REGULARITY OF DIRICHLET NEARLY MINIMIZING
MULTIPLE-VALUED FUNCTIONS
JORDAN GOBLET AND WEI ZHU
Abstract. In this paper, we extend the related notions of Dirichlet quasiminimizer, ω−minimizer and almost minimizer to the framework of multiplevalued functions in the sense of Almgren and prove Hölder regularity results.
We also give examples of those minimizers with various branch sets.
Introduction
In [2] F. J. Almgren, Jr. has developed a theory of functions defined on a bounded
open subset Ω of Rm with smooth boundary and taking values in the space QQ (Rn )
of unordered Q tuples of points of Rn (see [1] for a summary). In terms of Dirichlet’s
integral naturally defined for appropriate functions Ω → QQ (Rn ) he obtained the
following existence and regularity results:
Theorem. For each appropriate function v : ∂Ω → QQ (Rn ) there exists a function
u : Ω → QQ (Rn ) having boundary values v and of least Dirichlet integral among
such functions. Furthermore, each such minimizer u is Hölder continuous.
Almgren introduced the machinery of Dirichlet minimizing multiple-valued functions in order to study the partial regularity of mass-minimizing integral currents.
He proved that any m-dimensional mass-minimizing integral current is regular except on a set of Hausdorff dimension at most m − 2. In the present paper, we will
define and study the regularity of Dirichlet quasiminimizer as well as ω−minimizer
and almost minimizer in the setting of multiple-valued functions.
The notion of quasiminimizer, introduced by M. Giaquinta and E. Giusti [7],
embodies the notion of minimizer of variational integrals
Z
(1)
J(u, Ω) =
f (x, u, Du) dx
Ω
©
ª
where Ω is an open subset of R , u : Ω → Rn , Du = Dα ui , α = 1, . . . , m,
i = 1, . . . , n and f (x, u, p) : Ω × Rn × Rmn → R , but it is substantially more
general. By definition, a quasiminimizer u minimizes the functional J only up to a
multiplicative constant, that is, there exists K ≥ 1 such that
m
J(u, V ) ≤ K J(u + φ, V )
for each subdomain V that is compactly contained in Ω and for each φ ∈ C0∞ (V ). In
the scalar-valued case and under suitable conditions on f , Hölder continuity can be
Date: 7th June 2007.
Key words and phrases. Multiple-valued function, quasiminimizer, ω-minimizer, almost
minimizer.
1
2
JORDAN GOBLET AND WEI ZHU
derived directly from the quasiminimizing property. For more details of the statement above, as well as for more information on the properties of quasiminimizers,
we refer to [6], [7] and [8].
An alternative notion called ω-minimizer is considered in a paper by G. Anzellotti
[3]. A function u ∈ W 1,2 (Ω, Rn ) is a ω-minimizer, where ω : (0, R0 ) → R is a nonnegative function, for a functional J of the type (1) if one has
J(u, U m (x, r)) ≤ (1 + ω(r)) J(u + φ, U m (x, r))
for any ball U m (x, r) ⊂ Ω with r < R0 and for all functions φ ∈ W01,2 (U m (x, r), Rn ).
If the functional J is the Dirichlet integral and limr→0 ω(r) = 0 then Anzellotti
proves that u is locally Hölder continuous. In addition the first derivatives of u are
locally Hölder continuous, with exponent γ, in Ω provided 0 ≤ ω(r) ≤ cr2γ .
Motivated by the question of boundary regularity of Dirichlet minimizing multiplevalued functions (more precisely, subtracting a suitable extension of boundary value
from the original minimizer gives an almost minimizer), we also consider (c, α)almost minimizer in the following sense:
J(u, U m (x, r)) ≤ J(v, U m (x, r)) + c rm−2+α
for any ball U m (x, r) ⊂ Ω and for all functions v ∈ W 1,2 (U m (x, r), Rn ) such that
u = v on ∂U m (x, r). Readers are also referred to [10] for the discussion of almost
minimizer in the study of ferromagnetism in R3 .
This work is organized as follows. In Section 2 we define the notion of Dirichlet
quasiminimizing multiple-valued functions and Hölder regularity results are derived
from two Almgren’s estimates. We also construct a one-dimensional quasiminimizer
with a fractal branch set. In Section 3 we prove a similar result to Anzellotti’s
Theorem for ω-minimizing multiple-valued functions and we give an example of
ω−minimizer with a unique branch point. In Section 4 we prove a Hölder regularity
theorem for Dirichlet almost minimizing multiple-valued functions and construct a
one-dimensional almost minimizer with a fractal branch set.
1. Preliminaries
In our terminology and notations we shall follow scrupulously [2] and [5]. Throughout the whole text m, n and Q will be positive integers. We denote by U m (x, r)
and B m (x, r) the open and closed balls in Rm with center x and radius r.
1.1. The metric space QQ (Rn ). The space of unordered Q tuples of points in
Rn is
( Q
)
X
n
n
QQ (R ) =
[[ui ]] : u1 , . . . , uQ ∈ R
,
i=1
where [[ui ]] is the Dirac measure at ui . This space is equipped with the metric G;
à Q
!
à Q
!1/2
Q
X
X
X
2
G
[[ui ]],
[[vi ]] = min
|ui − vσ(i) |
i=1
i=1
σ
i=1
where σ runs through all the permutations of {1, . . . , Q}. A multiple-valued function in the sense of Almgren is a QQ (Rn )-valued function.
There are a bi-Lipschitz homeomorphism ξ : QQ (Rn ) → Q∗ ⊂ RP Q , where P is
a positive integer, and a Lipschitz retraction ρ : RP Q → Q∗ with ρ|Q∗ =IdQ∗ (see
Theorems 1.2(3) and 1.3(1) in [2]).
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 3
1.2. Affine approximations. The set of all affine maps Rm → Rn is denoted by
A(m, n). If v = (v1 , . . . , vn ) ∈ A(m, n), we put


¶2 1/2
n X
m µ
X
∂v
i

|v| = 
∈ R.
∂x
j
i=1 j=1
A function v : Rm → QQ (Rn ) is called affine if there are v1 , . . . , vQ ∈ A(m, n) such
³P
´1/2
PQ
Q
2
that v = i=1 [[vi ]]. Then we set |v| =
.
i=1 |vi |
If Ω ⊂ Rm is open and a ∈ Ω, a function u : Ω → QQ (Rn ) is said to be
approximately affinely approximable at a if there is an affine function v : Rm →
QQ (Rn ) such that
ap lim
x→a
G(u(x), v(x))
= 0.
|x − a|
Such a function v is uniquely determined and denoted by apAu(a). This is the case
if ξ ◦ u is approximately differentiable at a, and then
−1
(Lip(ξ))
|apD(ξ ◦ u)(a)| ≤ |apAu(a)| ≤ |apD(ξ ◦ u)(a)|
(see Theorem 1.4(3) in [2]).
1.3. The Sobolev space Y2 (U, QQ (Rn )). Suppose U is an open ball in Rm or
the entire space Rm .
(1) We denote by W 1,2 (U, Rn ) the Sobolev space of Rn -valued functions on U
which together with their first order distributional partial derivatives are
Lm square summable over U .
(2) A function u ∈ W 1,2 (U, Rn ) is said to be strictly defined if u(x) = y
whenever x ∈ U, y ∈ Rn and
Z
lim r−m
|u(z) − y| dLm z = 0.
r→0
B m (x,r)
Any v ∈ W 1,2 (U, Rn ) agrees Lm almost everywhere on U with a strictly
defined u ∈ W 1,2 (U, Rn ). If u ∈ W 1,2 (U, Rn ) is strictly defined, then
Z
−m
lim r
|u(z) − u(x)| dLm z = 0
r→0
B m (x,r)
for Hm−1 almost all x ∈ U (see Section 4.8 in [4]).
(3) The space Y2 (U, QQ¡(Rn )) consists
of those functions u : U → QQ (Rn ) for
¢
1,2
PQ
which ξ ◦ u ∈ W
U, R
. We say that u is strictly defined if ξ ◦ u is
strictly defined. If u, v ∈ Y2 (U, QQ (Rn )), we write u = v if u(x) = v(x) for
Lm almost all x ∈ U .
(4) Suppose u ∈ Y2 (U, QQ (Rn )). Then u is approximately affinely approximable Lm almost everywhere on U since ξ ◦ u is (see Section 6.1.3 in [4]).
The Dirichlet integral of u over U is defined by
Z
Dir(u; U ) =
|apAu(x)|2 dLm x.
U
4
JORDAN GOBLET AND WEI ZHU
1.4. The Sobolev space ∂Y2 (∂U, QQ (Rn )). Suppose U ⊂ Rm is an open ball.
(1) The space ∂Y2 (∂U, QQ (Rn )) will be the set of those functions v : ∂U →
QQ (Rn ) for which there is a strictly defined u ∈ Y2 (Rm , QQ (Rn )) such that
u(x) = v(x) for Hm−1 almost all x ∈ ∂U (cf. Theorem A.1.2(7) and Section
2.1(2) in [2]). If v, w ∈ ∂Y2 (∂U, QQ (Rn )), we write v = w if v(x) = w(x)
for Hm−1 almost all x ∈ ∂U .
(2) If v ∈ ∂Y2 (∂U, QQ (Rn )) we say that u has boundary values v if there is
w ∈ Y2 (Rm , QQ (Rn )) which is strictly defined such that w|U = u|U and
w|∂U = v. We then write u|∂U = v. If u is strictly defined, v agrees with u
Hm−1 almost everywhere on ∂U .
(3) One says that u : U → QQ (Rn ) is Dirichlet minimizing if and only if u ∈
Y2 (U, QQ (Rn )) and, assuming u has boundary values v ∈ ∂Y2 (∂U, QQ (Rn )),
one has
Dir (u; U ) = inf {Dir (w; U ) : w ∈ Y2 (U, QQ (Rn )) has boundary values v} .
If v ∈ ∂Y2 (∂U, QQ (Rn )) then there exists a Dirichlet minimizing u ∈
Y2 (U, QQ (Rn )) that has boundary values v (see Theorem 2.2(2) in [2]).
(4) For each v ∈ ∂Y2 (∂U, QQ (Rn )), the Dirichlet integral of v over ∂U is defined
by
Z
|apAv(x)|2 dHm−1 x
dir(v; ∂U ) =
∂U
where the affine approximation apAv is with respect to ∂U .
2. Dirichlet quasiminimizer
Throughout this section K will be a real number larger than 1.
Definition 1. A strictly defined function u ∈ Y2 (U m (0, 1), QQ (Rn )) is a Dirichlet
K-quasiminimizer if for every ball U ⊂ U m (0, 1),
Dir(u; U ) ≤ K Dir(v; U )
where v is a Dirichlet minimizing multiple-valued function having boundary values
u|∂U ∈ ∂Y2 (∂U, QQ (Rn )).
2.1. Regularity. For the convenience of the reader, we shall recall here a few
known results that we are going to need later. We begin with a "Dirichlet growth"
theorem guaranteeing Hölder continuity:
Theorem 3.5.2 in [15]. Suppose u ∈ W 1,p (B m (x0 , R)), 0 < σ < 1 and
Z
|∇u(x)|p dx ≤ C (r/δ)m−p+pσ , 0 ≤ r ≤ δ = R − |z − x0 |
B(z,r)
for every z ∈ B(x0 , R). Then
u ∈ C 0,σ (B m (x0 , r))
for each r < R.
We recall Almgren’s estimates for 2-dimensional and higher dimensional domains:
¡
¢
Section 2.7(2) in [2]. Suppose¡v ∈ ∂Y2 ∂U 2 (x, r),
QQ (Rn ) . Then there exists a
¢
multiple-valued function u ∈ Y2 U 2 (x, r), QQ (Rn ) such that
(1) u has boundary values v,
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 5
¡
¢
¡
¢
(2) Dir u; U 2 (x, r) ≤ Qr dir v; ∂U 2 (x, r) .
Theorem 2.12 in [2]. Let m ≥ 3 be an integer. If f ∈ Y2 (U m (x, r), QQ (Rn )) is
strictly defined and Dirichlet minimizing then
¶
µ
1 − εQ
Dir(f ; U m (x, r)) ≤
r dir(f ; ∂U m (x, r))
m−2
where 0 < εQ < 1 is a constant.
¡
¢
n
Theorem 2. Suppose that u ∈ Y2¡ U 2 (0, 1), Q
Q (R ) is a strictly defined Dirichlet
¢
K-quasiminimizer such that Dir u; U 2 (0, 1) > 0. Then the following assertions
are checked:
(1) Suppose z ∈ U 2 (0, 1). Then for L1 almost all 0 < r < 1 − |z|,
Dir(u; U 2 (z, r)) ≤ KQr
d
Dir(u; U 2 (z, s))|s=r .
ds
(2) For each z ∈ U 2 (0, 1), 0 < r < 1 − |z|, and 0 < s ≤ 1,
1
Dir(u; U 2 (z, sr)) ≤ s KQ Dir(u; U 2 (z, r)).
(3) f |B 2 (0,δ) is Hölder continuous with exponent
1
2KQ
for all 0 < δ < 1.
Proof. Let v ∈ Y2 (U 2 (z, r), QQ (Rn )) be a Dirichlet minimizing multiple-valued
function with boundary values u|∂U 2 (z,r) . It follows from Section 2.7(2) in [2] that
Dir(u; U 2 (z, r)) ≤ K Dir(v; U 2 (z, r)) ≤ KQr dir(u; ∂U 2 (z, r)).
Using the fact that the function φ(r) =Dir(u; U 2 (z, r)) is absolutely continuous
with
Z
φ0 (r) =
|apAu(x)|2 dH1 x ≥ dir(u; ∂U 2 (z, r))
∂U 2 (z,r)
1
for L almost all 0 < r < 1 − |z|, we obtain the first assertion. One derives (2) from
(1) by integration. In order to verify (3), one notes that conclusion (2) implies for
each z ∈ U 2 (0, 1), 0 < r < 1 − |z|, and 0 < s ≤ 1,
Dir(ξ ◦ u; U 2 (z, sr)) ≤
Lip(ξ)2 Dir(u; U 2 (z, sr))
1
≤
Lip(ξ)2 s KQ Dir(u; U 2 (z, r))
≤
Lip(ξ)2 s KQ Dir(ξ ◦ u; U 2 (z, r))
≤
Lip(ξ)2 Dir(ξ ◦ u; U 2 (0, 1)) s KQ .
1
1
Let us fix 0 < δ < 1. Theorem 3.5.2 in [15] applied to the function ξ ◦ u implies
that
1
ξ ◦ u ∈ C 0, 2KQ (B 2 (0, δ))
and the third assertion is checked since ξ is bi-Lipschitz.
¤
The regularity result is weaker for higher dimensional domains since we need an
extra assumption (2) on the size of K. This assumption on K is necessary since M.
Giaquinta presents in [6] a single-valued quasiminimizer for the Dirichlet integral
which is singular on a dense set.
6
JORDAN GOBLET AND WEI ZHU
Theorem 3. Let m ≥ 3 be an integer. Suppose that u ∈ Y2 (U m (0, 1), QQ (Rn )) is
a strictly defined Dirichlet K-quasiminimizer such that Dir(u; U m (0, 1)) > 0. If
(2)
1≤K<
1
(1 − εQ )
then there exists a constant 0 < σ = σ (K, εQ , m) < 1 such that the following
assertions are checked:
(1) Suppose z ∈ U m (0, 1). Then for L1 almost all 0 < r < 1 − |z|,
Dir(u; U m (z, r)) ≤
1
d
r
Dir(u; U m (z, s))|s=r .
(m − 2 + 2σ) ds
(2) For each z ∈ U m (0, 1), 0 < r < 1 − |z|, and 0 < s ≤ 1,
Dir(u; U m (z, sr)) ≤ sm−2+2σ Dir(u; U m (z, r)).
(3) f |B m (0,δ) is Hölder continuous with exponent σ for all 0 < δ < 1.
Proof. We fix 0 < σ < 1 requiring that
m−2
> m − 2 + 2σ.
K(1 − εQ )
Let v ∈ Y2 (U m (z, r), QQ (Rn )) be a Dirichlet minimizing multiple-valued function
with boundary values u|∂U m (z,r) . It follows from Theorem 2.12 in [2] that
Dir(u; U m (z, r))
≤ K Dir(v; U m (z, r))
µ
¶
1 − εQ
≤ K
r dir(v; ∂U m (z, r))
m−2
1
r dir(u; ∂U m (z, r)).
≤
(m − 2 + 2σ)
Using the fact that the function φ(r) =Dir(u; U m (z, r)) is absolutely continuous
with
Z
φ0 (r) =
|apAu(x)|2 dHm−1 x ≥ dir(u; ∂U m (z, r))
∂U m (z,r)
for L1 almost all 0 < r < 1 − |z|, we obtain the first claim. One derives (2) from
(1) by integration. In order to conclude, one notes that the second claim implies
for each z ∈ U m (0, 1), 0 < r < 1 − |z|, and 0 < s ≤ 1,
Dir(ξ ◦ u; U m (z, sr))
≤
Lip(ξ)2 Dir(u; U m (z, sr))
≤
Lip(ξ)2 sm−2+2σ Dir(u; U m (z, r))
≤
Lip(ξ)2 sm−2+2σ Dir(ξ ◦ u; U m (z, r))
≤
Lip(ξ)2 Dir(ξ ◦ u; U m (0, 1)) sm−2+2σ .
Let us fix 0 < δ < 1. Theorem 3.5.2 in [15] applied to the function ξ ◦ u implies
that
ξ ◦ u ∈ C 0,σ (B m (0, δ))
and the third assertion is checked since ξ is bi-Lipschitz.
¤
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 7
2.2. Branch set of quasiminimizer.
Definition 4. Assume the multiple-valued function u : U m (0, 1) → QQ (Rn ) is
continuous. Define the function σ : U m (0, 1) → {1, . . . , Q} as
σ(x) = card (spt (u(x)))
m
for all x ∈ U (0, 1). Then, the closed set
Σ = {x ∈ U m (0, 1) | σ is discontinuous at x}
is called the branch set of the continuous multiple-valued function u.
Almgren proved that the Hausdorff dimension of the branch set of a strictly
defined and Dirichlet minimizing u ∈ Y2 (U m (0, 1), QQ (Rn )) does not exceed m −
2 (see Theorem 2.14 in [2]). Here we will construct a one-dimensional Dirichlet
quasiminimizer with a fractal branch set. This construction can be adapted in
order to obtain a one-dimensional Dirichlet quasiminimizer with a branch set of
positive 1-dimensional Lebesgue measure.
The following result characterizes Dirichlet minimizing multiple-valued functions
in codimension one.
Theorem 2.16 in [2]. Suppose v ∈ ∂Y2 (∂U m (0, 1), QQ (R)) and v1 , . . . , vQ :
∂U m (0, 1) → R are defined by requiring for each x ∈ ∂U m (0, 1) that v(x) =
PQ
m
i=1 [[vi (x)]] and v1 (x) ≤ v2 (x) ≤ . . . ≤ vQ (x). Then vi ∈ ∂Y2 (∂U (0, 1), R)
m
for each i = 1, . . . , Q and u ∈ Y2 (U (0, 1), QQ (R)) is strictly defined and Dirichlet
PQ
minimizing with boundary values v if and only if u = i=1 [[ui ]] corresponding to
u1 , u2 , . . . , uQ ∈ Y2 (U m (0, 1), R) which are the unique harmonic functions having
boundary values v1 , v2 , . . . , vQ respectively.
This last theorem implies the following characterization of Dirichlet quasiminimizers in dimension and codimension one.
Theorem 5. A multiple-valued function u ∈ Y2 ((−1, 1), QQ (R)) is a Dirichlet
K-quasiminimizer if and only if for every interval (a, b) ⊂ (−1, 1),
Dir(u, (a, b)) ≤ K
G 2 (u(b), u(a))
.
b−a
Proof. Let (a, b) ⊂ (−1, 1). We define the functions v1 , . . . , vQ : ∂(a, b) → R by
PQ
requiring for each x ∈ ∂(a, b) that u(x) = i=1 [[vi (x)]] and v1 (x) ≤ v2 (x) ≤ . . . ≤
vQ (x). By Theorem 2.16 in [2], the Dirichlet minimizing multiple-valued function
coinciding with u on ∂(a, b) is given by
¸¸
Q ··
X
vi (b) − vi (a)
(· − a) + vi (a)
v(·) =
b−a
i=1
and
Z
Dir(v; [a, b]) =
a
b
|apAv(x)|2 dx =
Q
(b − a) X
|vi (b) − vi (a)|2
(b − a)2 i=1
=
G 2 (u(b), u(a))
.
b−a
¤
8
JORDAN GOBLET AND WEI ZHU
We will now produce a Dirichlet quasiminimizer with a fractal branch set. We
begin with some definitions:
(1) A diamond above an interval I = [a, b] ⊂ R is a multiple-valued function
u : [a, b] → Q2 (R) whose the graph is a parallelogram with the following
vertices [a, h], [(a + b)/2, h], [(a + b)/2, h + (b − a)/2], [b, h + (b − a)/2] where
h ∈ R.
(2) A pluri-diamond is a multiple-valued function u : [0, 1] → Q2 (R) which
admits a partition of [0, 1] in intervals {Ij } such that
- u is a diamond above Ij or a map of the form x → 2[[x + pj ]] above Ij
where pj ∈ R,
- u is continuous on [0, 1].
Figure 1 gives examples
√ of pluri-diamonds. It is clear that a pluri-diamond u is
Lipschitz with Lip(u) ≤ 2. We record two lemmas for later use:
Lemma 6. A pluri-diamond u is a Dirichlet 4-quasiminimizer.
Proof. Let umin , umax : [0, 1] → R be such that u = [[umin ]] + [[umax ]] and umin (x) ≤
umax (x) for all x ∈ [0, 1]. For every interval [a, b] ⊂ [0, 1], we have
G 2 (u(b), u(a)) ≥
1
2
(|umin (b) − umin (a)| + |umax (b) − umax (a)|)
2
ÃZ
!2
b
1
(b − a)2
0
0
=
umin (x) + umax (x) dx
≥
2
2
a
and
Rb
Rb
02
(b − a) a |apAu(x)|2 dx
(b − a) a u02
min (x) + umax (x) dx
=
G 2 (u(b), u(a))
G 2 (u(b), u(a))
≤
2 (b − a)2
≤4
G 2 (u(b), u(a))
hence u is a Dirichlet 4-quasiminimizer by Theorem 5.
¤
m
Lemma 23.7 in [17]. Let Ω be an open subset of R , 1 < p < ∞. If {ui } is a
bounded sequence in W 1,p (Ω) such that ui * u in Lp (Ω) then u ∈ W 1,p (Ω) and
k∇ukp ≤ lim inf k∇ui kp .
i→∞
We will now create a particular sequence of pluri-diamonds {ui }. We define
u : R → Q2 (R) by requiring that
- u1 is a pluri-diamond,
- u1 (1/3) = 0,
- u1 is a diamond only above [1/3, 2/3].
We define u2 : R → Q2 (R) by requiring that
- u2 is a pluri-diamond,
- u2 (1/3) = 0,
- u2 is a diamond only above the intervals [1/3, 2/3], [1/9, 2/9] and [7/9, 8/9].
The reader will easily imagine how the remaining terms of the sequence are
defined. For each i ∈ N, we define uimin , uimax : [0, 1] → R by requiring that
1
ui (x) = [[uimin (x)]] + [[uimax (x)]] and uimin (x) ≤ uimax (x)
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 9
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.3
0.2
0.1
0
0
0
−0.1
−0.1
−0.2
−0.2
−0.1
−0.2
−0.3
−0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−0.3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−0.3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 1. The pluri-diamonds u1 , u2 and u3 .
for all x ∈ [0, 1]. One can checks that
Lip(uimin ) = Lip(uimax ) = 1
for each i ∈ N. By Ascoli’s Theorem, we can extract two subsequences still denoted
{uimin } and {uimax } and there exists two 1-Lipschitz functions umin , umax : [0, 1] → R
such that {uimin } and {uimax } converge uniformly to umin and umax . We define the
Lipschitz multiple-valued function u : [0, 1] → Q2 (R) by
(3)
u = [[umin ]] + [[umax ]].
Proposition 7. The multiple-valued function u defined by (3) is a Dirichlet 4quasiminimizer and the branch set of u is the Cantor ternary set.
Proof. We estimate for any interval [a, b] ⊂ [0, 1],
Z b
Z b
2
2
|apAu(x)|2 dx =
(u0min (x)) + (u0max (x)) dx
a
a
≤
=
Z
b
lim inf
i→∞
a
Z
b
lim inf
i→∞
¡ i 0 ¢2 ¡ i
¢2
umin (x) + umax 0 (x) dx
|apAui (x)|2 dx
a
G 2 (ui (b), ui (a))
i→∞
b−a
i
|u (b) − uimin (a)|2 + |uimax (b) − uimax (a)|2
= 4 lim inf min
i→∞
b−a
|umin (b) − umin (a)|2 + |umax (b) − umax (a)|2
= 4
b−a
G 2 (u(b), u(a))
= 4
b−a
where the first inequality follows from Lemma 23.7 in [17] and the second inequality
is a consequence of Lemma 6. By Theorem 5, the multiple-valued function u is a
Dirichlet 4-quasiminimizer. We denote the Cantor ternary set by
≤
4 lim inf
T = ∩∞
i=1 Ti
where T1 = [0, 1/3] ∪ [2/3, 1], T2 = [0, 1/9] ∪ [2/9, 1/3] ∪ [2/3, 7/9] ∪ [8/9, 1], etc. We
can make the following observations:
10
JORDAN GOBLET AND WEI ZHU
(1) If x ∈ T then uimin (x) = uimax (x) for all i ∈ N0 hence umin (x) = umax (x).
(2) If x ∈
/ T then there exists i ∈ N0 such that x ∈
/ Tj for each j ≥ i. By
construction, there exists a constant η > 0 such that ujmax (x) − ujmin (x) = η
for each j ≥ i hence umin (x) 6= umax (x).
We deduce from (1) and (2) that
{x ∈ [0, 1] | card (spt(u(x))) = 1} = T
hence the branch set of u is T since it contains no interval.
¤
It is clear that this construction can be adapted in order to obtain a quasiminimizer with a fat Cantor branch set.
2.3. Application. Let a ∈ L∞ (U m (0, 1), R) and consider the following functional
Z
J(u; U m (0, 1)) =
a(x) |apAu(x)|2 dLm x
U m (0,1)
defined for each u ∈ Y2 (U m (0, 1), QQ (Rn )). One says that u : U m (0, 1) → QQ (Rn )
is J−minimizing if and only if u ∈ Y2 (U m (0, 1), QQ (Rn )) and, assuming u has
boundary values v ∈ ∂Y2 (∂U m (0, 1), QQ (Rn )), one has
J (u; U m (0, 1)) = inf{J (w; U m (0, 1)) : w ∈ Y2 (U m (0, 1), QQ (Rn ))
has boundary values v}.
Theorem 8. Let m ≥ 2 be an integer. Assume v ∈ ∂Y2 (∂U m (0, 1), QQ (Rn )) and
a(x) ≥ γ > 0 for almost all x ∈ U m (0, 1). Then there exists a strictly defined
J−minimizing u ∈ Y2 (U m (0, 1), QQ (Rn )) that has boundary values v and
(1) if m = 2, u|B 2 (0,δ) is Hölder continuous for all 0 < δ < 1;
(2) if m ≥ 3 and
kak∞
1
≤
γ
1 − εQ
then u|B m (0,δ) is Hölder continuous for all 0 < δ < 1.
Proof. The existence proof is a direct adaptation of the proof of Theorem 2.2(2) in
[2]. The regularity follows from the fact that u is a Dirichlet kakγ ∞ -quasiminimizer.
Consequently it remains to apply Theorem 2 or Theorem 3.
¤
3. Dirichlet ω−minimizer
Throughout this section ω : (0, ∞) → R will be a non-negative function.
Definition 9. A strictly defined function u ∈ Y2 (U m (0, 1), QQ (Rn )) is a Dirichlet
ω-minimizer if for every ball U m (x, r) ⊂ U m (0, 1),
Dir(u; U m (x, r)) ≤ (1 + ω(r)) Dir(v; U m (x, r))
where v is a Dirichlet minimizing multiple-valued function having boundary values
u|∂U m (x,r) ∈ ∂Y2 (∂U m (x, r), QQ (Rn )).
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 11
3.1. Regularity. The following regularity result is a direct consequence of Theorem 2 and Theorem 3.
Theorem 10. Let m ≥ 2 be an integer. Suppose that u ∈ Y2 (U m (0, 1), QQ (Rn ))
is a strictly defined ω-minimizer such that Dir(u; U m (0, 1)) > 0. If
limr→0 ω(r) = 0 then
0,σ
u ∈ Cloc
(U m (0, 1), QQ (Rn ))
for some 0 < σ < 1.
3.2. Branch set of ω−minimizer. We will construct a one-dimensional Dirichlet
ω−minimizer with a unique branch point at the origin. First of all, we show that
a one-dimensional single-valued ω−minimizer does not have to be a straight line
even if limr→0 ω(r) = 0.
Proposition 11. sin x : (−π/4, π/4) → R is a Dirichlet ω−minimizer with
ω(r) =
r2
− 1 & 0 as r & 0.
sin2 r
Proof. It is easy to check that
Z
1
x+r
Dir(sin(·); U (x, r)) =
(cos s)2 ds
x−r
sin 2(x + r) − sin 2(x − r)
+r
4
cos 2x sin 2r
=
+r
2
The Dirichlet minimizer v with boundary values v(x−r) = sin(x−r) and v(x+r) =
sin(x + r) is the straight line with slope
=
sin(x + r) − sin(x − r)
,
2r
hence
·
sin(x + r) − sin(x − r)
Dir(v; U (x, r)) =
2r
1
=
¸2
2r
2 cos2 x sin2 r
.
r
Now it suffices to show that
cos 2x sin 2r
2 cos2 x sin2 r
+ r ≤ (1 + ω(r))
,
2
r
for any U 1 (x, r) ⊂ (−π/4, π/4). Define the function
·
¸ ·
¸
cos 2x sin 2r
2 cos2 x sin2 r
W (x, r) =
+r /
.
2
r
(4)
Denote
W (x, r) =
where
H(x, r) =
1
H(x, r),
4 sin2 r
r cos(2x) sin(2r) + 2r2
.
cos2 x
12
JORDAN GOBLET AND WEI ZHU
We have
∂H
2r sin x
=
(2r − sin(2r)).
∂x
cos3 x
For r ∈ (0, 1],
∂H
≤ 0, for x ∈ (−π/4, 0],
∂x
∂H
≥ 0, for x ∈ [0, π/4).
∂x
Since W (x, r) is an even function of x, we conclude
W (x, r) ≤ W (π/4, r) =
r2
, ∀x ∈ (−π/4, π/4)
sin2 r
which proves (4).
¤
We are ready to construct a one-dimensional Q2 (R)−valued ω−minimizer with
a branch point and limr→0 ω(r) = 0.
Proposition 12. The multiple-valued function
u(x) = [[x]] + [[sin x]] : (−π/4, π/4) → Q2 (R)
is a ω−minimizer with
ω(r) =
r2
− 1 & 0 as r & 0.
sin2 r
The origin is a branch point of u.
Proof. Proposition 12 is a direct consequence of Proposition 11 since the Dirichlet
minimizer v with boundary values u(x − r) = [[x − r]] + [[sin(x − r)]], u(x + r) =
[[x + r]] + [[sin(x + r)]] is two straight lines connecting x − r with x + r and sin(x − r)
with sin(x + r).
¤
Let us mention that the authors ignore the maximum Hausdorff dimension of
branch sets of ω-minimizers.
4. Dirichlet almost minimizer
Throughout this section c ≥ 0 and 0 < α < 1 are real numbers.
Definition 13. A strictly defined function u ∈ Y2 (U m (0, 1), QQ (Rn )) is a Dirichlet
(c, α)-almost minimizer if for every ball U m (x, r) ⊂ U m (0, 1),
Dir(u; U m (x, r)) ≤ Dir(v; U m (x, r)) + crm−2+α
where v is a Dirichlet minimizing multiple-valued function having boundary values
u|∂U m (x,r) ∈ ∂Y2 (∂U m (x, r), QQ (Rn )) .
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 13
4.1. Regularity. For the convenience of the reader, we shall recall a few results
that we are going to need later.
Section 2.9 in [2]. Corresponding to numbers 0 < s0 < ∞, 1 < K < ∞, and (not
necessarily distinct points) q1 , . . . , qQ ∈ Rn we can find J ∈ {1, . . . , Q}, k1 , . . . , kJ ∈
{1, . . . , Q}, distinct points p1 , . . . , pJ ∈ {q1 , . . . , qQ }, and s0 ≤ r ≤ Cs0 such that
(1) |pi³− pj | > 2Kr for each 1 ´≤ i < j ≤ J,
PQ
PJ
1/2
,
(2) G
i=1 [[qi ]],
i=1 ki [[pi ]] ≤ C(Q)s0 /(Q − 1)
³ P
´
³ P
´
Q
J
(3) z ∈ QQ (Rn ) with G z, i=1 [[qi ]] ≤ s0 implies G z, i=1 ki [[pi ]] ≤ r,
³ ³P
´´
Q
(4) in case J = 1, diam spt
≤ C(Q)s0 /(Q − 1); here
i=1 [[qi ]]
£
¤1 £
¤2
£
¤Q−1
C(Q) = 1 + (2K)(Q − 1)2 + (2K)(Q − 1)2 + . . . + (2K)(Q − 1)2
.
Section 2.10 in [2]. Corresponding to
(1) J ∈ {1, 2, . . . , Q},
(2) k1 , k2 , . . . , kJ ∈ {1, 2, . . . , Q} with k1 + k2 + . . . + kJ = Q,
(3) distinct points p1 , p2 , . . . , pJ ∈ Rn ,
(4) 0 < s1 < s2 = 2−1 inf {|pi − pj | : 1 ≤ i < j ≤ J},
we set
P = QQ (Rn ) ∩ {
Q
X
[[qi ]] : q1 , . . . , qQ ∈ Rn with
i=1
card {i : qi ∈ B n (pj , s1 )} = kj for each j = 1, . . . , J}.
Then there exists a map Φ : QQ (Rn ) → P such that
´
³ P
J
(1) Φ(q) = q whenever q ∈ QQ (Rn ) with G q, i=1 ki [[pi ]] ≤ s1 ,
´
³ P
PJ
J
(2) Φ(q) = j=1 kj [[pj ]] whenever q ∈ QQ (Rn ) with G q, i=1 ki [[pi ]] ≥ s2 ,
´
³ P
J
(3) G (q, Φ(q)) ≤ G q, i=1 ki [[pi ]] for each q ∈ QQ (Rn ),
(4) Lip Φ ≤ 1 + Q1/2 s1 /(s2 − s1 ).
Theorem 2.12 in [2]. For z ∈ QQ (Rn ) with G(z, q0 ) > r,
(1) G(z, q) > s0 .
C(Q)s0
(2) G(z, q0 ) ≤ G(z, q) + G(q, q0 ) ≤ G(z, q) +
≤ [1 + C2 (Q)]G(z, q),
(Q − 1)1/2
C(Q)
where C2 (Q) =
.
(Q − 1)1/2
C(Q)s0
(3) G(z, q) ≤ G(z, q0 ) + G(q0 , q) ≤ G(z, q0 ) +
≤ [1 + C2 (Q)]G(z, q0 ).
(Q − 1)1/2
Additionally, we define
0 < s0 ≤ r ≤ s1 = K −1 s2 < s2 = 2−1 inf{|pi − pj | : 1 ≤ i < j ≤ J}.
Let Φ : QQ (Rn ) → QQ (Rn ) be the semi-retraction mapping constructed
in Section 2.10 in [2] corresponding to J, k1 , . . . , kJ , p1 , . . . , pJ , s1 , s2 above.
For each p ∈ QQ (Rn )
£
¤
G (Φ(p), p) ≤ G(q0 , p) = 2 G(q0 , p) − 2−1 G(q0 , p)
so that
14
JORDAN GOBLET AND WEI ZHU
(4) G(Φ(p), p) ≤ 2[G (p, q0 ) − s1 /2] .
Theorem A.1.6(17) in [2]. Suppose f ∈ Y2 (∂U m (0, 1), Rn ), q ∈ V, 0 < r < ∞,
and
A = ∂B m (0, 1) ∩ {x : |f (x) − q| > r}
with Hm−1 (A) ≤ mα(m)/4. Then
Z
2
2−1−4/(m−1) (2/π)2m−4+2/(m−1) (m − 1)β(m)−1
(|f (x) − q| − r) dHm−1
A
£
¤2/(m−1)
≤ Hm−1 (A)
dir(f ; A).
We are aiming to prove the following theorem:
Theorem 14. Suppose u ∈ Y2 (U m (0, 1), QQ (Rn )) is a strictly defined Dirichlet
(c, α)-almost minimizer such that Dir(u; U m (0, 1)) > 0. Then
0,σ
u ∈ Cloc
(U m (0, 1), QQ (Rn ))
for some 0 < σ < 1.
We first prove an energy growth estimate for points with small normalized energy.
This estimate is divided into two parts, on strong branch points (see definition
below) and non-strong-branch points. Then we show that the energy density is
zero for every point in the domain, which completes the interior regularity.
Definition 15. For a multiple-valued function u, we define the strong branch set
to be
Bu = {x ∈ U m (0, 1) : x is a Lebesgue point of ξ ◦ u,
ξ −1 ◦ ρ ◦ AVr,x (ξ ◦ u) = Q[[br ]], for any small enough radius r > 0, br ∈ Rn }.
R
where AVr,x (ξ ◦ u) := −∂U m (x,r) ξ ◦ u. Obviously, x ∈ Bu implies that u(x) = Q[[y]]
for some y ∈ Rn .
Lemma 16 (Hybrid Inequality). There is a positive constant C, depending only
on m, n, Q, c, α such that if 0 < λ < 1, 0 < ρ ≤ 1 and u is a strictly defined
(c, α)−almost minimizer, then
"
#
Z
(5)
Eρ/2 (u) ≤ λEρ (u) + C ρα + λ−1 ρ−m
|ξ ◦ u − µ|2 dx ,
U m (0,ρ)
for any constant vector µ ∈ RP Q , where Er (u) = r2−m Dir(u; U m (0, r)).
Proof. We first use Fubini’s Theorem as in Section 2.3 in [11], to obtain a radius
ρ/2 ≤ r ≤ ρ so that u|∂U m (0, r) ∈ ∂Y2 (∂U m (0, r), QQ (Rn )),
Z
Z
2
(6)
|∇tan u| dHm−1 ≤ 8
|Du|2 dx,
∂U m (0,r)
Z
U m (0,ρ)
Z
|ξ ◦ u − µ|2 dHm−1 ≤ 8
(7)
∂U m (0,r)
|ξ ◦ u − µ|2 dx.
U m (0,ρ)
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 15
Let h : U m (0, r) → QQ (Rn ) be Dirichlet minimizing with boundary values u|∂U m (0, r).
¿
À
Z
Z
∂(ξ ◦ h)
2
|Dh| dx =
ξ ◦ h,
dHm−1
∂r
U m (0,r)
∂U m (0,r)
¿
À
Z
∂(ξ ◦ h)
ξ ◦ h − µ,
=
dHm−1
∂r
∂U m (0,r)
#1/2
"Z
#1/2 "Z
¯ ¯2
¯ ∂h ¯
m−1
2
m−1
¯
¯
≤
|ξ ◦ h − µ| dH
¯ ¯ dH
∂U m (0,r)
∂B m (0,r) ∂r
"Z
#1/2 "Z
#1/2
¯ ¯2
¯ ∂h ¯
2
m−1
m−1
¯
¯
|ξ ◦ u − µ| dH
=
.
¯ ¯ dH
∂U m (0,r)
∂U m (0,r) ∂r
By Section 2.6 in [2],
¯ ¯2
Z
Z
Z
¯ ∂h ¯
2
m−1
¯ ¯ dHm−1 ≤
|∇tan h| dH
=
|∇tan u|2 dHm−1 .
¯ ¯
∂U m (0,r)
∂U m (0,r)
∂U m (0,r) ∂r
Therefore,
"Z
Z
2
|Dh| dx ≤
U m (0,r)
#1/2 "Z
2
|ξ ◦ u − µ| dH
#1/2
m−1
∂U m (0,r)
2
∂U m (0,r)
|∇tan u| dH
m−1
and
Eρ/2 (u) = (ρ/2)2−m Dir(u; U m (0, ρ/2))
≤ (ρ/2)2−m Dir(u; U m (0, r))
"Z
#
≤ (ρ/2)2−m
|Dh|2 dx + crm−2+α
U m (0,r)
"Z
#1/2 "Z
2−m
2
≤ (ρ/2)
∂U m (0,r)
|∇tan u| dH
#1/2
m−1
2
|ξ ◦ u − µ| dH
m−1
∂U m (0,r)
+ 2m−2 cρα .
Then we obtained the desired estimate by applying the inequality ab ≤ 12 δa2 +
1 δ −1 b2 , with δ = λ and using (6)(7) as follows
2
2m
Ã
!
Z
³ ρ ´2−m 1 Z
1 −1
2
m−1
2
m−1
Eρ/2 (u) ≤
δ
|∇ u| dH
+ δ
|ξ ◦ u − µ| dH
2
2 ∂U m (0,r) tan
2
∂U m (0,r)
+ 2m−2 cρα
!
à Z
Z
³ ρ ´2−m
2
2
−1
|ξ ◦ u − µ| dx + 2m−2 cρα
≤
4δ
|Du| dx + 4δ
2
U m (0,ρ)
U m (0,ρ)
Z
|ξ ◦ u − µ|2 dx + 2m−2 cρα
= λEρ (u) + 4m λ−1 ρ−m
U m (0,ρ)
"
#
Z
α
≤ λEρ (u) + C ρ + λ
where C = max{2m−2 c, 4m }.
2
−1 −m
|ξ ◦ u − µ| dx
ρ
U m (0,ρ)
¤
16
JORDAN GOBLET AND WEI ZHU
Let us introduce the family of functions that we will "blow up":
F = {u ∈ Y2 (U m (0, 1), QQ (Rn )) : u is (c, α) − almost minimizing and 0 ∈ Bu }
Lemma 17 (Energy Improvement). There are positive constants ²0 , r0 , η and θ < 1
so that, for any (c, α)−almost minimizer u ∈ F with Er0 (u) < ²20 , one has
Eθr (u) ≤ θω2.13 max {ηrα , Er (u)} , ∀0 < r < r0 ,
(8)
where the constant 0 < ω2.13 < 1 is defined by Theorem 2.13 in [2].
Proof. If this theorem were false, then for any fixed positive θ < 1/2, there would
exist (c, α)−almost minimizers ui ∈ F and ri → 0 for which
²2i := Eri (ui ) → 0
ri−α Eθri (ui ) → ∞
(9)
as i → ∞, but
Eθri (ui ) > θω2.13 ²2i
(10)
for all i. The above relation (9) clearly implies that
α
²−2
i ri → 0, as i → ∞.
We define the blowing-up sequence vi : U m (0, 1) → QQ (Rn ) by
£
¤
−1
vi (x) = µ(²−1
◦ ρ ◦ AVri ,0 (ξ ◦ ui ) ,
i )] ◦ ui (ri x) − ξ
where the subtraction makes sense because ξ −1 ◦ ρ ◦ AVri ,0 (ξ ◦ ui ) = Q[[bi ]] for some
bi ∈ Rn .
The Dirichlet energy of vi is clearly uniformly bounded by 1. As for the L2 norms,
we have
Z
¡
¢
G 2 ui (ri x) − ξ −1 ◦ ρ ◦ AVri ,0 (ξ ◦ ui ), Q[[0]] dx
U m (0,1)
Z
¡
¢
=
G 2 ui ◦ µ(ri )(x) − ξ −1 ◦ ρ ◦ AV1,0 (ξ ◦ ui ◦ µ(ri )), Q[[0]] dx
U m (0,1)
Z
−1 2
2
≤ Lip(ξ ) Lip(ρ)
|ξ ◦ ui ◦ µ(ri )(x) − AV1,0 (ξ ◦ ui ◦ µ(ri ))|2 dx
≤ C Lip(ξ
−1 2
U m (0,1)
2
) Lip(ρ) Dir (ξ ◦ ui ◦ µ(ri ); U m (0, 1))
= C Lip(ξ −1 )2 Lip(ρ)2 ²2i ,
where the second inequality comes from the Poincaré inequality (see Corollary 6.1
in [18]).
Using compactness theorem for multiple-valued functions (see Theorem 4.2 in [19]),
there is a subsequence of vi (still denoted as vi ) such that vi converges weakly to
v ∈ Y2 (U m (0, 1), QQ (Rn )). Moreover, by similar argument as in Section 6.4 of [18],
we can show that this convergence is actually strong and v is Dirichlet minimizing.
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 17
Let’s estimate the extra term in the hybrid inequality
Z
2
−
|ξ ◦ ui − AVr,0 (ξ ◦ ui ◦ µ(ri ))| dx
m
U (0,rri )
Z
2
=−
|ξ ◦ ui (ri x) − AVr,0 (ξ ◦ ui ◦ µ(ri ))| dx
m
U (0,r)
Z
2−m
|D(ξ ◦ ui ◦ µ(ri ))|2 dx (by Poincaré inequality)
≤ Cr
U m (0,r)
Z
2 2−m
= C(²i ) r
|Dvi |2 dx
U m (0,r)
Z
2 2−m
≤ C(²i ) r
|Dv|2 dx (by strong convergence)
U m (0,r)
2 2−m m−2+2ω2.13
≤ C(²i ) r
r
Dir(v; U m (0, 1)) (interior regularity of Dirichlet minimizer)
= C(²i )2 r2ω2.13 .
Applying the hybrid inequality to ui with ρ = 2θri , we get
"
#
Z
α
2
Eθri (ui ) ≤ λE2θri (ui ) + C (2θri ) + λ−1 −
|ξ ◦ ui − AV2θ,0 (ξ ◦ ui ◦ µ(ri ))| dx
U m (0,2θri )
£
¤
α
≤ λE2θri (ui ) + C (2θri ) + λ−1 C(²i )2 (2θ)2ω2.13
Choosing a positive integer k = k(θ) for which 2k θ ≤ 1 < 2k+1 θ, we iterate k − 1
more times to obtain (we suppress all universal constant as C)
Eθri (ui ) ≤ λk E2k θri (ui ) +
≤ λk 2m−2 ²2i +
k
X
k
¡
¢α X
¡ ¢2ω2.13
λj−1 C 2j θri +
λj−2 C(²i )2 2j θ
j=1
∞
X
j−1
λ
¡
¢α
C 2j θri +
j=1
j=1
∞
X
j−2
λ
2
C (²i ) (2j θ)2ω2.13
j=1
λ · 22ω2.13
2
Cλ−2 θ2ω2.13 (²i )2
= λk 2m−2 ²2i + C(θri )α
α +
1 − 2 λ 1 − λ · 22ω2.13
·
¸
¡
¢ 2α
λ · 22ω2.13
−2 2ω2.13
= λk 2m−2 + Cθα riα ²−2
+
Cλ
θ
(²i )2
i
1 − 2α λ 1 − λ · 22ω2.13
m + ω2.13
k
Taking λ = θ
, we have
α
λk · 2m−2 = θm+ω2.13 · 2m−2 = θm · 2m−2 · θω2.13 ≤ (1/2)m · 2m−2 θω2.13 ≤ θω2.13 /4
m + ω2.13
m + ω2.13
k
k
Since λ = θ
≤ (2−k )
= 2−(m+ω2.13 ) ,
m + ω2.13
−
λ · 22ω2.13
22ω2.13 C
−2 2ω2.13
k
θ
Cλ
θ
≤
θ2ω2.13
1 − 2ω2.13 −m
1 − λ · 22ω2.13
m + ω2.13
ω2.13 −
k
≡ Mθ
θω2.13 ,
where M =
22ω2.13 C .
1 − 2ω2.13 −m
Let’s choose θ small enough such that θ
ω2.13 −
m + ω2.13
k
≤ 1/4M . This is possible
18
JORDAN GOBLET AND WEI ZHU
because it is equivalent to
θω2.13 ≤ θ
m + ω2.13
k
/4M.
Notice that θ ≥ 2−1−k , the right side of above one is greater than
2−(k+1)(m+ω2.13 )/k /4M
which is bounded from below although when θ goes to zero, k goes to infinity.
α
Notice that ²−2
i ri → 0, as i → ∞, for i sufficiently large enough,we have
µ
¶
1 ω2.13 1 ω2.13 2
Eθri (ui ) ≤
θ
+ θ
²i < θω2.13 ²2i ,
4
4
contradicting the choice of ui in (10).
¤
Iteration of (8) as Section 3.5 of [11] leads to the following energy decay estimate.
Theorem 18 (Energy Rdecay for strong branch points). If u ∈ F is (c, α)-almost
minimizing, with r02−m U m (0,r0 ) |Du|2 ≤ ²20 , then
Z
2−m
r
|Du|2 ≤ Crα , for 0 ≤ r ≤ r0
U m (0,r)
where ²0 is as in the Energy Improvement.
Now we turn to non-strong-branch points and are going to prove an energy decay
estimate by induction on Q. In particular, without loss of generality, we assume
ξ −1 ◦ ρ ◦ AV1,0 (ξ ◦ u) =
6 Q[[y]] for any y ∈ Rn .
PJ
PQ
Let q ∗ = ρ◦AV1,0 (ξ◦u), q = ξ −1 (q ∗ ) = i=1 [[qi ]], and q0 = i=1 ki [[pi ]] is obtained
from q using Section 2.9 in [2]. By the argument in Section 3 of [18], J ≥ 2.
Lemma 19 (Construction of a comparison function). Assume m, n, Q ≥ 2 and u ∈
Y2 (U m (0, 1), QQ (Rn )) is a strictly defined (c, α)−almost minimizer. Let tQ > 0 be
a real number such that
·
¸2
2Lip(ρ)Lip(ξ −1 ) (1 + C2 (Q))
Lip(ξ)2 tm−1
≤ mα(m)/4.
Q
s0
If u satisfies dir(u; ∂U m (0, 1)) < tm−1
, then there is a comparison function g ∈
Q
Y2 (U m (0, 1), QQ (Rn )) satisfying the followings:
(1) g = u on ∂U m (0, 1),
PJ
(2) g|U m (0,1−tQ ) = i=1 gi , where gi ∈ Y2 (U m (0, 1 − tQ ), Qki (Rn )) is Dirichlet
PJ
minimizing and i=1 ki = Q,
(3) Dir(g; U m (0, 1) ∼ U m (0, 1 − tQ )) ≤ δQ dir(g; ∂U m (0, 1)),
for some constant δQ .
Proof. For each x ∈ B m (0, 1) ∼ B m (0, 1 − tQ ), we define τ : B m (0, 1) ∼ B m (0, 1 − tQ ) →
R and F, G, H : B m (0, 1) ∼ B m (0, 1 − tQ ) → RP Q by
τ (x) = t−1
Q (1 − |x|),
F (x) = ξ ◦ u(x/|x|),
H(x) = ξ ◦ Φ ◦ u(x/|x|),
G(x) = (1 − τ (x))F (x) + τ (x)H(x).
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 19
Define g : B m (0, 1) ∼ B m (0, 1 − tQ ) → QQ (Rn ) by
g|B m (0, 1) ∼ B m (0, 1 − tQ ) = ξ −1 ◦ ρ ◦ G|B m (0, 1) ∼ B m (0, 1 − tQ ).
PJ
On ∂B m (0, 1), g(x) = u(x). On ∂B m (0, 1 − tQ ), g(x) = Φ ◦ u(x/|x|) = i=1 hi ,
where hi ∈ ∂Y2 (∂B m (0, 1 − tQ ), Qki (Rn )), for each i = 1, . . . , J.
Let gi ∈ Y2 (U m (0, 1 − tQ ), Qki (Rn )) be a Dirichlet minimizing function with boundPJ
ary hi and g|B m (0, 1 − tQ ) = i=1 gi . This completes (1) and (2).
By the definition of g,
Z
|∇g|2 dx
U m (0,1)∼U m (0,1−tQ )
Z
¯
¯
¯∇(ξ −1 ◦ ρ ◦ G(x))¯2 dx
=
U m (0,1)∼U m (0,1−tQ )
£
¤2
≤ Lip(ξ −1 )Lip(ρ)
Z
2
|∇G(x)| dx
U m (0,1)∼U m (0,1−tQ )
£
¤2
= Lip(ξ −1 )Lip(ρ)
#
¯
¯
Z
¯ ∂G ¯2
2
¯ dx +
¯
|∇Tan G(x)| dx
¯
¯
U m (0,1)∼U m (0,1−tQ )
U m (0,1)∼U m (0,1−tQ ) ∂r
"Z
Also, by the definition of G, we can compute
¯
¯
¯ ¯
¯
¯ ∂G ¯ ¯¯ −1
¯ ¯ −1
¯
¯
¯
¯ ∂r ¯ = ¯tQ (F (x) − H(x))¯ = ¯tQ (ξ ◦ u(x/|x|) − ξ ◦ Φ ◦ u(x/|x|))¯ ,
and
¯
¯
¯
¯
¯∇∂U m (0,|x|) G(x)¯ = |x|−1 ¯∇∂U m (0,1) G(x)¯
¯
¯
¯ 1 − τ (x)
¯
τ (x)
= ¯¯
∇∂U m (0,1) F (x) +
∇∂U m (0,1) H(x)¯¯
|x|
|x|
¯
1 ¯¯
∇∂U m (0,1) (ξ ◦ u(x/|x|)) − ∇∂U m (0,1) (ξ ◦ Φ ◦ u(x/|x|))¯
≤
1 − tQ
¯
¯
1
≤
Lip(ξ) (1 + Lip(Φ)) ¯∇∂U m (0,1) u(x/|x|)¯ .
1 − tQ
Therefore,
(11)
Z
|∇G(x)|2 dx
U m (0,1)∼U m (0,1−t
Z
Q)
2
≤ t−2
Q
[ξ ◦ u(x/|x|) − ξ ◦ Φ ◦ u(x/|x|)] dx
U m (0,1)∼U m (0,1−t
−2
+ (1 − tQ )
Q)
[Lip(ξ)(1 + Lip(Φ))]
U m (0,1)∼U m (0,1−tQ )
Z
≤ t−1
Q
+
Z
2
¯
¯
¯∇∂U m (0,1) (u(x/|x|))¯2 dx
2
[ξ ◦ u(x/|x|) − ξ ◦ Φ ◦ u(x/|x|)] dHm−1
∂U m (0,1)
tQ
2
[Lip(ξ)(1 + Lip(Φ))] dir(u; ∂U m (0, 1)).
(1 − tQ )2
Now let us estimate
Z
2
[ξ ◦ u(x/|x|) − ξ ◦ Φ ◦ u(x/|x|)] dHm−1 .
∂U m (0,1)
20
JORDAN GOBLET AND WEI ZHU
Let Z = ∂U m (0, 1) ∩ {x : G(u(x), q0 ) > s1 }
¾
s1
(by Theorem 2.12 in [2])
Z ⊂ ∂U (0, 1) ∩ x : G(u(x), q) >
1 + C2 (Q)
½
¾
s1
m
∗
⊂ ∂U (0, 1) ∩ x : |ξ ◦ u(x) − q | >
Lip(ξ −1 )(1 + C2 (Q))
½
¾
s0
m
∗
⊂ ∂U (0, 1) ∩ x : |ξ ◦ u(x) − q | >
≡ A.
2Lip(ξ −1 )(1 + C2 (Q))
½
m
Therefore,
µ
H
m−1
(A)
Z
s0
−1
2Lip(ξ )(1 + C2 (Q))
¶2
|ξ ◦ u(x) − q ∗ |2 dHm−1
≤
∂U m (0,1)
(12)
Z
=
∂U m (0,1)
|ξ ◦ u(x) − ρ ◦ AV1,0 (ξ ◦ u)|2 dHm−1
Z
2
2
≤ Lip(ρ)
∂U m (0,1)
|ξ ◦ u(x) − AV1,0 (ξ ◦ u)| dHm−1
and
Hm−1 (Z) ≤ Hm−1 (A)
·
¸2 Z
2Lip(ρ) · Lip(ξ −1 )(1 + C2 (Q))
2
≤
|ξ ◦ u(x) − AV1,0 (ξ ◦ u)| dHm−1
s0
m
∂U (0,1)
·
¸2
−1
2Lip(ρ) · Lip(ξ )(1 + C2 (Q))
≤
Lip(ξ)2 dir(u; ∂U m (0, 1)),
s0
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 21
where the last inequality follows from [2] §A.1.6(3).
This permits us to estimate
Z
2
|ξ ◦ u − ξ ◦ Φ ◦ u| dHm−1
∂U m (0,1)
Z
2
=
|ξ ◦ u − ξ ◦ Φ ◦ u| dHm−1
Z
Z
2
[G(u, Φ ◦ u)] dHm−1
≤ Lip(ξ)2
Z
Z h
s1 i 2
2
≤ 4Lip(ξ)
G(u(x), q0 ) −
dHm−1 (by Theorem 2.12(4) in [2])
2
Z
Z h
s1 i 2
dHm−1
≤ 4Lip(ξ)2
(1 + C2 (Q)) G(u(x), q) −
2
Z
(because on Z, G(u(x), q0 ) > s1 ≥ s0 and Theorem 2.12(2) in [2])
¸2
Z ·
£
¤2
s1
−1
∗
≤ 4 Lip(ξ)Lip(ξ ) (1 + C2 (Q))
|ξ ◦ u(x) − q | −
dHm−1
2Lip(ξ −1 ) (1 + C2 (Q))
Z
¸2
Z ·
£
¤2
s0
≤ 4 Lip(ξ)Lip(ξ −1 )(1 + C2 (Q))
|ξ ◦ u(x) − q ∗ | −
dHm−1
2Lip(ξ −1 ) (1 + C2 (Q))
Z
(because s0 ≤ s1 )
£
¤2
£
¤2/(m−1)
≤ 4 Lip(ξ)Lip(ξ −1 )(1 + C2 (Q)) C1 Hm−1 (A)
dir(ξ ◦ u; A)
(by Theorem A.1.6(17) in [2] and the estimate of Hm−1 (A) in (12))
³
´
C1 = [2−1−4/(m−1) (2/π)2m−4+2/(m−1) (m − 1)β(m)−1 ]−1
≤ C (ξ, Φ, m, Q) dir (u; ∂U m (0, 1))
1+
2
m−1
In summary, we have
Z
2
|∇g| dx
U m (0,1)∼U m (0,1−t
Q)
"
1+
m
m
t−1
Q dir(g; ∂U (0, 1))
≤ C(ξ, Φ, m, n, Q)
"
= C(ξ, Φ, m, n, Q)
m
m
t−1
Q dir(g; ∂U (0, 1))
2
− 1 + tQ dir(g; ∂U m (0, 1))
#
#
2
− 1 + tQ dir(g; ∂U m (0, 1))
≤ 2C(ξ, Φ, m, n, Q) tQ dir(g; ∂U m (0, 1))
= δQ dir(g; ∂U m (0, 1)),
where δQ = 2C(ξ, Φ, m, n, Q)tQ .
¤
Remark.
(1) Here is the scaled version: Assume (c, α)−almost minimizer u ∈
Y2 (U m (0, r), QQ (Rn )) satisfies
dir(u; ∂U m (0, r)) < rm−3 tm−1
,
Q
then the constructed comparison function g with g = u on ∂U m (0, r) satisfies
Dir(g; U m (0, r) ∼ U m (0, r(1 − tQ ))) ≤ δQ rdir(u; ∂U m (0, r)).
22
JORDAN GOBLET AND WEI ZHU
(2) The smallness assumption of dir(u; ∂U m (0, r)) can be replaced by the smallness of Dir(u; U m (0, r)). This is because we choose g ∈ Y2 (U m (0, r), QQ (Rn ))
with g = u on ∂U m (0, r). By the squeeze formula ([2] §2.6) , we have
2rdir(u; ∂U m (0, r)) = 2rdir(g; ∂U m (0, r))
= (m − 2)Dir(g; U m (0, r)) + rDir(g; U m (0, r))0
≤ (m − 2)Dir(u; U m (0, r)) + rDir(g; U m (0, r))0
So in the blowing-up analysis, by choosing a good slicing by ∂U m (0, r) and
rescaling, we can get small Dir(g; U m (0, r))0 and hence the smallness of
dir(u; ∂U m (0, r)).
Theorem 20 (Energy decay for non-strong-branch points). Let m, n ≥ 2. There
exists a small number ² = ²(Q, m, n, c, α) such that any (c, α)−almost minimizer
u ∈ Y2 (U m (0, 1), QQ (Rn )) with 0 ∈
/ Bv such that r02−m Dir(u; U m (0, r0 )) < ² for
some 0 < r0 ≤ 1 satisfies
Dir(u; U m (0, r)) ≤ Crm−2+β , 0 < r ≤ r0
for some universal positive constants C, β.
Proof. Let g be the comparison function on B m (0, r),
Dir(u; U m (0, r)) ≤ Dir(g, U m (0, r(1 − tQ ))) + δQ rdir(g; ∂U m (0, r)) + crm−2+α
= Dir(g; U m (0, r(1 − tQ ))) + δQ rdir(u; ∂U m (0, r)) + crm−2+α
≤ Dir(g, U m (0, r(1 − tQ ))) + δQ rDir(u, U m (0, r))0 + crm−2+α
Applying the interior regularity of Dirichlet minimizing function to g|B m (0, r(1 − tQ )),
Dir(u; U m (0, r)) ≤ rm−2+2ω2.13 + δQ rDir(u; U m (0, r))0 + crm−2+α .
Denote ω = min{2ω2.13 , α}, N = 1 + c, D(r) = Dir(u; U m (0, r)), we have
D(r) ≤ δQ rD0 (r) + N rm−2+ω ,
i.e.
D0 −
D
N m−3+ω
+
r
≥ 0.
δQ r δQ
−1
Multiplying r δQ ,

d
dr

−1

D(r)r δQ +

m−2+ω− 1 
N
δQ  ≥ 0.
¶r
µ

1
δQ m − 2 + ω −
δQ
Therefore,
−1
D(r)r δQ +
m−2+ω− 1
N
N
δQ ≤ M := D(1)+ µ
¶r
¶
µ
1
1
δQ m − 2 + ω −
δQ m − 2 + ω −
δQ
δQ
1
D(r) ≤ M r δQ +
N
rm−2+ω
1 − δQ (m − 2 + ω)
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 23
½
Letting C = max M,
¾
½
¾
N
, β = min 1 − (m − 2), ω (which
δQ
1 − δQ (m − 2 + ω)
is positive since we can choose δQ small enough) finishes the proof.
¤
In spirit of Morrey’s growth lemma, it only remains to show Theorem 21 to prove
Theorem 14.
Theorem 21. For any (c, α)−almost minimizer u,
lim inf r2−m Dir(u; U m (x, r)) = 0, ∀x ∈ U m (0, 1).
r↓0
Remark. It is well-known (e.g [13] lemma 2.1.1) that the above one holds Hm−2
a.e. Also, it suffices to prove this for x = 0.
Lemma 22 (Monotonicity Formula). If u is a (c, α)−almost minimizer, then for
any 0 < t < s < 1, we have
¯ ¯2
Z s
Z
¯ ∂u ¯
¯ ¯ dHm−1 dr − (m − 2)c (sα − tα )
r2−m
¯ ¯
α
m
t
∂U (0,r) ∂r
≤ s2−m Dir(u; U m (0, s)) − t2−m Dir(u; U m (0, t)).
(m − 2)c α
r is nondecreasing in r. Hence the
α
Z
Θu (0) := lim r2−m
|Du|2 dx
In particular, r2−m Dir(u; U m (0, r)) +
limiting density
r↓0
U m (0,r)
exists.
Proof. For any 0 < r < 1, define a function g : B m (0, 1) → QQ (Rn ) by setting
¶
 µ
f r x
|x| ≤ r
|x|
g(x) =

f (x)
otherwise.
By definition of almost minimizer, we have
Z
|Du|2 dx
U m (0,r)
Z
m−2+α
≤ cr
+
|Dg|2 dx
U m (0,r)
Z
r
2
m−2+α
= cr
+
|∇tan u| dHm−1
m − 2 ∂U m (0,r)
"
#
¯ ¯2
Z
Z
¯ ∂u ¯
r
d
m−2+α
2
m−1
¯
¯
= cr
+
|ρ=r
|Du| dx −
¯ ¯ dH
m − 2 dρ
U m (0,ρ)
∂U m (0,r) ∂r
Multiplying by (m − 2)r1−m to both sides,
¯ ¯2
Z
¯ ∂u ¯
2−m
¯ ¯ dHm−1 − (m − 2)crα−1 ≤ d |ρ=r (ρ2−m Dir(u; U m (0, ρ))).
r
¯ ¯
dρ
m
∂U (0,r) ∂r
Then integrate this from t to s.
¤
24
JORDAN GOBLET AND WEI ZHU
From the monotonicity formula, following the standard technique (e.g. [16]),
we obtain a tangent map v ∈ Y2 (B m (0, 1), QQ (Rn )) of u at the origin. v is a
homogeneous degree zero Dirichlet minimizing function with
θv (0) = θu (0).
From the interior regularity, v has to be constant hence θu (0) = θv (0) = 0. This
finishes the proof of Theorem 21 and also the proof of Theorem 14.
4.2. Branch set of almost minimizer. We will now produce a Dirichlet almost
minimizer with a fractal branch set. We begin with some definitions:
(1) A losange above an interval I = [a, b] ⊂ R is a multiple-valued function
u : [a, b] → Q2 (R) whose the graph is a parallelogram with the following
vertices [a, 0], [b, 0], [(a + b)/2, (b − a)/2], [(a + b)/2, (a − b)/2] .
(2) A pluri-losange is a multiple-valued function u : [0, 1] → Q2 (R) which
admits a partition of [0, 1] in intervals {Ij } such that u is a losange above
Ij or a map of the form x → 2[[0]] above Ij for each j.
Figure 2 gives examples
√ of pluri-losanges. It is clear that a pluri-losange u is
Lipschitz with Lip(u) ≤ 2 and one readily checks that a pluri-losange is a Dirichlet
(2, 1/2)−almost minimizer. We also notice that a pluri-losange is not a Dirichlet
quasiminimizer or an ω-minimizer.
We will now create a particular sequence of pluri-losanges {ui }. We define u1 :
[0, 1] → Q2 (R) by requiring that
- u1 is a pluri-losange,
- u1 (1/3) = 0,
- u1 is a losange only above [1/3, 2/3].
We define u2 : [0, 1] → Q2 (R) by requiring that
- u2 is a pluri-losange,
- u2 (1/3) = 0,
- u2 is a losange only above the intervals [1/3, 2/3], [1/9, 2/9] and [7/9, 8/9].
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
−0.1
−0.1
−0.1
−0.2
−0.2
−0.2
−0.3
−0.3
−0.3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 2. The pluri-losanges u1 , u2 and u3 .
The reader will easily imagine how the remaining terms of the sequence are
defined. By mimicking the arguments used in Section 2.2, we can extract a subsequence which converges uniformly to a Lipschitz (2, 1/2)−almost minimizer u :
[0, 1] → Q2 (R) whose the branch set is the Cantor ternary set. This construction
can also be adapted in order to obtain an almost minimizer with a fat Cantor
branch set.
REGULARITY OF DIRICHLET NEARLY MINIMIZING MULTIPLE-VALUED FUNCTIONS 25
5. Acknowledgements
The authors are thankful to Thierry De Pauw and Robert Hardt for suggesting
the problem and helpful discussions.
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[13] F.H. Lin, X.Y. Yang, Geometric measure theory: an Introduction, International Press, 2003
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integrals of multiple valued functions, Trans. Amer. Math. Soc. 280 (1983), no. 2, 589-610.
[15] C. B. Morrey, Jr., Multiple integrals in the calculus of Variations, Springer-Verlag, Berlin
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Jordan Goblet, Département de mathématique, Université catholique de Louvain,
Chemin du cyclotron 2, 1348 Louvain-La-Neuve (Belgium)
E-mail address: [email protected]
Wei Zhu, Department of Mathematics, Rice University, 6100 Main Street, Houston, TX 77005 (U.S.A)
E-mail address: [email protected]