδ uniformly for the left-hand side and
Dϕk → DΦδ weak* in L∞ for the right-hand side to obtain (4.1).
For every s ∈ (r − δ, r) and x ∈ ∂B(y, s) we have DΦδ (x − y) = 1δ νB(y,s) and thus
the last identity can be written as
Z
Z r
Z
1
Φδ (|x − y|) dJf (x) =
hf1 (x) cof Df (x), νB(y,s) i dHn−1 (x) ds.
Ω
r−δ δ ∂B(y,s)
By the degree formula Proposition 2.1 we obtain
Z
Z
Z r
1
(4.2)
Φδ (|x − y|) dJf (x) =
deg(f, B(y, s), z) dz ds
Ω
r−δ δ Rn
provided that Ln (f (∂B(y, s))) = 0 for a.e. s ∈ (0, r). By [25, Proposition 4.10 (3)] we
know that each nonconstant, sense-preserving, open and discrete mapping has locally
bounded multiplicity. Denote by N the bound on the multiplicity of N (f, B(y, r)).
Clearly
we cannot find uncountably many sets Sα ⊂ Rn such that Ln (Sα ) > 0 and
P
α χSα ≤ N . It follows that the set of bad radii
RB := s ∈ (0, r) : Ln (f (∂B(y, s))) > 0
is at most countable and (4.2) is legitimate.
Let us pick z ∈
/ f (∂B(y, r)) which is true for a.e. z ∈ Rn by the assumption of the
theorem. This point has at most N preimages in B(y, r). For every s ∈ (0, r) \ RB
such that z ∈
/ f (∂B(y, s)) we thus have
0 ≤ deg(f, B(y, s), z) ≤ deg(f, B(y, r), z),
10
LUIGI D’ONOFRIO, STANISLAV HENCL, JAN MALÝ AND ROBERTA SCHIATTARELLA
where we used the fact that f is sense-preserving (i.e. deg(f, B(y, r) \ B(y, s), z) ≥ 0)
and [25, Proposition 4.4 D4 ].
This gives us an majorant and we can apply the Lebesgue dominated convergence
theorem in (4.2) to obtain our thesis once we verify that
(4.3)
lim deg(f, B(y, s), z) = deg(f, B(y, r), z) for a.e. z ∈ Rn .
s→r+
Let us pick z ∈
/ f (∂B(y, r)) which is true for a.e. z ∈ Rn by the assumption of the
theorem. We know that
(4.4)
δ := dist z, f (∂B(y, r)) > 0
and using uniform continuity of f we know that for s close enough to r we have
s
δ
(4.5)
dist f (y − x) , f (y − x) < for every x ∈ B(0, r).
r
2
Let us consider the homothety
s
F (x, t) := tf (y − x) + (1 − t)f (y − x) .
r
This is clearly a continuous mapping from B(0, r) × [0, 1] to Rn and for every x ∈
∂B(0, r) and every t ∈ [0, 1] we have using (4.5)
δ
dist F (x, t), f (∂B(y, r)) < .
2
By (4.4) it follows that F (x, t) 6= z and hence using the stability of the degree with
respect to homotopy we obtain
lim deg(f, B(y, s), z) = deg(F (·, 0), B(0, r), z)
s→r+
= deg(F (·, 1), B(0, r), z) = deg(f, B(y, r), z).
Now (4.3) and our claim follows.
Moreover, if the measure of the image of branching set satisfies Ln (f (Bf )) = 0,
then using our assumption Ln (f (∂B(y, r))) = 0 by [25, Proposition 4.10 (1)] we have
that N (f, B(y, r), z) = deg(f, B(y, r), z) for a.e. z ∈ Rn .
Acknowledgement. The authors would like to thank Carlo Sbordone for pointing
their interest to the problem.
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Dipartimento di Studi Econimici e Giuridici, Università “Parthenope ”Via Generale
Parisi 13, 80100 Napoli, Italy
E-mail address: [email protected]
Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00
Prague 8, Czech Republic
E-mail address: [email protected]
Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00
Prague 8, Czech Republic
E-mail address: [email protected]
Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Università di Napoli
“Federico II”, Via Cintia, 80126 Napoli, Italy
E-mail address: [email protected]