Extremal Mappings of Finite Distortion
K. Astala
1
T. Iwaniec
G. Martin
J. Onninen ∗
Introduction
The theory of mappings of finite distortion has arisen out of a need to extend the
ideas and applications of the classical theory of quasiconformal mappings to the
degenerate elliptic setting. There one finds concrete applications in materials
science, particularly nonlinear elasticity and critical phase phenomena, and the
calculus of variations.
In this paper we refine and extend these connections by initiating the study
of extremal problems for mappings with finite distortion.
There are many natural reasons for studying such problems. First, we eventually hope to lay down the analytical foundations for approaches to compactifying the moduli spaces for holomorphic dynamical systems such as Teichmüller
spaces where it is our expectation that a compactification will be by mappings
of finite distortion whose distortion function lies in some natural integrability
class.
Secondly, we find there are many new and unexpected phenomena concerning
existence, uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex. These seem to differ markedly
from phenomena observed when studying multi–well type functionals.
Thus our primary aim here is to extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead
of its L ∞ -norm. Let us indicate the sorts of results we shall prove here by a
surprising example in two dimensions. Suppose fo : S → S is a homeomorphism of the circle. We ask what is the mapping f : D → D of the disk with
f |S = fo , whose distortion function is minimal in L 1 –norm. We show that if
fo−1 ∈ W 1/2,2 (S), then there is a unique extremal which is a real analytic diffeomorphism with nonvanishing Jacobian determinant. The condition on fo is
∗ Astala was supported in part by the Academy of Finland, projects 34082 and 41933,
Iwaniec by the National Science Foundation grants DMS-0301582 and DMS-0244297, Martin
by the N.Z. Marsden Fund and the N.Z. Royal Society (James Cook Fellowship), and Onninen
by the National Science Foundation grant DMS-0400611.
AMS (2000) Classification. Primary 30C60, 35J15, 35J70
Keywords Extremal, Quasiconformal, Calculus of Variations
1
sharp. Contrast this with the classical theory [30] where the mapping fo must
be assumed quasisymmetric (a much stronger assumption). Then there is an
extremal quasiconformal mapping with boundary values fo , but it is not always
unique and it is seldom smooth. Indeed, even when fo is quasisymmetric, the
L 1 –minimiser for the distortion function will almost never be quasiconformal.
This result is doubly surprising when we note that there is no general modulus
of continuity (so no compactness) nor general improved regularity theory for
mappings with only integrable distortion.
Away from the L 1 –theory and closer to the L ∞ –theory we have mappings
whose distortion function is exponentially integrable. In this case there are
substantial results concerning modulus of continuity and regularity available,
[15]. From these one obtains existence of minimisers and some improved regularity for the boundary value problem as discussed above. However establishing
uniqueness seems difficult.
There are also other natural problems that we shall consider such as the
generalizations to the L 1 setting of the famous Grötzsch problem concerning
extremal mappings between n–dimensional boxes, see Sections 6 and 7.
Finally we wish to make an observation. The principal advantage of minimising over families of homeomorphisms in the above variational problems lies
in the fact that the inverse maps are also extremal for their own variational integrals. Sometimes this associated problem is easier to solve than the original one
as it may involve minimising a convex functional. For instance, in n–dimensions,
the L 1 minimisation problem leads to the n–harmonic equation for the inverse
of an extremal mapping.
2
Formulation of the General Problem
We begin by defining the class of mappings which will most concern us here. A
1,1
mapping f : Ω → Ω0 between subdomains of Rn and of Sobolev class Wloc
(Ω, Ω0 )
is said to have finite distortion if there is a measurable function 1 6 K(x) < ∞
such that
n
|Df (x)| 6 K(x) J(x, f )
(2.1)
and if J(z, f ) ∈ L1loc (Ω). Here |Df (x)| is the operator norm, |Df | = max{|Df v| :
|v| = 1}, of the linear differential map Df (x) and J(x, f ) = det Df (x) is its
Jacobian determinant. The reader may observe that for homeomorphisms the
condition J(z, f ) ∈ L1loc (Ω) is largely redundant. The smallest function K(x)
for which the distortion inequality (2.1) holds is called the outer distortion of
f , defined by
n
|Df (x)|
(2.2)
K(x, f ) =
J(x, f )
at points where Df (x) exists and is nonsingular and we set K(x, f ) = 1 elsewhere. The operator norm of the differential matrix at (2.2) has the disadvantage of being insufficiently regular to deal with variational equations. We
2
therefore introduce the outer distortion function
|| Df (x) || n
J(x, f )
K (x, f ) =
(2.3)
at points where Df (x) exists and is nonsingular and set K (x, f ) = 1 elsewhere.
Here || A || 2 = n1 tr(At A) is the mean Hilbert-Schmidt norm. Notice that in two
dimensions
1
1
K (x, f ) =
K(x, f ) +
,
(n = 2)
(2.4)
2
K(x, f )
is a convex function of K(x, f ) and therefore
1
1
kK (x, f )k∞ =
kK(x, f )k∞ +
,
2
kK(x, f )k∞
(n = 2)
so that minimising either kK (x, f )k∞ or kK(x, f )k∞ leads to the same quasiconformal minimisers. Formula (2.4) fails in general for n ≥ 3.
More generally, to every strictly increasing convex function Ψ : [1, ∞) →
[1, ∞), Ψ(1) = 1, we associate two further distortion functions
KΨ (x, f ) = Ψ K(x, f )
and K Ψ (x, f ) = Ψ K (x, f )
We can now formulate the major minimisation problem we shall address.
Let F consist of homeomorphisms f : Ω → Ω0 of finite distortion such that
Z
K Ψ (x, f ) dx < ∞
(2.5)
Ω
Problem 1. Given fo ∈ F find a mapping f ∈ F which coincides with fo on
∂Ω and minimises the integral at (2.5).
We shall also consider related problems where we do not prescribe the precise
boundary values or where we minimise integral averages of other distortion
functions. We now give a fuller discussion of these.
3
Distortion Functions
Distortion functions are designed to measure the deviation from conformality
of a given mapping f : Ω → Ω0 by considering the linear differential map
Df (x) : Rn → Rn . We shall denote the set of all n × n-matrices by Rn×n , and
n×n
those with positive determinant by R+
. It will be convenient to include the
zero matrix in the domain of distortion functions. The conformal matrices are
CO+ (n) = {A : |A|n = det A}
3
equivalently the defining equation could be || A || n = det A. In two dimensions
it is advantageous to use complex variables. A general linear transformation
takes the form
a, b ∈ C
Az = az + bz
The determinant, the operator norm and mean Hilbert-Schmidt norm are then
det A = |a|2 − |b|2 ,
|A| = |a| + |b|,
|| A || 2 = |a|2 + |b|2
To every pair of ordered `-tuples I = (i1 , ..., i` ) and J = (j1 , ..., j` ), 1 6 i1 < ... <
i` 6 n and 1 6 j1 < ... < j` 6 n, there corresponds the ` × `-subdeterminant of
a matrix A = Aij ∈ Rn×n , namely
Aij11

AIJ = det  ...
Aij`1

The `-th exterior power of A is the
n
`
×
···
···
n
`

Aij1`
.. 
. 
Aij``
-matrix
n
n
A`×` = AIJ ∈ R( ` )×( ` )
indexed by `-tuples I = (i1 , ..., i` ) and J = (j1 , ..., j` ). We shall need the
following well known identities from exterior algebra
`×`
[AB]
= A`×` B `×` ,
−1 `×` `×` −1
A
= A
,
t `×` `×` t
A
= A
det A`×` = (det A)`
Then Cramer’s rule states
At A] = (det A) I
where A] is the cofactor matrix. To every pair (i, j) there corresponds the (i, j)cofactor, which is the product of (−1)i+j with the (n−1)×(n−1) subdeterminant
obtained by deleting the i-th row and the j-th column in A. Cramer’s rule can
n
n
be formulated by using ` × `-minors. Given A`×` ∈ R( ` )×( ` ) , we define the
n
n
cofactor matrix A`×`
∈ R( ` )×( ` ) . The cofactor of the element AIJ is the product
]
i1 +...+i` +j1 +...+j`
of (−1)
with the (n − `) × (n − `)- subdeterminant obtained by
suppressing the i1 , ..., i` -th rows and j1 , ..., j` -th columns. Cramer’s rule reads
as
`×` t `×`
A
A] = (det A) I
Thus in particular, if det A 6= 0, then
h
it
`×`
A]`×` = (det A) A−1
4
(3.1)
There is an isometry, called the Hodge star operator, which assigns to A`×`
the
]
(n − `)-exterior power of A. This isometry arises from identifying the `-tuple
I = (i1 , ..., i` ) with its complementary (n − `)-tuple. Hence
`×` (n−`)×(n−`) (3.2)
and || A]`×` || = || A(n−`)×(n−`) ||
A] = A
These identities be can easily derived by reducing A to a diagonal matrix.
The following distortion functions will interest us most as they have the very
important property of being polyconvex, see Appendix 12.2. For each A ∈ Rn×n
+
and ` = 1, 2, ..., n − 1, we define
n
`×` n−`
n
A || A`×` || n−`
,
K ` (A) =
K` (A) =
`
`
(det A) n−`
(det A) n−`
We extend this definition to Rn×n
∪ {0} by setting K` (0) = K ` (0) = 1. Notice
+
that K` (A) > 1 and K ` (A) > 1, equality occurring if and only if A ∈ CO+ (n).
The two outer distortion functions discussed previously in (2.2) and (2.3)
arise when ` = 1 as
K (A) = K 1 (A)n−1 and K (A) = K 1 (A)n−1
The corresponding inner distortion functions are
KI (A) = Kn−1 (A) = K(A−1 ), K I (A) = K n−1 (A) = K (A−1 )
When n = 2 the inner distortions are the same as the outer distortions.
1,1
Given the above, a mapping f ∈ Wloc
(Ω, Ω0 ) will have finite distortion if
n×n
1
Df (x) ∈ R+ ∪ {0} for almost every x ∈ Ω and J(x, f ) = det Df ∈ Lloc
(Ω)
n×n
Having defined distortion functions for matrices in R+ ∪ {0} we define for
mappings f : Ω → Ω0 of finite distortion:
n
D`×` f (x) n−`
K` (x, f ) = K` Df (x) =
(3.3)
`
J(x, f ) n−`
n
|| D`×` f (x) || n−`
K ` (x, f ) = K ` Df (x) =
`
J(x, f ) n−`
(3.4)
The reader may find more about distortion functions in Appendix 12.1.
4
Variations of Weakly Differentiable Homeomorphisms
As this paper is largely concerned with Sobolev classes of homeomorphisms we
shall need to establish the existence of suitable variations of such mappings.
These variations may be required to preserve quasiconformality as well as some
other natural properties of mappings. The following theorem captures all those
needs.
5
THEOREM 4.1. Let f : B → Rn be a homeomorphism of the Sobolev class
1,1
Wloc
(B, Rn ) and let a ∈ B be a Lebesgue point of Df such that J(a, f ) > 0. Then
there exists a diffeomorphism h : B → B, referred to as change of variables, such
that the composite mapping f˜(x) = f (h(x)) satisfies:
(i) f˜(x) = f (x) near ∂B
(ii) The origin is a Lebesgue point of Df˜
(iii) Df˜(0) = I
The proof for the construction of this change of variables consists of three
lemmas that are of independent interest.
LEMMA 4.1. Given a point a ∈ B and r > |a| there exists a diffeomorphism
Φ : Rn → Rn such that
Φ(0) = a,
DΦ(0) = I,
and Φ(x) = x for |x| > r
Proof. Such a diffeomorphism Φ will be a perturbation of the identity map,
namely
Φ(x) = x + η(x)a
where η ∈ C0∞ (Br ) satisfies:
0 6 η 6 1,
η(0) = 1,
and
|| ∇η || ∞ <
1
|a|
Its differential is a matrix of the form
DΦ(x) = I + a ⊗ ∇η
Since η assumes its largest value 1 at the origin we infer that ∇η(0) = 0. Hence
DΦ(0) = I, as claimed. The Jacobian determinant of Φ is computed in Br as
det DΦ = 1 + ha, ∇ηi > 1 − |a| |∇η| > 1 − |a| || ∇η || ∞ > 0
(4.2)
and det DΦ = 1 outside Br . In particular, Φ is a local diffeomorphism of Rn
into itself. Since Φ(x) = x outside the ball Br , by topological arguments we
conclude that Φ : Rn → Rn is one-to-one.
LEMMA 4.2 (Decomposition of Matrices). Given A ∈ Rn×n
+ and > 0, there
n×n
exist A1 , A2 , ..., Ak ∈ R+ such that
|I − Ai | 6 for i = 1, 2, ..., k
and
A = A1 · A2 · ... · Ak
6
Proof. As Rn×n
is a connected matrix Lie group we can decompose A as
+
A = eX1 eX2 · ... · eXm
(4.3)
where X1 , X2 , ..., Xm ∈ Rn×n see [11, Corollary 2.31.]. Next choose δ = N1 ,
where N is a large positive integer such that e δ|Xk | − 1 6 for k = 1, 2, ..., m.
Hence
A
=
δ X2
δ Xm
e| δ X1 {z
...e δ X}1 · e
...e δ X}2 ·... · e
...e δ Xm}
|
|
{z
{z
N −times
N −times
def
== A1 · A2 · ... · Ak ,
N −times
where k = mN
(4.4)
For every Ai we have
|I − Ai | 6 max 1 − e δ Xk 6 max 1 − e δ|Xk | 6 16k6m
16k6m
as desired.
LEMMA 4.3. Given A ∈ Rn×n
and r > 0, there exists a diffeomorphism
+
n
n
Ψ : R → R such that
Ψ(0) = 0,
DΨ(0) = A,
and Ψ(x) = x for |x| > r
Proof. First assume that A is sufficiently close to the identity matrix, say
|I − A| <
1
4
(4.5)
We construct Ψ as a perturbation of the identity Ψ(x) = x − (x − Ax) η where
η ∈ C0∞ (Br ), 0 6 η(x) 6 1, η(0) = 1, and || ∇η || ∞ < 2r . We find the differential
of Ψ as follows
DΨ(x) = I − (I − A)η − (x − Ax) ⊗ ∇η
Clearly, DΨ(0) = A. In order to see that det DΨ(x) 6= 0 we view DΨ(x) as
a small perturbation of I. For |x| 6 r we have the following estimate of the
perturbation term
|η(x)(I − A) + (x − Ax) ⊗ ∇η| 6 η(x) |I − A| + |x − Ax| |∇η(x)|
6 |I − A| + r |I − A| || ∇η || ∞
3
6 3 |I − A| <
4
(4.6)
If follows that det DΨ(x) > 4−n . As in Lemma 4.1 we conclude that Ψ is a
diffeomorphism of Rn onto itself. We now can free ourselves from the assumption
7
(4.5) by using the decomposition at Lemma 4.2. Accordingly we put A =
A1 ·A2 ·...·Ak where |I − Ai | < 41 for i = 1, 2, ..., k. Let Ψ1 , Ψ2 , ..., Ψk : Rn → Rn
be the diffeomorphisms constructed above, that is,
Ψi (0) = 0, Ψi (x) = x for |x| > r and DΨi (0) = Ai for i = 1, 2, ..., k
The composition Ψ = Ψ1 ◦ Ψ2 ◦ ... ◦ Ψk : Rn → Rn satisfies all the assertions of
Lemma 4.3.
Proof of Theorem 4.1. We define h as composition of Φ from Lemma 4.1 and Ψ
−1
from Lemma 4.3. This latter map Ψ is determined by taking A = [Df (a)] ∈
Rn×n
+ . Actually, the composite map
h = Φ ◦ Ψ : R n → Rn ,
h(0) = a
is a diffeomorphism of the entire space Rn . It maps the unit ball onto itself,
and is the identity near ∂B. The chain rule applies to f˜ as follows
Df˜(x) = Df (h(x)) Df (x)
for almost every x ∈ B
In particular, the origin is a Lebesgue point of Df˜. Moreover,
Df˜(0) = Df (a) Dh(0) = I
because
−1
Dh(0) = DΦ(0) DΨ(0) = DΨ(0) = [Df (a)]
5
Sublinear Growth, the Failure of Minimization
Given a mapping fo : B → Rn we now study minima of the variational integral
Z
min − Ψ [KI (x, f )] dx,
(5.1)
B
in the class Fo = F(fo ) of all W (B, Rn ) homeomorphisms f : B → Rn which
coincide with fo on ∂B. We demonstrate that these integrals almost never
attains the minimum value if Ψ exhibits sublinear growth, meaning that
1,n
Ψ(t)
=0
(5.2)
t→∞
t
In many ways this situation is reminiscent to the well known Lavrentiev phenomenon in the Calculus of Variations [21]. For simplicity we consider quasiconformal boundary data. The reader may easily generalise this situation.
lim
8
THEOREM 5.3. Let Ψ ∈ C [1, ∞) be a positive strictly increasing function of
sublinear growth. Given a quasiconformal map fo : B → Rn we have
Z
inf − Ψ [KI (x, f )] dx = Ψ(1)
(5.4)
f ∈Fo
B
In particular, the minimization problem (5.1) has no solution in Fo if fo : ∂B →
Rn admits no conformal extension to B.
Proof. Let a ∈ B be a Lebesgue point for both Df (x) and Ψ [KI (x, fo )] such
that det Dfo (a) > 0. We shall make use of Theorem 4.1 to modify fo without
changing its boundary values and quasiconformality. For this modified map,
still denoted by fo , the origin becomes a Lebesgue point of both functions Dfo
and Ψ [KI (x, fo )]. What we have gained here is that fo is an isometry near the
origin; we have Dfo (0) = I and Ψ [KI (0, fo )] = Ψ(1). For every 0 < < r < 1
we consider the radial mapping g : B → B defined by the rule
g(x) = x ρ(|x|)
where
(
ρ(|x|) =
−r
1−r
+
1−
1−r |x|
if |x| 6 r
if r 6 |x| 6 1
The distortion function of g can be explicitly computed [15, p. 114] giving the
following estimate.
(
1
for |x| 6 r
KI (x, g) 6
(5.5)
2
for
r 6 |x| 6 1
1−r
Note that g(x) = x for |x| 6 r. Now the following composite map is a
legitimate competitor for our variational integral
f (x) = fo (g(x))
The chain rule is valid for quasiconformal mappings, so we can write
Df (x) = Dfo (g(x)) · Dg(x) = Dfo (x),
for |x| 6 r
Hence we obtain the following estimate of the inner distortion function of f
(
KI (x, fo ) for |x| 6 r
(5.6)
KI (x, f ) 6
2K
for r 6 |x| 6 1
1−r
where K = || KI (x, fo ) || ∞ . We may now evaluate the variational integral
Z
Z
Z
2K
Ψ [KI (x, f )] dx 6
Ψ [KI (x, fo )] dx +
Ψ
dx
1−r
B
|x|6r
r6|x|<1
9
Hence
Z
Z
− Ψ [KI (x, f )] dx 6 −
B
Ψ [KI (y, fo )] dy + (1 − rn )Ψ
|y|6
2K
1−r
(5.7)
Using the supposed sublinear growth of Ψ we find that the latter term goes to
zero as r → 1. In the first term we let go to zero. Since the origin is the
Lebesgue point of Ψ [KI (x, fo )] the integral mean converges to Ψ [KI (0, fo )] =
Ψ(1). In conclusion, the infimum at (5.4) does not exceed Ψ(1). On the other
hand the integrand is always greater than or equal to Ψ(1). Thus (5.4) holds.
Finally, any minimizer of finite distortion must satisfy the point-wise equation
Ψ [KI (x, f )] = Ψ(1) a.e. in B
or, equivalently, KI (x, f ) = 1 a.e. in B because Ψ is strictly increasing. Thus
f is conformal by the Liouville Theorem [15]. This shows that for arbitrary
boundary values, other than those of a conformal mapping of course, there can
be no minimizers to the problem (5.1).
6
The L 1 -Grötzsch problem, n = 2
In this section we present an L 1 -variant of the celebrated Grötzsch extremal
problem for mappings between rectangles, [8]. In this section we confine our
discussion to two dimensions. Let Q be the unit square in the plane i.e.
Q = [0, 1] × [0, 1] ⊂ R2
and Q0 be a rectangle
Q0 = [0, 2] × [0, 1] ⊂ R2
We shall show that
ZZ
2
K (z, f ) |dz| =
min
f ∈F
Q
5
4
(6.1)
1,1
where F consists of homeomorphisms f : Q → Q0 in the Sobolev class Wloc
(Q, R2 )
of integrable distortion taking vertices into vertices. Our goal is to show that
this free boundary value problem has still unique extremal. Before jumping into
the proof of this result we demonstrate that uniqueness is lost for K(z, f ).
THEOREM 6.2. The minimization problem
ZZ
2
min
K(z, f ) |dz|
f ∈F
Q
has infinitely many extremals.
10
(6.3)
Proof. Suppose f ∈ F, we first show that
ZZ
2
26
K(z, f ) |dz|
(6.4)
Q
To see this we note
Z
26
1
|Df (x + iy)| dx
for almost all y ∈ [0, 1]
0
and then after integrating over y we find that
ZZ
ZZ p
p
2
2
26
|Df (z)| |dz| =
K(z, f ) J(z, f ) |dz|
Q
Q
Upon squaring, Hölder’s inequality implies
ZZ
ZZ
2
2
46
K(z, f ) |dz| ·
J(z, f ) |dz|
Q
(6.5)
Q
1,1
Since f : Q → Q0 is a homeomorphism of the Sobolev class Wloc
(Q, R2 ), it
follows that
ZZ
J(x, f ) dx 6 |Q0 |
Q
where equality holds if f maps sets of zero measure into sets of zero measure.
The claim (6.4) follows.
For every 0 6 a < 1, the piece-wise linear map
(
x + iy
if z ∈ [0, a] × [0, 1]
g(z) =
a
2−a
if z ∈ [a, 1] × [0, 1]
1−a x − 1−a + iy
is a minimizer. Indeed, g is the identity if 0 6 x 6 a, while for x 6 a < 1 we
have
!
2−a
0
2−a
2−a
1−a
Dg(z) =
|Dg| ≡
hence K(z, g) ≡
1−a
1−a
0
1
The integral of K(z, g) does not depend on a, indeed
ZZ
2−a
2
=2
K(z, g) |dz| = a · 1 + (1 − a)
1
−a
Q
This completes the proof of Theorem 6.2.
The situation is dramatically different for the distortion function K.
11
THEOREM 6.6. The minimization problem (6.1) has a unique extremal.
Proof. Let f : Q → Q0 be any admissible mapping, that is, f ∈ F. Using
complex notation
f (z) = u(x, y) + iv(x, y),
z = x + iy
we observe that for almost every 0 < y < 1
Z 1
ux (x + iy) dx = u(1 + iy) − u(iy) = 2,
0
while for almost every 0 < x < 1
Z 1
vy (x + iy) dx = u(x + i) − u(x) = 1
0
Further integration yields
ZZ
ux dx dy = 2
ZZ
and
vy dx dy = 1
Q
Q
We combine these equations in one weighted sum and use Schwarz inequality
to obtain
ZZ
√ ZZ q
5 =
(2ux + vy ) dx dy 6 5
u2x + vy2 dx dy
(6.7)
Q
Q
√ ZZ q
6
5
u2x + vy2 + u2y + vx2 dx dy
(6.8)
Q
Z
Z
Z
Z
p
p
√
√
2
=
10
|| D] f || = 10
K (z, f ) J(z, f ) |dz|
Q
Q
Upon squaring both sides, Hölder’s inequality implies
ZZ
ZZ
2
2
25 6 10 |Q0 |
K (z, f ) |dz| = 20
K (z, f ) |dz|
Q
Q
In other words,
ZZ
2
K (z, f ) |dz| >
Q
5
4
for all f ∈ F
As expected, equality occurs when f (z) = 2x + iy. We need only show that the
equation
ZZ
5
2
K (z, f ) |dz| =
for f ∈ F
yields f = 2x + iy
4
Q
To this end, we must examine when equalities occurs in the chain of the above
estimates. First, (6.8) holds as equality if and only if
uy ≡ vx ≡ 0
12
meaning that u depends only on x and v depends only on y. Recall that (6.7)
came from using Schwartz inequality. This forces the following relation
2vy = ux
a.e. in Q
As ux and vy depend on different variables we infer that ux and vy are constant
functions. This leaves only one possibility
u(x, y) = 2x
and
v(x, y) = y
(6.9)
as claimed.
7
The Grötzsch Problem, n > 2
In higher dimensions there are several distortion functions to investigate. Given
` = 1, 2, ..., n − 1 we shall consider the minimization problem
Z
min
K ` (x, f ) dx
(7.1)
f ∈F
Q
where F consists of homeomorphisms f = (f 1 , ..., f n ) : Q → Q0 of the Sobolev
1,p
class Wloc
(Ω), p > `, with finite distortion for which K ` (x, f ) is integrable.
Here Q and Q0 are rectangular boxes,
Q = [0, a1 ] × ... × [0, an ] ⊂ Rn ,
Q0 = [0, a01 ] × ... × [0, a0n ] ⊂ Rn
(7.2)
We shall assume that f maps every (n − 1)-dimensional face
[0, a1 ] × ... × [0, ak−1 ] × {a} × [0, ak+1 ] × ... × [0, an ]
(7.3)
where a = 0 or a = ak into the corresponding face
[0, a01 ] × ... × [0, a0k−1 ] × {a0 } × [0, a0k+1 ] × ... × [0, a0n ]
(7.4)
where a0 = 0 or a0 = a0k , respectively. This actually implies that f maps every
`-dimensional face, ` = 0, 1, ..., n, of Q into the corresponding `-dimensional face
of Q0 .
One example of a mapping in F is the linear map
g(x) = (λ1 x1 , ..., λn xn )
with λk =
a0k
ak
THEOREM 7.5. For each ` = 1, 2, ..., n − 1, the minimization problem (7.1)
has exactly one solution, namely the linear map g.
13
Proof. We consider the `-th exterior power of the differential matrix
D`×` f (x) ∈ R
(nl)×(nl)
defined for almost every x = (x1 , ..., xn ) ∈ Q. To every multi-index I =
(i1 , i2 , ..., i` ), 1 6 i1 < ... < i` 6 n, there corresponds the diagonal entry of
D`×` f , namely
∂(f i1 , ..., f i` )
∂(xi1 , ..., xi` )
We shall first prove the following inequality
Z ∂(f i1 , ..., f i` ) dx
λi1 · ... · λi` 6 − Q ∂(xi1 , ..., xi` )
(7.6)
To simplify the writing consider the case (i1 , i2 , ..., i` ) = (1, 2, ..., `). Fix the
remaining variables x`+1 = c`+1 , ..., xn = cn and define the mapping
F (x1 , x2 , ..., x` ) = f 1 (x1 , ..., x` , c`+1 , ..., cn ), ..., f ` (x1 , ..., x` , c`+1 , ..., cn )
on the closed `-dimensional rectangle
def
U == [0, a1 ] × ... × [0, a` ] ⊂ R`
This mapping is valued in the closed `-dimensional rectangle
def
U0 == [0, a01 ] × ... × [0, a0` ] ⊂ R`
By elementary topological arguments we see that F : U → U0 is a continuous
surjective map, for every parameter
def
c = (c`+1 , ..., cn ) ∈ W == [0, a`+1 ] × ... × [0, an ]
Moreover, for almost every c ∈ W this map F belongs to the Sobolev class
1,p
Wloc
(U, U0 ), where we emphasize that p > ` = dim U. Now the ` × `-minor
∂(f 1 , ..., f ` )/∂(x1 , ..., x` ) is nothing other than the Jacobian determinant of F .
At this point we appeal to geometric measure theory, see for instance [2, theorem
8.3], to deduce that
Z ∂(f 1 , ..., f ` ) dx1 ...dx` > |U0 |
U ∂(x1 , ..., x` )
Integrating with respect to the parameter c ∈ W, by Fubini theorem, we obtain
Z
∂(f 1 , ..., f ` ) 0
0
0
(7.7)
∂(x1 , ..., x` ) dx > |U | |W| = a1 · · · a` · a`+1 · · · an
U×W
14
which is the same as (7.6).
We now multiply (7.6) by the product λi1 · · · λi` and sum with respect to all
multi-indices
Z
X
X
∂(f i1 , ..., f i` ) 2
dx (7.8)
(λi1 · · · λi` ) 6 −
λi1 · · · λi` ∂(xi1 , ..., xi` ) Q
16i1 <...<i` 6n
16i1 <...<i` 6n
Z 6−
2
Q
X
21 (λi1 · · · λi` )
16i1 <...<i` 6n
Hence,
X
X
16i1 <...<i` 6n
2
(λi1 · · · λi` )
16i1 <...<i` 6n
21
1
∂(f i1 , ..., f i` ) 2 2
dx (7.9)
∂(xi , ..., xi ) 1
`
1
∂(f i1 , ..., f i` ) 2 2
(7.10)
Q 16i <...<i 6n ∂(xi1 , ..., xi` )
1
`
1
Z X
∂(f i1 , ..., f i` ) 2 2
6 −
(7.11)
∂(xj , ..., xj ) Q
1
`
16i <...<i 6n
Z 6 −
X
1
`
16j1 <...<j` 6n
The reader may wish to notice that we have added the missing off–diagonal
entries of D`×` f to give us the Hilbert-Schmidt norm of D`×` f .
12
Z
X
1
2
(λ
·
·
·
λ
)
6
− || D`×` f (x) || dx
i1
i`
n
`
Q
16i1 <...<i` 6n
Z
n` Z
n−`
n
6
− J(x, f ) dx
− K ` (x, f ) dx
Q
Q
The latter follows by Hölder’s inequality and the identity
`
`
|| D`×` f || = J(x, f ) n K ` (x, f )1− n
In conclusion,
Z
− K ` (x, f )dx
> (λ1 · · · λn )
`
`−n
·
Q
1
n
`
Z
= − K ` (x, g)dx
X
2
n
2n−2`
(λi1 · · · λi` )
16i1 <...<i` 6n
(7.12)
Q
Thus the linear map g is a minimizer. As in the previous section we shall
establish the uniqueness of the minimizer by examining all the steps in the
above computation. First, (7.11) holds as equality if and only if the matrix
D`×` f is diagonal. We now need an algebraic lemma.
LEMMA 7.1. Let A ∈ Rn×n be a nonsingular matrix whose `-th exterior power
(n`)×(n`)
A`×` ∈ R
is diagonal. Then A is diagonal.
15
Proof. Set A = Aij and note that det A`×` = (det A)` 6= 0. Our goal is to
show that Aα
β = 0 whenever α 6= β. We choose a multi-index I = {i1 , ..., i` },
1 6 i1 < ... < i` 6 n containing α but not β. We shall actually prove that the
whole vector
b = Aβi1 , Aβi2 , ..., Aβi` ∈ R`
in which Aβα is one of the coordinates, is equal to zero. To this end consider the
` × `-submatrix of A


Aii11 Aii12 · · · Aii1`
 .
..
.. 
..

.
M =
.
.
. 
 .
i`
i`
i`
Ai1 Ai2 · · · Ai`
This is a nonsingular matrix because det M is an entry of a nonsingular diagonal
matrix A`×` . We may solve the equation M a = b for a = (a1 , a2 , ..., a` ) ∈ R` .
By Cramer’s formula the coordinates of a are given by
ak =
det Mk
,
det M
k = 1, 2, ..., `
(7.13)
where Mk is
from M by replacing its k-th column by the
a matrix obtained
vector b = Aβi1 , Aβi2 , ..., Aβi` . Since β does not belong to the set I, det Mk is
an out-diagonal entry of A`×` and, therefore, is equal to zero. We conclude that
each ak = 0, hence b = M −1 a = 0, completing the proof of Lemma 7.1.
Using this lemma we infer that Df is also diagonal, the case det Df = 0
being trivial. Obviously, the differential matrix can be diagonal only when each
of the coordinate functions depend on its own variable, f 1 = f 1 (x1 ), ..., f n =
f n (xn ). Furthermore, (7.9) becomes equality only when the minors
i ` ∂(f i1 , ..., f i` ) ∂f i1 · · · ∂f =
∂xi ∂(xi , ..., xi ) ∂xi 1
1
`
`
remain, at almost every point, in the same proportion as the products λi1 · · · λi` .
This is because Schwarz inequality has been used in (7.9). In other words, there
is a measurable function λ = λ(x) such that
i i ∂f 1 ∂f ` ∂xi · · · ∂xi = λi1 · · · λi` λ(x)
1
`
The left hand side depends only on the variables xi1 , ..., xi` , forcing λ to be a
constant. On the other hand each coordinate function f i = f i (xi ), defined for
0 6 xi 6 ai , must be monotone because f (x1 , x2 , ..., xn ) = (f 1 (x1 ), ..., f n (xn )) is
16
a homeomorphism. We then see that the derivative of each coordinate function
is constant. This leaves the only possibility that
f (x) = (λ1 x1 , ..., λn xn )
as desired.
8
Linear Boundary Values
We then turn to Problem 1. from the introduction, that is, minimizing of the
integral (2.5) under fixed boundary values fo . We show first that in the case
when boundary data is linear the problem has a unique minimizer in the full
1,1
class Wloc
(Ω, Rn ).
THEOREM 8.1. Let Ω ⊂ Rn be a bounded domain with (n − 1)-rectifiable
boundary and let Ψ : [0, ∞) → [0, ∞) be convex with Ψ0 (1) ≥ 1.
Given any homeomorphism f : Ω → Ω0 of finite distortion, which coincides
on ∂Ω with an orientation preserving affine map fo : Rn → Rn , we have
Z
Z
K Ψ (x, f ) dx ≥
K Ψ (x, fo ) dx
(8.2)
Ω
Ω
Equality occurs if and only if f = fo on Ω.
Proof. We begin with the subgradient inequality
Ψ(t) − Ψ(t0 ) ≥ Ψ0 (t0 )(t − t0 )
valid for every t, t0 ∈ [1, ∞). Here Ψ0 (t0 ) ≥ Ψ0 (1) > 0 is any subgradient of Ψ
at t0 .
We use the subgradient inequality at the point
t0 = K (Dfo ) =
|| Dfo || n
> 1,
det Dfo
where the inequality holds since fo is nonsingular with det Dfo > 0.
We claim that for almost every x ∈ Ω we have
K Ψ Df (x) − K Ψ (Dfo ) ≥
h Γ, Df (x) − Dfo i +
where
Γ = Γ(Dfo ) = Ψ0 (t0 )
(8.3)
γ det Dfo − det Df (x)
|| Dfo || n−2
Dfo ∈ R n×n
+
det Dfo
17
and
γ = γ(Dfo ) = Ψ0 (t0 )
|| Dfo || n
∈ R n+
(det Dfo )2
1,1
In fact, as f ∈ Wloc
(Ω) it has partial derivatives at almost every x ∈ Ω,
and at such points x we have two possibilities. Since f is a mapping of finite
distortion, either || Df (x) || = 0 or det Df (x) > 0.
In the first case, by definition, K Ψ (Df (x)) = 1 so that
K Ψ (Df (x)) − K Ψ (Dfo ) = Ψ(1) − Ψ K (Dfo )
(8.4)
0
0
≥ Ψ (t0 ) 1 − K (Dfo ) > − Ψ (t0 )K (Dfo )
since Ψ0 (t0 ) > 0. As h Γ, Dfo i = nΨ0 (t0 )K (Dfo ) and γ det Dfo = Ψ0 (t0 )K (Dfo ),
the lower bound (8.3) follows whenever || Df (x) || = 0 or det Df (x) = 0.
In the second case , if x ∈ Ω is such a point that det Df (x) > 0, we have
from the subgradient inequality:
K Ψ Df (x) − K Ψ (Dfo ) ≥ Ψ0 (t0 ) K Df (x) − K (Dfo )
(8.5)
We then need the inequality (14.6) from the appendix which implies
|| Dfo || n
|| Df (x) || n
−
(8.6)
K Df (x) − K (Dfo ) =
det Df (x)
det Dfo
|| Dfo || n || Dfo || n−1 || Df (x) || − || Dfo || −
det
Df
(x)
−
det
Df
≥n
o
det Dfo
(det Dfo )2
Furthermore, we estimate by using Schwarz inequality
E
1 D Dfo
|| Df (x) || − || Dfo || ≥
, Df (x) − Dfo
n || Dfo ||
where we recall that hA, Bi = T rhAt Bi and || A || 2 =
holds at x if and only if
1
t
n T rhA Ai.
(8.7)
Equality here
Df (x) = λ(x)Dfo , where the scalar function λ(x) ≥ 0
(8.8)
Now combining the estimates (8.5), (8.6) and (8.7) yields the desired bound
and shows that (8.3) holds at almost every point x ∈ Ω.
For the next step we wish to integrate the terms of (8.3) over the domain Ω.
1,1
However, since f is only assumed to be in Wloc
(Ω, Rn ) we restrict ourselves to
an increasing sequence of compact subdomains Ωj b Ω, j = 1, 2, . . . , such that
∪Ωj = Ω and supj |∂Ωj | < ∞. This is possible since ∂Ω is (n − 1)-rectifiable.
Concerning the first term in the right hand side of (8.3) we observe, using
Stokes’ formula, that
Z
Z
[Df (x) − Dfo ] dx ≤ C
|f − fo | → 0
(8.9)
Ωj
∂Ωj
18
as j → ∞. Furthermore, in (8.3) the matrix function Df (x) − Dfo appears in
an inner product with a constant matrix. We then infer that upon integration
this first term converges to zero as j → ∞.
As comes for the second term in the right hand side of (8.3) we argue by
using the inequality
Z
Z
det Df (x) dx ≤ |f (Ωj )| ≤ |f (Ω)| = |fo (Ω)| =
det Dfo dx
Ωj
Ω
Note that it is at this point that we have exploited the assumption that f is a
1,1
homeomorphism in Wloc
(Ω, Rn ). In conclusion, the limit of the integral of this
second term is nonnegative. This shows that
Z
Z
K Ψ (x, f ) dx ≥
K Ψ (x, fo ) dx
(8.10)
Ω
Ω
In order to have equality in (8.10) we must have equality in (8.4) and in (8.7),
at almost all points x ∈ Ω. This forces first det Df (x) > 0 and then (8.8) to
hold almost everywhere. Equation (8.8), on the other hand, reads also as
Dg(x) = λ(x)I,
where
g = fo−1 ◦ f
This implies that λ = λo is a constant and that g(x) = λ0 I. Thus f (x) =
λ0 fo (x) + const. Finally, since f = fo on ∂Ω we conclude that f = fo on Ω, as
claimed.
9
An Identity
Variations of the identity we are about to formulate are fairly well known in
geometric function theory, see [2], [20]. However what we need is not explicitly stated there and for the convenience of the reader, and because it is quite
important in what follows, we present a complete proof of it. Throughout this
section Ω and Ω0 will be bounded domains in Rn , n > 2.
1,n
THEOREM 9.1. Let f ∈ Wloc
(Ω, Ω0 ) be a homeomorphism of finite distortion
with
Z
KI (x, f ) dx < ∞
(9.2)
Ω
Then the inverse map h : Ω0 → Ω belongs to W 1,n (Ω0 , Ω) and
Z
Z
n
|Dh(y)| dy =
KI (x, f ) dx
Ω0
Ω
19
(9.3)
Proof. Fix a test mapping ϕ ∈ C0∞ (Ω0 , Rn ). For k = 1, 2, ..., n we consider the
1,n
Sobolev mappings Fk ∈ Wloc
(Ω, Rn )
Fk (x) = f 1 , ..., f k−1 , ω , f k+1 , ..., f n ,
ω(x) =
n
X
xi ϕi f (x)
(9.4)
i=1
As the k-th coordinate function has compact support we see that
Z
J(x, Fk ) dx = 0
(9.5)
Ω
A lengthy, though standard, computation shows that the vector field
V (x) = (J(x, F1 ), ..., J(x, Fn ))
can be written as:
]
D f (x) ϕ(f (x)) + J(x, f ) (Dt ϕ)(f (x)) x
R
Since Ω V (x) dx = 0 ∈ Rn , we obtain
Z
Z
t
]
(D ϕ)(f (x)) x J(x, f ) dx = −
D f (x) ϕ(f (x)) dx
Ω
Ω
At this point we appeal to an old result of Reshetnyak [28, Corollary 1, p.
182] concerning change of variables via a homeomorphism of the Sobolev class
1,n
Wloc
(Ω, Rn ). We make the substitution y = f (x) in the integral of the left hand
side to obtain
Z
Z
t
]
(D ϕ)(y) h(y) dy = −
D f (x) ϕ(f (x)) dx
(9.6)
Ω0
Ω
1,n
So far we have only exploited the fact that f ∈ Wloc
(Ω, Rn ) is a homeomorphism. Before using the identity (9.6) let us remind that f , being a mapping of
finite distortion, satisfies the condition:
D] f (x) = 0
Hence
Z
t
0 (D ϕ)(y) h(y) dy Ω
if and only if
Z
J(x, f ) = 0
]
D f (x) |ϕ(f (x))| dx
6
ZΩ p
n−1
n
KI (x, f ) J(x, f ) n |ϕ(f (x))| dx
=
Ω
Z
6
n1 Z
n−1
n
n
KI (x, f ) dx
|ϕ(f )| n−1 J(·, f )
Ω
=
Ω
1
n
|| KI (x, f ) || L 1 (Ω) || ϕ ||
20
n
L n−1 (Ω0 )
(9.7)
where we have made the substitution y = f (x) again. This estimate tells us
exactly that h ∈ W 1,n (Ω0 , Ω) and its differential satisfies
Z
Z
n
|Dh(y)| dy 6
KI (x, f ) dx
Ω0
Ω
Now, knowing that h ∈ W 1,n (Ω0 , Ω) it is legitimate to use change of variables
to obtain the identity
Z
Z
Z
n
n
|Dh(y)| dy =
|Dh(f (x))| J(x, f ) dx =
KI (x, f ) dx
(9.8)
Ω0
Ω
Ω
completing the proof of Theorem 9.1.
n
Remark 9.1. We still owe the reader an explanation why |Dh(f (x))| J(x, f ) =
KI (x, f ) almost everywhere. To see this we first observe that both f and h are
differentiable almost everywhere. This elegant result belongs to Väisälä [31].
The chain rule Dh(f (x)) Df (x) = I shows that J(x, f ) > 0 and J(y, h) > 0
almost everywhere since both f and h preserve sets of zero measure. Now, the
formula is a direct consequence of the definition of the inner distortion,
def
KI (x, f ) ==
10
]
D f (x)n
J(x, f )n−1
n
n
= D−1 f (x) J(x, f ) = |Dh(f (x))| J(x, f )
(9.9)
The L 1 -Theory with Dirichlet Data
The minimisation problem with general boundary values is of course considerably deeper than the problem for linear data.
We first study homeomorphisms of finite distortion f : Ω → Ω0 between
bounded domains Ω , Ω0 ⊂ Rn , such that the inner distortion function of f is
integrable.
1,n
In view of Theorem 9.1, for f ∈ Wloc
(Ω, Ω0 ) the minimization of
Z
KI (x, f ) dx
(10.1)
Ω
is closely related to a well-known variational problem for the inverse mapping
h = f −1 : Ω → Ω0 .
The Dirichlet Problem
Given a mapping ho ∈ W 1,n (Ω0 , Rn ) minimize
the energy
Z
n
|Dh(y)| dy < ∞
Ω0
21
over the class of all mappings h ∈ ho + W01,n (Ω0 , Rn ).
While the existence of the minimizer is guaranteed by the principles of convex
analysis the uniqueness is a delicate issue due to the lack of strict convexity of
the operator norm. One way out of this is to replace the operator norm by the
mean Hilbert-Schmidt norm. Therefore, we consider the following variational
integral
Z
|| Dh(y) || n dy
E[h] =
Ω0
The advantage of using this functional is that the minimization problem

R
n

 min Ω0 || Dh(y) || dy


h ∈ ho + W01,n (Ω0 , Rn )
admits a unique solution for all Dirichlet data ho ∈ W 1,n (Ω0 , Rn ). Moreover,
the minimizer is the unique solution to the n-harmonic equation
h ∈ ho + W01,n (Ω0 , Rn )
Div || Dh || n−2 Dh = 0,
This equation simply means that
Z
|| Dh || n−2 hDh | Dϕi = 0
(10.2)
Ω0
for every test mapping ϕ ∈ C0∞ (Ω0 , Rn ).
To formulate an analogue of Theorem 9.1 we make use of the mean HilbertSchmidt variant of the inner distortion
n
|| D] f (x) ||
= Kn−1 Df (x) ,
KI (x, f ) =
n−1
J(x, f )
(10.3)
c.f. Section 3.
1,n
THEOREM 10.4. Let f ∈ Wloc
(Ω, Ω0 ) be a homeomorphism of finite distortion with
Z
KI (x, f ) dx < ∞
Ω
Then the inverse map h : Ω → Ω belongs to W 1,n (Ω0 , Ω) and
Z
Z
n
|| Dh(y) || dy =
KI (x, f ) dx
0
Ω0
Ω
22
(10.5)
Proof. The chain of inequalities at (9.7) together with Theorem 9.1 guarantees
that h lies in the Sobolev class W 1,n (Ω0 , Ω). The only thing we have to worry
here is the identity (10.5). This follows by replacing (9.9) with the following
computation
n
n
KI (x, f ) = || D−1 f (x) || J(x, f ) = || Dh(f (x)) || J(x, f )
which proves the identity (10.5).
To apply Theorem 10.4 to our variational problem we need to recall some
of the considerable literature on the existence and topological properties of
harmonic maps between planar domains. We refer the reader to the recent book
of Duren [6]. In particular, according to the fundamental theorem of Radó [27],
Kneser [18] and Choquet [3], if Ω ⊂ R2 is a bounded convex domain, then each
homeomorphism ho : ∂Ω0 → ∂Ω has a unique continuous extension h : Ω0 → Ω
which is univalent and maps Ω0 harmonically to Ω. Then, by a theorem of Lewy
[23] the univalent harmonic map has a non-vanishing Jacobian. Its inverse is
therefore a real analytic diffeomorphism.
1,2
We now consider the class F = F(Ω, Ω0 ) of Wloc
(Ω, R2 )-regular homeomor0
phisms f : Ω → Ω of finite distortion for which K (z, f ) is integrable in Ω.
THEOREM 10.6. Let Ω ⊂ R2 be a convex domain and fo ∈ F(Ω, Ω0 ). Then
the minimization problem
ZZ
2
min
K (z, f ) |dz| ,
f = f0 on ∂Ω
f ∈F
Ω
has a unique solution. This extremal map is a C ∞ -diffeomorphism whose inverse is harmonic in Ω0 .
Proof. Let H = H(Ω0 , Ω) denote the class of inverse mappings h = f −1 : Ω0 → Ω
where f ∈ F(Ω, Ω0 ). Thus, in particular, h0 = f0−1 ∈ H(Ω0 , Ω). In light of
Theorem 11.1 we are reduced, equivalently, to the Dirichlet problem
ZZ
|| Dh || 2 ,
h = h0 on ∂Ω
min
f ∈H
Ω
The existence and uniqueness of the minimizer in the Sobolev class ho +W01,2 (Ω0 , R2 )
is well known. The only point to make is that such a minimizer lies in H(Ω0 , Ω)
by Radó-Kneser-Choquet and Lewy theorems.
COROLLARY 10.1. The extremal map f : Ω → Ω0 solves the quasilinear
Beltrami equation
∂f
∂f
= µ f (x)
(10.7)
∂z
∂z
23
where µ : Ω0 → B is an anti–analytic function valued in the unit disk.
Proof. As f is a diffeomorphism with positive Jacobian,
2
2
J(z, f ) = |fz | − |fz | > 0
and thus |fz /fz | < 1. Let us define µ : Ω0 → B by the equation (10.7). We need
only show that µ is anti–analytic. For this reason we consider the inverse map
h(ξ) = f −1 (ξ) to write
hξ
µ(ξ) = −
hξ
Hence
∂µ
=−
∂ξ
hξ
hξ
=
−hξξ hξ + hξ hξξ
=0
2
hξ
(10.8)
as claimed.
Now the following is immediate.
COROLLARY 10.2 (maximum principle). Let K (z, f ) denote the inner distortion of the extremal map. Then
max K (x, f ) 6 max K (x, f )
∂U
U
for every U ⊂ Ω.
11
The Traces of Mappings with Integrable Distortion
Theorem 10.6 demands, for the sake of completeness, that we give necessary
and sufficient conditions for a homeomorphism f0 : ∂Ω → ∂Ω0 to admit an
1,2
extension f : Ω → Ω0 which lies in F(Ω, Ω0 ); the class of Wloc
(Ω, R2 )-regular
homeomorphisms f : Ω → Ω0 of finite distortion for which K (z, f ) is integrable
in Ω. Recall that for the L ∞ minimisation problem the requisite notion is that
of quasisymmetry, though in this case there is a surprise, see Theorem 11.15.
THEOREM 11.1. Suppose Γ = ∂Ω and Γ0 = ∂Ω0 are C 1 -regular Jordan
curves. A necessary and sufficient condition that a homeomorphism fo : Γ → Γ0
should extend to an f : Ω → Ω0 , f ∈ F(Ω, Ω0 ), is that the double integral
ZZ
1
def
LΓ (fo ) == −
log |fo (z) − fo (w)| dz dw
(11.2)
π
Γ×Γ
24
converges absolutely. That is,
ZZ
log |fo (z) − fo (w)| |dz| |dw| < ∞
Γ×Γ
Among all such extensions of fo there is one which maps Ω diffeomorphically
onto Ω0 .
Using the Riemann mapping theorem we reduce to the case when both Ω and
Ω are disks in C. Because of C 1 -regularity of Γ = ∂Ω and Γ0 = ∂Ω0 neither the
hypotheses nor the assertion of this theorem will be affected by such a change
of variables. Therefore, we shall have established Theorem 11.1 once we prove
the following more precise special case of it.
0
THEOREM 11.3. Theorem 11.1 holds when Ω0 is a disk and Ω has the additional property of being convex. In this case, for every extension of f0 to an
f ∈ F(Ω, Ω0 ), we have
ZZ
ZZ
1
−
log |fo (z) − fo (w)| dz dw 6
K (z, f ) |dz|2
(11.4)
π
Γ×Γ
Ω
Equality occurs only when f = h−1 , where h : Ω0 → Ω is the unique harmonic
extension of f0−1 : Γ0 → Γ. This extremal extension turns out to be a diffeomorphism.
Remark. Our results are reminiscent of ideas of Douglas [5], characterizing
boundary functions whose harmonic extension have finite Dirichlet energy. Douglas’ condition for ho = fo−1 reads as
ZZ
h (ξ) − h (ζ) 2
o
o
|dξ||dζ| < ∞
ξ−ζ
Γ0 ×Γ0
Proof. Observe that (11.4) is invariant under translation and rescaling of Ω0 ,
so we may assume that Ω0 is the unit disk, Ω0 = D ⊂ C. Consider the inverse
homeomorphism ho : ∂D → ∂Ω. As shown in the previous section, fo admits an
extension to f ∈ F(Ω, Ω0 ) if and only if the Poisson extension h : D → Ω has
finite energy. Moreover, in this case the inverse map f = h−1 provides us with
one of the desired extensions of fo to f ∈ F(Ω, Ω0 ).
We begin with integral representation formulas of the complex derivatives
of the harmonic map h in terms of fo : Γ → ∂D;
Z
∂h
1
dz
=
(11.5)
∂a
2πi Γ fo (z) − a
25
∂h
1
=
∂a
2πi
Z
dz
Γ
(11.6)
a − fo (z)
for every a ∈ D. We give the proof only for (11.5), the second identity follows
in much the same way. Consider an exhaustion of Ω by smooth domains Ω1 b
Ω2 b ... ⊂ Ω, such that
a ∈ f (Ω1 ) b f (Ω2 ) b ... ⊂ D
(11.7)
We have
Z
Γ
dz
fo (z) − a
Z
Z
dz
dh(ξ)
= lim
n→∞ ∂Ω f (z) − a
n→∞ C ξ − a
n
n
Z
Z
hξ dξ
hξ dξ
= lim
+ lim
n→∞ C ξ − a
n→∞ C ξ − a
n
n
=
lim
(11.8)
where the curves Cn = f (∂Ωn ) approach ∂D, uniformly as n → ∞. We do
not claim here that the lengths of Cn stay bounded. Since hξ is an analytic
function in D the first integral is independent of the curve Cn and equals 2πi ∂h
∂a ,
by Cauchy’s formula. Concerning the second integral, it would be equal to zero
if Cn was a circle. Indeed, we would have
Z
Z
hξ dξ
ξ hξ dξ
=
= 0,
2
ξ
−
a
|ξ|=ρ
|ξ|=ρ ρ − aξ
by Cauchy’s theorem for anti–analytic functions. The above arguments suggest
imbedding every f (Ωn ) in a disk, say f (Ωn ) ⊂ Dn b D. We can now express
the curve integral by the area integral by using Stoke’s formula
"
#
Z
Z
Z
ZZ
hξ dξ
hξ dξ
hξ dξ
hξ dξ
=
−
=
d
ξ−a
Cn ξ − a
Cn ξ − a
∂Dn ξ − a
Dn \f (Ωn )
ZZ
hξ dξ ∧ dξ
= −
(11.9)
2
Dn \f (Ωn ) (ξ − a)
Hölder’s inequality yields
Z
hξ dξ 6C
Cn ξ − a ! 12
ZZ
2
|Dh|
→0
as n → ∞
D\f (Ωn )
completing the proof of (11.5).
Having disposed of formulas (11.5) and (11.6) we can since |Dh| ∈ L 2 (D)
compute the Dirichlet integral of h over an arbitrary disk Dr = {ξ : |ξ| < r},
with 0 < r < 1,
2
2
ZZ
ZZ
ZZ
∂h ∂h + 2
|Dh|2 = 2
∂a Dr
|a|6r
|a|6r ∂a
26
The computation of the first integral goes as follows.
2
ZZ
∂h dz dw
= 1
∂a 4π 2
Γ×Γ [fo (z) − a] [fo (w) − a]
Hence, by Fubini’s theorem
2
ZZ
ZZ
∂h = 1
4π 2
Γ×Γ
|a|6r ∂a
|da|2
ZZ
|a|6r
!
[fo (z) − a] [fo (w) − a]
dz dw
A tedious (but elementary) computation, developing the integrand as a power
series, yields an explicit expression for the area integral
ZZ
|da|2
= −π log(1 − r2 ξζ)
|a|6r (ξ − a)(ζ − a)
where |ξ| = |ζ| = 1. We substitute this value into the latter formula to obtain
2
ZZ
ZZ
∂h 2
= − 1
log
1
−
r
f
(z)f
2
(w)
dz dw
o
o
2π
|a|6r ∂a
Γ×Γ
Similar arguments to those above show that
2
ZZ
ZZ
∂h 2
= − 1
2
log
1
−
r
f
(z)f
(w)
dz dw
o
o
2π
|a|6r ∂a
Γ×Γ
These two equations add up to
ZZ
ZZ
2
1
|Dh|2 = −
log 1 − r2 fo (z)fo (w) dz dw
2π
Dr
Γ×Γ
which, in view of the identity (10.5) can be stated as
ZZ
ZZ
ZZ
1
2
2
|| Dh || = 2
K (z, f ) |dz|2
−
log r − fo (z)fo (w) dz dw =
π
Dr
f −1 (Dr )
Γ×Γ
It is now clear that the integral in the left hand side increases with r. Letting
r go to 1 we see that the limit exists if and only if K (z, f ) is integrable. The
only point remaining concerns the equivalence of the following two properties
of the boundary map fo : Γ → ∂D; the existence of this limit and the absolute
convergence of the integral
ZZ
log |fo (z) − fo (w)| |dz| |dw|
(11.10)
Γ×Γ
It is clear that, regardless of the regularity of Γ, the absolute convergence of the
integral at (11.10) implies
ZZ
ZZ
lim
log |fo (z) − fo (w)| dz dw
log r2 − fo (z)fo (w) dz dw =
r%1
Γ×Γ
Γ×Γ
(11.11)
27
by Lebesgue Dominated Convergence Theorem. For the converse, we need C 1 regularity of Γ. Suppose that the above limit exists. Since the integrand is
invariant under the interchange of variable z and w, we may replace the complex
area element dz dw by the real one Re (dz dw). However, on C 1 -regular curves
this latter element is comparable with |dz| |dw| when z is sufficiently closed to
w, say
1
(11.12)
|dz| |dw| 6 Re (dz dw) 6 2 |dz| |dw|
2
provided |z − w| 6 . The interested reader may wish to observe that this
estimate fails for the cube Ω = [0, 1] × [0, 1]. Indeed, near the corner (0, 0) we
may take z = x and w = iy to obtain
Re (dz dw) = −Re (i dx dy) = 0
(11.13)
On the other hand (11.12), with 2 replaced by some positive number, remains
valid e.g. for polygons with obtuse angles. Now, for C 1 -regular curves, in view
of (11.12), the existence of the limit and equality at (11.11) becomes equivalent
to the absolute convergence of the integral at (11.2), as desired.
Example 11.1. Consider a homeomorphism of the unit circle onto itself fo :
∂D → ∂D given by
fo eiθ = e i Φ(θ) for − π < θ 6 π
(11.14)
where
π2
Φ(θ) = π sgn θ e 1− θ2
The reader may wish to verify that the double integral
Z πZ 0
log eiΦ(α) − eiΦ(β) dα dβ
0
−π
diverges. As a corollary, we see that fo has no homeomorphic extension into
the unit disk with integrable distortion.
Note that in the proof of Theorem 11.3 we did not really have to use C 1 regularity of the convex domain Ω; we could have used the limit formula at
(11.11) instead of the integral at (11.4). As Ω is convex its boundary Γ = ∂Ω
is Lipschitz and, therefore, a rectifiable Jordan curve. We leave the details of
such extension of Theorem 11.3 to arbitrary convex domains to the reader.
Finally in this section, we wish to observe that given quasiconformal boundary data, even for the disk D, the minimiser of the L 1 –problem is seldom
quasiconformal.
28
THEOREM 11.15. Let fo : D → D be quasiconformal and Fo as in Theorem
10.6. Then the unique minimiser of the problem
ZZ
2
min
K (z, f ) |dz| ,
f = f0 on ∂D
(11.16)
f ∈F
D
is quasiconformal if and only if fo is bi–Lipschitz.
Proof. Set go = fo−1 . We know that the minimiser f exists and its inverse h is
the unique harmonic extension of go |∂D. If f is quasiconformal, then so too is
h. However a theorem of Pavlović [26] states that the Poisson extension of the
boundary values go of a quasiconformal mapping is quasiconformal if and only
if the map go is bi–Lipschitz. Thus fo is bi–Lipschitz.
Actually [26] points out the quasiconformality of the Poisson extension is
equivalent to the boundary values themselves being bi–Lipschitz or that the
Hilbert Transform of their derivative lies in L ∞ .
12
Exponentially Integrable Distortion
In this section we shall be concerned with the variational integrals (2.5) when
Ψ(t) = eλt−λ for some fixed parameter λ > 0. If Ω and Ω0 are bounded domains
we shall consider the family F of all homeomorphisms f : Ω → Ω0 of a finite
distortion such that
Z
eλ Ko (x,f ) dx < ∞
(12.1)
Ω
We fix fo ∈ F and denote by Fo the class of all f ∈ F which coincide with fo on
∂Ω. For convenience we assume, moreover, that fo extends to a neighborhood
of Ω as a homeomorphism of finite distortion satisfying (12.1).
THEOREM 12.2. The minimization problem
Z
min
eλ Ko (x,f ) dx = m
f ∈Fo
(12.3)
Ω
has a solution.
As an interesting first point to make we show that the inverse mapping of any
f ∈ Fo has better Sobolev regularity than f itself, namely f −1 ∈ W 1,n (Ω0 , Ω).
This is an improvement of Theorem 10.4, since we do not assume that f ∈
1,n
Wloc
(Ω, Ω0 ).
29
THEOREM 12.4. Let f : Ω → Ω0 be a homeomorphism of finite distortion
with
Z
eλ Ko (x,f ) dx < ∞
(12.5)
Ω
Then f lies in the Orlicz-Sobolev space W 1,P (Ω, Ω0 ), P (t) =
Z
Ω
1 0
|| Df || n
6
|Ω | +
n
log(e + || Df || )
λ
Z
tn
log(e+t) ,
and
eλKo (x,f ) dx
(12.6)
Ω
Moreover, the inverse map h : Ω0 → Ω belongs to W 1,n (Ω0 , Ω) and
Z
Z
|| Dh(y) || n dy =
KI (x, f ) dx
Ω0
(12.7)
Ω
Proof. We follow the arguments of Theorems 10.4 and 9.1. In the proof of
Theorem 9.1 we defined auxiliary mappings
Fk (x) = f 1 , ..., f k−1 , ω , f k+1 , ..., f n ,
ω(x) =
n
X
xi ϕi f (x)
i=1
1,n
Recall that the assumption f ∈ Wloc
(Ω, Rn ) was used to ensure that Fk ∈
1,n
Wloc (Ω, Rn ). This regularity of Fk was important so as to have the identity
Z
J(x, Fk ) dx = 0
(12.8)
Ω
This time the assumptions of Theorem 12.4 only guarantee that f ∈ W 1,P (Ω, Ω0 ).
1,1
Indeed, for homeomorphisms of the Sobolev class Wloc
(Ω, Ω0 ) we have
Z
J(x, f ) dx 6 |f (Ω)| = |Ω0 |
(12.9)
Ω
The distortion inequality
|| Df || n 6 Ko (x, f ) J(x, f )
yields
1
|| Df || n
6
J(x, f ) + eλ Ko (x,f )
n
log (e + || Df || )
λ
where we have employed the elementary inequality
ab 6 a log(a + 1) + eb − 1
a, b > 0
(12.10)
eλ Ko (x,f ) dx < ∞
(12.11)
Hence
Z
Ω
|| Df || n
1
6 |Ω0 | +
log (e + || Df || n )
λ
30
Z
Ω
1,P
1,n−1
1,P
Thus f ∈ Wloc
(Ω, Ω0 ) ⊂ Wloc
(Ω, Ω0 ). Hence also Fk ∈ Wloc
(Ω, Rn ).
Although the Jacobian determinant of Fk changes sign, the Jacobian is still
locally integrable. Indeed,
J(x, Fk ) dx = df 1 ∧ ... ∧ df k−1 ∧ dω ∧ df k+1 ∧ ... ∧ df n
where
n
X
(12.12)
n
n
X
X
∂ϕi j
dω =
df
ϕ f (x) dxi +
xi
∂yj
i=1
i=1
j=1
i
Hence,
J(x, Fk ) dx
= df 1 ∧ ... ∧ df k−1 ∧
n
X
ϕi f (x) dxi ∧ df k+1 ∧ ... ∧ df n
i=1
"
+
n
X
i=1
i
xi
∂ϕ f (x)
∂yk
#
df 1 ∧ ... ∧ df n
(12.13)
1,n−1
which is in L 1 (Ω) because f ∈ Wloc
(Ω, Ω0 ) and J(x, f ) ∈ L 1 (Ω). Now
Lemma 7.8.1 in [15] comes to the rescue as the identity (12.8) still holds.
For (12.7) we appeal to the computation in the proof of Theorem 9.1. The
only point is to justify change of variables, for which we need condition (N ).
This condition has been established in [17] for mappings of exponentially integrable distortion, completing the proof of Theorem 12.4.
Proof of Theorem 12.2. Let {fj } be a minimizing sequence. That is,
• fj : Ω → Ω0 are homeomorphisms which coincide with fo : Ω → Ω0 on ∂Ω
1,1
• fj ∈ Wloc
(Ω, Ω0 )
• || Dfj (x) || n 6 Ko (x, fj ) J(x, fj )
R
• lim Ω eλ Ko (x,fj ) dx = m
a.e.
j→∞
As in (12.11) this yields uniform estimates of the differentials in the Orlicz space
L P (Ω),
Z
Z
|| Dfj || n
1 0
6 |Ω | +
eλ Ko (x,fj ) dx 6 C for j = 1, 2, ...
n
λ
Ω log (e + || Dfj || )
Ω
From these estimates we deduce that {fj } are equicontinuous on Ω. The simplest
way to see this is as follows. We extend each fj beyond Ω by setting

f
on Ω
j
f˜j =
fo
beyond Ω
31
The extension is possible since fj = fo on ∂Ω and fo is defined in a neighborhood
of Ω, as a homeomorphism of finite distortion satisfying (12.1). We can now use
the local estimates of [13, Theorem 1.4] to conclude with equicontinuity of {fj }
on Ω.
Ascolli’s theorem gives us a subsequence, again denoted by fj , such that
fj ⇒ f
uniformly on Ω
(12.14)
Hence
Dfj * Df
weakly in L P (Ω, Rn×n )
Consider then the inverse mappings hj : Ω0 → Ω. Theorem 12.4 yields
Z
Z
|| Dhj || n =
KI (x, fj ) dx 6 C,
k = 1, 2, ...
Ω0
Ω
As above, the homeomorphisms {hj } which coincide with the given ho = fo−1
on ∂Ω0 share a uniform modulus of continuity on Ω0 . Hence, we find that
hj ⇒ h
uniformly on Ω0
(12.15)
This together with (12.14) implies that f is a homeomorphism, with h as its
inverse. It remains to show that
Z
eλ Ko (x,f ) dx = m
(12.16)
Ω
To this end, we observe that the integrand is polyconvex, see Appendix 12.2.
Precisely, we have the following pointwise inequality
eλ Ko (X) > eλ Ko (A) + λeλ Ko (A) [Ko (X) − Ko (A)]
>
eλ Ko (A) + λeλ Ko (A) ·
−λeλ Ko (A)
E
n || A || n−1 D A
,X − A
det A
|| A ||
|| A || n
(det X − det A)
(det A)2
(12.17)
for matrices A, X ∈ Rn×n
+ . Given any > 0 we consider the set Ω b Ω on
which
1
J(x, f ) > and
|| Df (x) || 6
Moreover,
[
Ω = x ∈ Ω; J(x, f ) > 0
>0
32
For x ∈ Ω , we can write (see (14.8))
eλ Ko (x,fj ) > eλ Ko (x,f ) +
E
|| Df (x) || n−1 D Df (x)
, Dfj (x) − Df (x)
J(x, f )
|| Df (x) ||
n
|| Df (x) ||
[J(x, fj ) − J(x, f )]
(12.18)
J(x, f )2
+n λeλ Ko (x,f )
− λeλ Ko (x,f )
For notational simplicity we introduce measurable bounded coefficients A ∈
L ∞ (Ω , Rn×n ) and α ∈ L ∞ (Ω ) which enter the right hand side. Integrating
over Ω yields
Z
Z
Z D
E
λ Ko (x,fj )
λ Ko (x,f )
e
dx >
e
dx +
A(x), Dfj (x) − Df (x) dx
Ω
Ω
Ω
Z
−
α(x) [J(x, fj ) − J(x, f )] dx
(12.19)
Ω
The last two integrals converge to zero as j → ∞. By the definition of outer
distortion
Ko (x, f ) = 1 6 Ko (x, fj ) whenever J(x, f ) = 0
With this convention in mind we can write
Z
Z
eλ Ko (x,f ) dx +
eλ Ko (x,f ) dx
J(x,f )=0
Ω
"Z
λ Ko (x,fj )
e
6 lim inf
j→∞
dx +
Ω
Z
6 lim inf
j→∞
Z
#
e
λ Ko (x,fj )
dx
J(x,f )=0
eλ Ko (x,fj ) dx = m
(12.20)
Ω
Letting → 0, we conclude that
Z
eλ Ko (x,f ) dx 6 m
Ω
Remark 12.1. It is shown in [19] that in fact J(x, f ) > 0 almost everywhere, so
the addition of the integral over the set where J(x, f ) = 0 is redundant.
13
Variational Equations
Suggested by many problems in the Calculus of Variation we strongly believe
that if Ψ ∈ C ∞ [1, ∞) then the extremals are continuously differentiable, as in
the case Ψ(t) = t.
33
Conjecture 13.1. Suppose Ψ is C ∞ -smooth. Then every homeomorphism of
finite distortion on a domain Ω that minimizes the variational integral
Z
KΨ (x, f )dx
Ω
subject to given boundary values is a C 1,α -diffeomorphism in Ω.
Unfortunately, we do not even know whether the minimizers enjoy partial regularity as in the Quasiconvex Calculus of Variations [7].
Under the conjecture we have the following computation. We begin with a
variation of the complex Beltrami coefficient
fz
fz
def
µ(z, f ) ==
where we assume that f : Ω → Ω0 is an orientation preserving diffeomorphism.
Let η ∈ C0∞ (Ω) be a complex valued test function. For all sufficiently small
complex parameters λ we still have J(z, f + λη) > 0, and f + λη enjoys the
same boundary values as f . The complex differential of µ(z, f ), denoted by µ̇ =
µ̇(z, f ), is a C-linear operator on C0∞ (Ω). It acts on a test function η ∈ C0∞ (Ω)
by the rule
fz ηz − fz ηz
∂µ(z, f + λ η) (13.1)
=
µ̇[η] = µ̇(z, f )[η] =
2
∂λ
(fz )
λ=0
Now consider the function
2
2
κ = κ(z, f ) = |µ(z, f )| =
|fz |
2
|fz |
(13.2)
Its complex differential is computed by using the chain rule
κ̇ = κ̇(z, f ) = µ µ̇
(13.3)
More explicitly, for each η ∈ C0∞ (Ω)
κ̇[η] = κ̇(z, f )[η] = κ
ηz
ηz
−
fz
fz
(13.4)
Next recall the linear distortion function
K(z, f ) =
2
2
2
2
|fz | + |fz |
|fz | − |fz |
=
|| Df (z) || 2
1 + κ(z, f )
=
J(z, f )
1 − κ(z, f )
(13.5)
Again by the chain rule we find that
K̇ = K̇(z, f ) =
34
2 κ̇(z, f )
(1 − κ)2
(13.6)
that is
2κ
(1 − κ)2
K̇[η] = K̇(z, f )[η] =
ηz
ηz
−
fz
fz
(13.7)
More generally, for every convex Ψ : [1, ∞) → [1, ∞) we have the corresponding
distortion function
!
2
2
|fz | + |fz |
KΨ = KΨ (z, f ) = Ψ K(z, f ) = Ψ
(13.8)
2
2
|fz | − |fz |
whose complex differential equals
K̇Ψ [η]
=
=
13.1
ηz
1+κ
2κ
ηz
−
1 − κ (1 − κ)2 fz
fz
1+κ
fz ηz − κ fz ηz
2
Ψ0
2
(1 − κ)2
1−κ
|fz |
K̇Ψ (z, f )[η] = Ψ0
(13.9)
The Lagrange-Euler Equations
We want now to discuss the minimizers f of the general variational integrals
ZZ
2
KΨ (z, f ) |dz|
(13.10)
Ω
If f is also a C 1 (Ω)-diffeomorphism, then
Z
∂
2
K (z, f + λη) |dz| = 0 at λ = 0
∂λ Ω Ψ
(13.11)
for every test function η ∈ C0∞ (Ω). This gives
h
1
Ψ0
(1 − κ)2
1+κ
1−κ
fz i
2
|fz |
κ
=
Ψ0
fz (1 − κ)2
h
z
1+κ i
1−κ z
or equivalently
∂
∂z
A(κ)
fz
=
∂
∂z
A(κ)
fz
(13.12)
where
A(κ) = Ψ
0
1+κ
1−κ
2κ
= Ψ0
(1 − κ)2
2
2
2
2
|fz | + |fz |
|fz | − |fz |
!
2
2
2 |fz | |fz |
2
2
|fz | − |fz |
2
Let us now introduce the so-called conjugate stationary solution g = g(z)
in order to express (13.12) as a first order system. From now on we need to
assume that Ω is a simply connected domain.
35
A(κ)
1
Note that A(κ)
fz and fz are continuous in Ω. Therefore there is g ∈ C (Ω),
unique up to a constant, such that
∂g
A(κ)
=
∂z
fz
∂g
A(κ)
=
∂z
fz
Notice that g need not be a homeomorphism even supposing that f were. However we do have the following.
LEMMA 13.1. The minimizer f and its conjugate stationary function g have
the same complex Beltrami coefficient,
gz = µ(z)gz with gz fz = gz fz = A(κ) > 0
for almost every z ∈ Ω.
To every extremal mapping f : Ω → Ω0 there corresponds a holomorphic
function F : Ω0 → C defined by
F (ξ) = g f −1 (ξ)
where g denotes the conjugate function to f . Indeed, since f is a homeomorphism we can always express g(z) = F (f (z)), for some mapping F : Ω0 → C.
Now, F is holomorphic because f and g have the same Beltrami coefficient. The
chain rule gives the following relations
gz = F 0 f (z) fz
gz = F 0 f (z) fz
Hence
F0 =
gz
A(κ)
gz
=
=
fz
fz
fz fz
We recall the derivatives of the inverse map h(ξ) = f −1 (ξ),
hξ (ξ) = −fz (z) J(ξ, h)
hξ (ξ) = fz (z) J(ξ, h)
and compute
k
hξ
|F 0 (ξ)|
fz fz
fz
=k
=
=−
F 0 (ξ)
|fz | |fz |
hξ
fz
where
k = k(z, f ) = |µ(z, f )| = |µ(ξ, h)| = k(ξ, h)
(13.13)
COROLLARY 13.1. Let f : Ω → Ω0 be a C 1 -diffeomorphism which is a
minimizer for the variational integral (13.10) subject to given boundary values.
Then its inverse h : Ω0 → Ω satisfies the Beltrami equation
ϕ(ξ) ∂h
∂h
= k
|ϕ(ξ)| ∂ξ
∂ξ
36
(13.14)
where ϕ(ξ) = −F 0 (ξ) is a holomorphic function in Ω0 and k = k(ξ, h) is as in
(13.13).
It is appropriate at this stage to recall that mappings satisfying (13.14) with
a constant 0 6 k < 1 are referred to as Teichmüller mappings [22], p. 231. For
this reason we shall call h the pseudo-Teichmüller mapping. What is so special
about the Beltrami coefficient of h is that its argument is a harmonic function.
Note that in the case when Ψ(t) = t and h is harmonic, as in Theorem 10.4,
we find that
φ(ξ) = hξ hξ
where both hξ and hξ are analytic functions.
The conjugate stationary solutions reduced (13.12) to a first order equation
for the inverse. As an alternative development we note that nondivergence forms
of equation (13.12) are also interesting. These are actually systems of second
order PDEs for the real and imaginary part of f . Since the integrand at (13.10)
is polyconvex such systems must satisfy the Legendre-Hadamard ellipticity condition, see [4], [10], [25] for more details. Let us consider the simplest case of
the distortion function
2
K(z, f ) =
2
|| Df (z) || 2
|fz | + |fz |
=
2
2
J(z, f )
|fz | − |fz |
(13.15)
In this case Ψ(t) = t and hence
2
A(κ) = 2
2 |fz | |fz |
2
2
|fz | − |fz |
The Euler-Lagrange equation reads as:
!
2
∂
|fz | fz
∂
=
∂z J(z, f )2
∂z
(13.16)
2
2
|fz | fz
J(z, f )2
!
(13.17)
Lengthly computation reduces (13.17) to an elegant nondivergence equation
fz z = α fz z + β fz z
(13.18)
where
µ(z, f )
fz fz
2 =
2
2 =β
1 + |µ(z, f )|
|fz | + |fz |
see (14.18). This quasilinear system is elliptic, meaning that
α=
|α(z)| + |β(z)| =
2k
<1
1 + k2
It is somewhat peculiar that (13.18) turns out to be C-linear with respect to the
second order derivatives. We refer to the appendix for a detailed computation.
37
13.2
Equations for the Inverse Map
Suppose that a C 1 -diffeomorphism f : Ω → Ω0 is a minimizer of the variational
integral
!
ZZ
ZZ
2
2
|fz | + |fz |
2
2
KΨ (z, f ) |dz| =
Ψ
|dz|
(13.19)
2
2
|fz | − |fz |
Ω
Ω
This just amounts to saying that the inverse map h : Ω0 → Ω minimizes the
integral:

2 
2
ZZ
ZZ 2 |h
|
+
hξ 
ξ

2
2
2
J(ξ, h) KΨ (ξ, h) |dξ| =
Ψ
|hξ | − hξ 2  |dξ|
0
0
2
Ω
Ω
|hξ | − hξ (13.20)
This time the variation of the integrand reads as
!
η
η
2
κ
J
ξ
ξ
J˙ Ψ + J Ψ0 K̇ [η] = hξ ηξ − hξ ηξ Ψ +
−
Ψ0
(13.21)
(1 − κ)2 hξ
hξ
whose complex conjugate gives the following divergence form of the LagrangeEuler equation:
i i ∂ h
∂ h
0
0
Ψ − (K − 1)Ψ hξ = 0
(K + 1)Ψ − Ψ hξ +
(13.22)
∂ξ
∂ξ
First we view the square brackets as given measurable coefficients
∂ ∂ N hξ = 0
M hξ +
∂ξ
∂ξ
(13.23)
where
M (ξ) = (K + 1)Ψ0 (K) − Ψ(K) and N (ξ) = Ψ(K) − (K − 1)Ψ0 (K),
K = K(ξ, h). From this point of view, if M = M (ξ) and N = N (ξ) happen to
be smooth, then the equation (13.23) is elliptic. Indeed,
(M + N ) hξξ + Mξ hξ + Nξ hξ = 0,
where M + N = 2 Ψ0 (K) > 0
(13.24)
On the other hand we may consider (13.23) as the Lagrange-Euler equation
of a quadratic energy integrand
ZZ 2
2
2
E[h] =
M hξ + N |hξ | |dξ|
Ω0
38
However, inequality M + N > 0 is insufficient for this functional to be convex.
We must assume that both coefficients M (ξ) and N (ξ) are nonnegative. This
happens if and only if
Ψ(K)
K−16 0
6K+1
(13.25)
Ψ (K)
The case Ψ(K) = K was already investigated in Theorem 10.6 where we have
shown that h then satisfies the Laplace equation. Practically, there are no other
examples since (13.25) forces almost linear growth of Ψ.
That is why the variational approach has to be abandoned when Ψ has overlinear growth. There is, however, an interesting and promising nondivergence
form of (13.22). A tedious computation leads to a second order equation
hξξ = α hξξ + β hξξ + γ hξξ + δ hξξ
(13.26)
where the complex coefficients α, β, γ and δ depend in a rather explicit way, by
means of Ψ, only on the first order derivatives hξ and hξ .
We shall not bother with these explicit formulas here but refer the interested
reader to our appendix. What is perhaps more interesting is the ellipticity
condition
1 + α(ξ)λ + β(ξ)λ > γ(ξ)λ + δ(ξ)λ
(13.27)
for every complex number λ of modulus 1. This means that our second order
equation (13.26) is not only elliptic but also lies in the same homotopy class
as the Laplacian [1]. Verification of the ellipticity condition at (13.27) is again
postponed to Appendix.
14
14.1
Appendix
More about Distortion Functions
For nonsingular matrix A ∈ Rn×n
we obtain from (3.1) and (3.2) that
+
p
`
K` (A−1 ) =
p
n−`
Kn−` (A)
Various bounds of norms of the matrices A`×` will be useful.
LEMMA 14.1 (Hadamard-type inequlity). For every 1 6 ` 6 κ 6 n and
A ∈ Rn×n , we have
κ×κ κ1
1
A
6 A`×` ` ,
and
39
1
|| Aκ×κ || κ 6 || A`×` ||
1
`
Proof. It involves no loss of generality in assuming that A is diagonal, say
A = diag{λ1 , ..., λn }, where 0 < λ1 6 ... 6 λn . The first inequality reduces,
equivalently, to
1
1
(λn λn−1 · · · λn−κ+1 ) κ 6 (λn λn−1 · · · λn−`+1 ) `
which is easy to verify. The second inequality has been already pointed out in
[14, Lemma 2.1] with a proof based on symmetric averages, see also [24].
Hadamard’s inequalities give sharp relations between the distortion functions.
LEMMA 14.2. We have the following chains of inequalities
K1 6 K2 6 ... 6 K` 6 ... 6 Kn−2 6 Kn−1
and
n−2
n−`
`
K1n−1 > K2 2 > ... > K`
(14.1)
2
1
n−2
n−1
> ... > Kn−2
> Kn−1
(14.2)
Similarly,
K1 6 K2 6 ... 6 K` 6 ... 6 Kn−2 6 Kn−1
and
n−2
n−`
2
(14.3)
1
n−2
n−1
> Kn−1
Kn−1
> K2 2 > ... > K` ` > ... > Kn−2
1
(14.4)
Proof. Let A ∈ Rn×n
+ . We may assume that det A = 1. For (14.2) we apply
Lemma 14.1. Given 1 6 ` 6 κ 6 n, we have
[K` (A)]
n−`
`
n
n
n−κ
= A`×` ` > Aκ×κ κ = [Kκ (A)] κ
as desired. The same arguments give inequalities at (14.4) where, instead of
operator norm, the mean Hilbert-Schmidt norm has been used in the definition
of the distortion functions.
For the inequality at (14.1) we argue in much the same way. This time, we
express K` in terms of the inverse matrix. Since det A = 1, we have
K` (A)
=
6
n
`
(n−`)×(n−`) n−`
Kn−` A−1 n−` = A−1
n
n
−1 (n−κ)×(n−κ) n−κ
= Aκ×κ n−κ = Kκ (A)
A
(14.5)
by (3.1) and (3.2). Again, the same arguments give the inequalities at (14.3),
completing the proof.
40
14.2
Polyconvexity
A matrix function Ξ : Rn×n → R is said to be polyconvex if it can be written as
Ξ = F (A1×1 , A2×2 , ..., An×n )
where (A1×1 , A2×2 , ..., An×n ) is a list of all possible minors of A ∈ Rn×n and
the function F is convex. The list of minors can be identified with a point in
RN , where
2
2 2
(2n)!
n
n
n
=
+ ... +
+
−1
N=
n
2
n! n!
1
Thus F : RN → R.
Our basic examples of polyconvex functions are the distortion functions
K1 , ..., Kn−1 : Rn×n
→R
+
Precisely,
K` (A) =
|| A`×` ||
n
n−`
`
(det A) n−`
is a convex function of det A and the `-minors. For this, we observe that the
α
function xyβ of two variables x, y ∈ R+ is convex whenever α > β + 1 > 1, see
[15, Lemma 8.8.2]. In particular,
xα
aα
aα−1
aα
− β > α β (x − a) − β β+1 (y − b)
β
y
b
b
b
(14.6)
Hence,
K` (X) − K` (A) >
n
n−`
|| A`×` ||
det A
−
`
n−`
|| A`×` ||
det A
1
|| A`×` ||
det A
|| A`×` ||
det A
>
n−1
`
−
`
n−`
`
n−`
|| X || `×` − || A`×` ||
n
n−`
(det X − det A)
`
n−`
A`×`
, X `×` − A`×`
|| A`×` ||
(det X − det A)
PK (A) = P (K1 (A), ..., Kn−1 (A)) : Rn×n
→ [1, ∞)
+
where
P : [1, ∞) × ... × [1, ∞) → [1, ∞)
|
{z
}
41
n
n−`
More generally,
(n−1)−times
(14.7)
is a given convex function non-decreasing in each variable, when all the other
variables are held fixed. That is, the partials Pi = ∂P/∂i are non-negative. In
particular, we have
PK (X) − PK (A) >
>
n−1
X
`=1
n
X
Π` (A) [K` (X) − K` (A)]
G` (A), X `×` − A`×`
(14.8)
`=1
14.3
Second Order Elliptic System
The variational equations are very useful in the study of extremal problems.
Both the extremal mapping and its inverse turn out to satisfy their own second
order system of PDEs. It is important to examine the ellipticity of such systems.
We include in this appendix a brief discussion of the systems of two equations
with two unknowns. Using complex numbers the system reduces to one complex
equation. The general form of the second order elliptic operator is
L = a(z)
∂2
∂2
∂2
∂2
∂2
∂2
+ b(z)
+ c(z)
+ d(z)
+ e(z)
+ f(z)
∂z ∂z
∂z ∂z
∂z ∂z
∂z ∂z
∂z ∂z
∂z ∂z
For a fixed z ∈ Ω ⊂ C, the coefficients give rise to a point (a, b, c, d, e, f) ∈ C6 .
The ellipticity conditions determine an open set E ⊂ C6 . This set, as shown by
Bojarski [1], consists of six components represented by

∂2
∂2


∂z
∂z
∂z ∂z





∂2
∂2
(14.9)
∂z ∂z
∂z ∂z






 ∂2
∂2
∂z ∂z
∂z ∂z
In contrast with the scalar elliptic equations the elliptic systems with measurable
coefficients are not well developed yet, see [12], [9] for details. It turns out
2
that the systems in the same homotopy class as the Laplacian 4 = 4 ∂z∂ ∂z (or
its complex conjugate 4 ) are best suited to the regularity properties of the
solutions. In [1] it is shown that such systems take the form
fz z = α(z) fz z + β(z) fz z + γ(z) fz z + δ(z) fz z
where the ellipticity condition reads as
1 + α(z) λ + β(z) λ > γ(z) λ + δ(z) λ
42
for all λ ∈ C with modulus 1. Next two subsections are devoted to a computation
showing that variational equations for the extremal mappings belong to this
homotopy class.
14.3.1
The Lagrange-Euler Equation in Nondivergence Form
We begin this section by issuing a warning. The computations below can only be
rigorously justified for specific extremals only after we have established sufficient
regularity.
We are going to express the variational equation (13.12) in nondivergence
form. In our computation the explicit formula for A = A(κ) is irrelevant. Before
formulating the equation we need to introduce the following coefficients
a(z) =
fz
|fz |
b(z) =
fz
|fz |
Hence, the complex dilatation of f is:
µ(z) =
fz
= abk
fz
where k = |fz |/|fz |.
LEMMA 14.3. Equation (13.12) is equivalent to:
0
2
κA0 ∂ 2 f
κA
2κ
∂ f
(1 − κ)
= µ
−
+
A ∂z ∂z
A
1 − κ ∂z ∂z
2
κA0
∂2f
+ µ
−
+
1−κ
A
∂z ∂z
0
2
κA
∂ f
1 + κ)
+ a2 κ
−
+
A
1−κ
∂z ∂z
2
1 + κ κA0
∂ f
2
+ b
−
1−κ
A
∂z ∂z
(14.10)
Proof. For notational simplicity we introduce the function A(k) = A(k 2 ) = A(κ)
and collect first some formulas. The chain rule gives
∂A
∂k
= A0 (k)
(14.11)
∂z
∂z
∂A
∂k
= A0 (k)
(14.12)
∂z
∂z
The derivatives of k become linear forms of the second derivatives of f , namely
∂k
∂z
∂k
2 |fz |
∂z
2 |fz |
= b fz z + b fz z − kafz z − ka fz z
(14.13)
= b fz z + b fz z − kafz z − ka fz z
(14.14)
43
Equation (13.12) expands into
fz fz2 kz − fz fz2 kz A0 (k) =
fz2 fz z − fz2 fz z A(k)
(14.15)
which in view of formulas (14.13), (14.14) takes the form
−2k fzz + (1 + k 2 ) ab fzz +
2A
fzz − ka2 fzz +
+ k 2 ab 1 +
kA0
2A
fzz − kb2 fzz
+ ab 1 −
kA0
= 0
(14.16)
Finally, we solve this linear equation for fzz in terms of fzz and fzz to conclude
with (14.10).
It is interesting to know when the equation (14.10) is linear over the field of
complex numbers. This happens if and only if
κA0 (κ)
1+κ
=
= K(z, f )
A(κ)
1−κ
(14.17)
in which case the equation reduces to
(1 + κ)fzz = µ fzz + µ fzz
(14.18)
as claimed at (13.18). Note that the only solution to (14.17), up to a multiplicative constant, is
2κ
A(κ) =
(1 − κ)2
as in (13.16).
Recall that the formula (14.10) is the variational equation for the functional
ZZ
2
KΨ (z, f ) |dz|
(14.19)
Ω
where
KΨ = Ψ(K) = Ψ
1+κ
1−κ
(14.20)
and Ψ : [1, ∞) → [1, ∞) is convex, Ψ(1) = 1. In this case
1+κ
2κ
A(κ) = Ψ0
1 − κ (1 − κ)2
We want hence to express
κA0 (κ)
A(κ)
in terms of Ψ. Elementary computation yields
Ψ00
2κA0 (κ)
= 2K + K2 − 1
A(κ)
Ψ0
44
On substituting this into (14.10) we obtain
2
fz ∂ 2 f
2 K Ψ0
∂ f
2
+ (K − 1) Ψ00
= 2Ψ0 + K2 − 1 Ψ00
K+1
∂z∂z
fz ∂z∂z
fz ∂ 2 f
+ 2Ψ0 − K2 − 1 Ψ00
fz ∂z∂z
fz ∂ 2 f
+ (K − 1)2 Ψ00
fz ∂z∂z
2
fz ∂ 2 f
(14.21)
− (K − 1) Ψ00
fz ∂z∂z
14.3.2
Nondivergence Equation for the Inverse Map
We begin with the system in divergence form
[Ψ(K) − (K + 1)Ψ0 (K)] hζ = [Ψ(K) − (K − 1)Ψ0 (K)] hζ
ζ
(14.22)
ζ
which we have derived at (13.22). To make further calculation we need the
following identity.
h
i
h
i
2
2 |hζ | Kζ = (K + 1)2 hζζ hζ + hζζ hζ − (K − 1)2 hζζ hζ + hζζ hζ (14.23)
As K =
1+κ
1−κ ,
we find that Kζ =
∂κ
|hζ |
∂ζ
2
2
= |hζ |
=
2
∂κ
(1−κ)2 ∂ζ
|hζ |2
and we compute
!
|hζ |2
ζ
(hζζ hζ + hζζ hζ ) − κ(hζζ hζ + hζζ hζ )
from which formula (14.23) can be obtained.
Now, formally differentiating the equations in divergence form gives us the
second order system
0 = 4Ψ0 (K)|hζ |2 hζζ + (K + 1)3 hζζ |hζ |2 + hζζ h2ζ Ψ00 (K)
− (K + 1)(K − 1)2 hζζ hζ hζ + hζζ hζ hζ Ψ00 (K)
− (K − 1)(K + 1)2 hζζ hζ hζ + hζζ hζ hζ Ψ00 (K)
+ (K − 1)3 hζζ h2ζ + hζζ |hζ |2 Ψ00 (K)
We write Φ(K) = Ψ0 (K)/Ψ00 (K) and collect terms to find
45
0
= hζζ [ 4|hζ |2 Φ + (K + 1)3 |hζ |2 + (K − 1)3 |hζ |2 ]
− hζζ [ (K + 1)(K − 1)2 hζ hζ + (K − 1)(K + 1)2 hζ hζ ]
0
−
hζζ [ (K + 1)(K − 1)2 hζ hζ ]
+
hζζ [ (K − 1)3 h2ζ ]
+
hζζ [ (K + 1)3 h2ζ ]
−
hζζ [ (K − 1)(K + 1)2 hζ hζ ]
= hζζ [ 4Φ + (K + 1)3 κ + (K − 1)3 ]
hζ
− hζζ (K + 1)(K − 1)2 + (K − 1)(K + 1)2
hζ
hζ
− hζζ (K + 1)(K − 1)2
hζ
h
ζ
+ hζζ (K − 1)3
hζ
h2
ζ
+ hζζ (K + 1)3
|hζ |2
hζ
2
− hζζ (K − 1)(K + 1)
hζ
We simplify these terms and remove the common factor of K − 1,
hζ
4Φ
2
0 = hζζ
+ 2(K + 1) − hζζ 2K(K + 1)
K−1
hζ
hζ
hζ
+ hζζ (K − 1)2
− hζζ (K + 1)(K − 1)
hζ
hζ
hζ
hζ
− hζζ (K + 1)2
+ hζζ (K + 1)2
hζ
hζ
Now to simplify matters we make a couple of substitutions and write everything
in terms of k = k(ζ, h); here recall that we have used the notation κ = k 2 above.
We put
hζ
hζ
,
b=
a=
|hζ |
|hζ |
and recall that
K=
1 + k2
,
1 − k2
hence K + 1 =
2
1 − k2
46
and K − 1 =
2k 2
1 − k2
Our equations now have a common factor 4(1 − k 2 )2 and when this is removed
and we substitute
2(1 − k 2 )2 Ψ0
2(1 − k 2 )2 Φ
=
P =
k2
k 2 Ψ00
they read as
0
=
[P + (1 + k 4 )] hζζ − (1 + k 2 )kab hζζ − k 3 ab hζζ
+ k 4 a2 hζζ + b2 hζζ − kab hζζ
(14.24)
The complex conjugate equation reads as
0
=
[P + (1 + k 4 )] hζζ − (1 + k 2 )kab hζζ − k 3 ba hζζ
2
+ k 4 a2 hζζ + b hζζ − kba hζζ
(14.25)
We multiply (14.24) by [P + (1 + k 4 )] and equation (14.25) by (1 + k 2 )kab and
add so as to eliminate the term hζζ . We obtain
0
=
hζζ [ P + (1 + k 4 )]2 − (1 + k 2 )2 k 2 ]
+
hζζ [ −P − (1 + k 4 ) + k 2 (1 + k 2 ) ] k 3 ab
+
hζζ [ P + (1 + k 4 ) − (1 + k 2 ) ] k 4 a2
+ hζζ [ P + (1 + k 4 ) − (1 + k 2 )k 2 ] b2
+
hζζ [ −P − (1 + k 4 ) + (1 + k 2 ) ] kab
which simplifies to
LEMMA 14.4. The inverse mapping satisfies the equation
[ P 2 + 2P (1 + k 4 ) + (1 − k 2 )(1 − k 6 ) ] hζζ
=
[ P + 1 − k 2 ) ] k 3 ab hζζ
+
[ P − k 2 (1 − k 2 ) ] kab hζζ
+
[ k 2 (1 − k 2 ) − P ] k 4 a2 hζζ
+
[ k 2 − P − 1 ] b2 hζζ
where
(14.26)
2(1 − k 2 )2 Ψ0
.
k 2 Ψ00
In a moment we are going to try to determine if this system is elliptic and
in order to do this we need to simplify these equations as much as possible. For
this reason we introduce the new variable
P =
D=
2Ψ0 (K)
(K2 − 1)Ψ00 (K)
47
and remove a further common factor to obtain the system
0
=
[ D2 (1 − k 2 ) + 2D(1 + k 4 ) + (1 − k 6 ) ] hζζ
+
[ D + 1 ] k 3 ab hζζ − [ k 2 − D ] kab hζζ
+
[ D − k 2 ] k 4 a2 hζζ + [ D + 1 ] b2 hζζ
We now recall that an operator of the form θhζζ + αhζζ + βhζζ + γhζζ + δhζζ
is elliptic and lies in the same homotopy class as the Laplacian if and only if
| θ + αλ + βλ | > | γλ + δλ |
for every λ ∈ C with |λ| = 1. For us the desired inequality reads, after a little
simplification, as
| (D − k 2 )k 4 a2 λ + (D + 1)b2 λ | <
| D2 (1 − k 2 ) + 2D(1 + k 4 ) + 1 − k 6 − (D + 1)k 3 abλ + (k 2 − D)kabλ |
In order to simplify this notation we introduce the new complex variable
ξ = abλ,
|ξ| = 1
and then we need to show for D > 0,
| (D − k 2 )k 4 ξ + (D + 1)ξ | <
| D2 (1 − k 2 ) + 2D(1 + k 4 ) + 1 − k 6 − (D + 1)k 3 ξ + (k 2 − D)kξ(14.27)
|
Before establishing this inequality we suggest that the reader will check the
limiting and easier case D = 0 (that is K = ∞) by verifying
|1 − k 6 − k 3 ξ + k 3 ξ| = |ξ − k 6 ξ|
We now turn to establishing the ellipticity condition at (14.27). We write ξ =
x + iy for real x and y, with x2 + y 2 = 1. We put this into the inequality and
separate out the real and imaginary parts and then compute the square of the
absolute values. We are thus asked to verify that
[ (D − k 2 )k 4 + (D + 1) ]2 x2 + [ (D − k 2 )k 4 − D − 1 ]2 y 2 <
[ D2 (1 − k 2 ) + 2D(1 + k 4 ) + 1 − k 6 − kD(1 + k 2 )x]2 + [ 2k 3 + Dk(k 2 − 1) ]2 y 2
We next write y 2 = 1 − x2 to obtain a quadratic polynomial in x,
[ D2 (1−k 2 )+2D(1+k 4 )+1−k 6 ]2 −2kD(1+k 2 )[ D2 (1−k 2 )+2D(1+k 4 )+1−k 6 ]x
48
+k 2 D2 (1 + k 2 )x2 + [ 2k 3 + Dk(k 2 − 1) ]2
−[ 2k 3 + Dk(k 2 − 1) ]2 x2 − [ (D − k 2 )k 4 + D + 1 ]2 x2
−[ (D − k 2 )k 4 − D − 1]2 + [ (D − k 2 )k 4 − D − 1 ]2 x2 > 0
Next we group together the coefficient of x2 and find it is negative. That is we
see
k 2 D2 (1+k 2 )−[ 2k 3 +Dk(k 2 −1) ]2 −[ (D−k 2 )k 4 +D+1 ]2 +[ (D−k 2 )k 4 −(D+1) ]2
= −D2 k 2 (1 + k 2 )(1 + 2k 2 ) < 0
This then implies that the extreme case is when x = 1 and y = 0. Thus the
inequality (14.27) reduces to showing that
|D2 (1 − k 2 ) + 2D(1 + k 4 ) + 1 − k 6 − kD(1 + k 2 )| > |(D − k 2 )k 4 + D + 1|
That is
D2 (1 − k 2 ) + D(2 + 2k 4 − k 3 − k) > D(k 4 + 1)
Rearranging and simplifying, this reduces to verifying that
D2 (1 − k 2 ) + D(1 − k 3 )(1 − k) > 0
which in view of the fact that 0 ≤ k < 1 is clear.
49
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Department of Mathematics, University of Helsinki, P.O. Box 4, FIN00014, University of Helsinki, FINLAND
E-mail address: [email protected]
Department of Mathematics, Syracuse University, Syracuse, NY 13244,
USA
E-mail address: [email protected]
Department of Mathematics, University of Auckland, Auckland, NEW
ZEALAND
E-mail address: [email protected]
Department of Mathematics, University of Michigan, 525 East University,
Ann Arbor, MI 48109, USA
E-mail address: [email protected]
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