harmonic maps between alexandrov spaces -...

J Geom Anal (2017) 27:1355–1392DOI 10.1007/s12220-016-9722-y

Harmonic Maps Between Alexandrov Spaces

Jia-Cheng Huang1 · Hui-Chun Zhang1

Received: 13 July 2015 / Published online: 27 June 2016© Mathematica Josephina, Inc. 2016

Abstract In this paper, we shall discuss the existence, uniqueness and regularity ofharmonic maps from an Alexandrov space into a geodesic space with curvature � 1in the sense of Alexandrov.

Keywords Harmonic maps · Alexandrov spaces · Positive upper curvature bounds ·Global regularity

Mathematics Subject Classification 58E20 · 53C43

1 Introduction

After a remarkable work [13], the theory of harmonic maps into or between singularspaces has been studied extensively. In [17], Korevaar and Schoen introduced a con-cept of energy and Sobolevmaps for ametric space target, and developed a satisfactoryexistence and regularity theory for the Dirichlet problem of energy minimizers when-ever the target space has non-positive curvature in the sense of Alexandrov. See also[6–8,15,16,20] for related results. In general, in absence of conditions on the curvatureof the target, one does not have either existence or regularity of the minimizers.

Let (X, |· , · |), (Y, d) be two metric spaces and let � be a bounded domain (con-nected open subset) of X .μ is a Radon measure on X . Given p � 1, ε > 0 and a Borelmeasurable map u : � → Y , the approximating energy functional Eu

p,ε of u is givenas follows. For each compactly supported continuous function ϕ ∈ Cc(�), we set

B Jia-Cheng [email protected]

Hui-Chun [email protected]

1 Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

123

http://crossmark.crossref.org/dialog/?doi=10.1007/s12220-016-9722-y&domain=pdf

1356 J.-C. Huang, H.-C. Zhang

Eup,ε(ϕ) := Cn,p

∫�

ϕ(x)dμ(x)

∫Bε(x)∩�

d p(u(x), u(y))

εn+pdμ(y),

where Cn,p is a normalized constant. The p-th energy functional of u is defined by

Eup(ϕ) := lim sup

ε→0Eu

p,ε(ϕ), ∀ϕ ∈ Cc(�).

We say that u ∈ W 1,p(�, Y ) if u ∈ L p(�, Y ) and it has finite p-energy

supϕ∈Cc(�), 0�ϕ�1

Eup(ϕ) < ∞.

When � is a domain of a smooth manifold, and (Y, d) is an arbitrary metric space,Korevaar–Schoen [17] proved that, for any ϕ ∈ Cc(�), the limit limε→0 Eu

p,ε(ϕ)

exists. This is extended to the case where � is a domain of a Lipschitz manifold [12],or a domain of a polyhedra [8], or� is a domain of anAlexandrov spacewith curvaturebounded below [19].

Given a domain � of an Alexandrov space with curvature bounded below and ametric space (Y, d). A map u : � �→ Y is called a harmonic map if it is a localenergy minimizer of Eu

2 . Our purpose in this paper is to study the Dirichlet problemof harmonic maps, including the existence and regularity, from a bounded domain ofan Alexandrov space with curvature bounded below into a complete geodesic spaceof curvature � 1 in the sense of Alexandrov.

When the target (Y, d) is a non-positively curved space, by utilizing the convexityof distance function d(·, ·) on the product space Y × Y , the Dirichlet problem ofharmonic maps has been solved for a large variety of domains by [6,8,10,16,17,20],including a domain of an Alexandrov space.

Consider the case when (Y, d) has curvature � 1 in the sense of Alexandrov. If� is a smooth domain of a Riemannian manifold, the Dirichlet problem of harmonicmaps from � to Y has been solved by Serbinowski [26] for the case where the imagesof the maps are contained in a geodesic ball of Y with radius < π/2. In [17,26],the essential tool is a concept of directional energy, which is a generalization of thedirectional derivative of functions, defined by aC1-vector field on�. In fact, the energyof a Sobolev map in [17] is able to be represented as an integration of its directionalenergy. J. Eells and B. Fuglede [8,10,11] extended this method to the case where �

is a domain of a Riemannian polyhedra.It iswell known that the set of singular pointsmight be dense in a generalAlexandrov

space with curvature bounded from below [21].When� is a domain of an Alexandrovspace, since the C1 (even Lipschitz continuous) vector fields do not make sense, itseems difficult to employ directly the method of directional energy in the setting ofAlexandrov spaces. We need more arguments to conclude the following existenceresult when the domain is in an Alexandrov space. For a subset A ⊂ X and a mapu : A → Y , we denote by u(A) := {u(x) : x ∈ A}.Theorem 1.1 Let X be an Alexandrov space with curvature � k, � ⊂ X be a boundeddomain, and let Y be a complete geodesic space with curvature � 1 (in the sense of

123

Harmonic Maps Between Alexandrov Spaces 1357

Alexandrov). Given a ball Bρ(Q) ⊂ Y with ρ < π/2, and ϕ ∈ W 1,2(�, Y ) withϕ(�) ⊂ Bρ(Q), we write

W 1,2ϕ (�, Bρ(Q)) := {v ∈ W 1,2(�, Y ) : d(v, ϕ) ∈ W 1,2

0 (�), v(�) ⊂ Bρ(Q)}.

Then W 1,2ϕ (�, Bρ(Q)) has a unique element u of least 2-energy. It is called the har-

monic map on � which agrees ϕ on ∂�.

Next, we will consider the regularity of harmonic maps given in the above Theorem1.1. See, for example, [6,8,9,16,17,29] for the relational results for the case that thetarget has non-positive curvature. Let u : � → Y be a harmonic map from a smoothRiemannian domain� to a metric space with curvature�1, it is proved in [26] that u islocally Lipschitz continuous and globally Hölder continuous. J. Eells and B. Fuglede[8–11] proved the global Hölder continuity for harmonic maps from a domain withregular boundary of a Riemannian polyhedra to a metric space with curvature �1.

Here, we shall prove the global Hölder continuity for harmonicmaps from a domainof an Alexandrov space to a metric space with curvature � 1. Let � be a domain of anAlexandrov space X . It is said to satisfy the measure density condition, if there existsa constant C > 0 such that

μ(Br (x) ∩ �) � Cμ(Br (x)) (1.2)

for all x ∈ � and all 0 < r < min{1,Diam(�)}. Here Diam(�) := supx,y∈� |xy|.Theorem 1.3 Let X be an Alexandrov space with curvature � k and let � ⊂ X bea bounded domain such that both domains � and X\� satisfy the measure densitycondition, and let Y be a complete geodesic space with curvature � 1 (in the sense ofAlexandrov). Assume that w ∈ W 1,2(X, Y ) is Hölder continuous on � and the imagew(X) is contained in a geodesic ball Bρ(Q) ⊂ Y with radius ρ < π/2. Suppose thatu is the harmonic map on � which agrees w on ∂�. Then u is Hölder continuous on�.

Remark 1.4 (1) If� ⊂ M is a domain of a smoothmanifoldwithLipschitz continuousboundary ∂�, then both � and M\� satisfy the measure density condition.

(2) Let p be any point in an Alexandrov space X with curvature bounded below.Then there exists a number εp > 0 such that, for any r ∈ (0, εp), bothdomains Br (p) and X\Br (p) satisfy the measure density condition (see [28,from line −4 on p. 472 to line 3 on p. 473].

Recall that every harmonic map u : � ⊂ X → Y must be locally Lipschitzcontinuous in one of the following cases:

(a) X is an Alexandrov space with curvature bounded from below, and Y is a non-positively curved metric space (see [29]);

(b) X is a smooth manifold, Y is a metric space with curvature � 1, and the imageu(�) is contained in a geodesic ball with radius < π/2 (see [26]).

123


An interesting problem is to improve the Hölder continuity in Theorem 1.3 to theLipschitz continuity. A similar problem is asked in [20].

Organization of the paper. In Sect. 2, we will provide some necessary materials onAlexandrov spaces, Sobolev spaces for maps, and prove some necessary properties.In particular, we will prove that the energy measure of a W 1,p-map (p > 1) mustbe absolutely continuous with respect to the Hausdorff measure. In Sect. 3, we willdiscuss the existence of harmonic maps and prove Theorem 1.1. The regularity ofharmonic maps is included in the last section.

2 Preliminaries

2.1 Alexandrov Spaces

Let k ∈ R and l ∈ N. Denote by Mlk the simply connected, l-dimensional space form

of constant sectional curvature k. The space M2k is called k-plane.

Let (X, | · , · |) be a complete metric space. A rectifiable curve γ connecting twopoints p, q is called a geodesic if its length is equal to |pq| and it has unit speed. Ametric space X is called a geodesic space if, for every pair of points p, q ∈ X , thereexists some geodesic connecting them.

Fix any k ∈ R. Given three points p, q, r in a geodesic space X , we can take atriangle pqr in k-plane M

2k such that | pq| = |pq|, |qr | = |qr | and |r p| = |r p|.

If k > 0, we add the assumption that |pq| + |qr | + |r p| < 2π/√

k. The triangle pqr ⊂ M

2k is unique up to a rigid motion. We call it comparison triangle. We

let � k pqr denote the angle at the vertex q of the triangle pqr , and we call it ak-comparison angle.

Definition 2.1 Let k ∈ R. A geodesic space X is called an Alexandrov space withcurvature � k if it satisfies the following properties:

(1) it is locally compact;(2) for any point x ∈ X , there exists a neighborhood U of x such that the following

condition is satisfied: for any two geodesics γ (t) ⊂ U and σ(s) ⊂ U withγ (0) = σ(0) := p, the k-comparison angle

�κγ (t)pσ(s)

is non-increasing with respect to each of the variables t and s.

It iswell known that theHausdorff dimension of anAlexandrov spacewith curvature�k, for some constant k ∈ R, is always an integer (see, for example, [4] or [1]). LetX be an n-dimensional Alexandrov space with curvature �k. We denote by μ the n-dimensional Hausdorff measure on X . The Bishop inequality and the Bishop–Gromovinequality are satisfied on X , i.e., for every x ∈ X , the ratio

μ(Br (x))

V kr

� 1

123


and

μ(Br (x))

V kr

is non-increasing in r , where V kr is the volume of a ball of radius r in the space form

Mnk , cf. [1]. In particular, the doubling condition is satisfied on X .Let p ∈ X , given two geodesics γ (t) and σ(s) with γ (0) = σ(0) = p, the angle

� γ ′(0)σ ′(0) := lims,t→0

�κγ (t)pσ(s)

is well defined. We denote by �′p the set of equivalence classes of geodesic γ (t) with

γ (0) = p, where γ (t) is equivalent to σ(s) if � γ ′(0)σ ′(0) = 0. The completion ofmetric space (�′

p,� ) is called the space of directions at p, denoted by�p. The tangent

cone at p, Tp, is the Euclidean cone over �p. For two tangent vectors u, v ∈ Tp, their“scalar product” is defined by (see Sect. 1 in [23])

〈u, v〉 := 1

2(|u|2 + |v|2 − |uv|2).

For any δ > 0, we denote

X δ := {x ∈ X : vol(�x ) > (1 − δ) · vol(Sn−1)

},

where Sn−1 is the standard (n − 1)-sphere, and vol denotes the (n − 1)-dimensional

Hausdorff measure. X δ an open set (see [1]). The set Sδ := X\X δ is called the δ-singular set. Each point p ∈ Sδ is called a δ-singular point. The set

SX :=⋃δ>0

Sδ

is called singular set. A point p ∈ X is called a singular point if p ∈ SX . Otherwise itis called a regular point. Equivalently, a point p is regular if and only if Tp is isometricto R

n ([1]). It is proved in [1] that the Hausdorff dimension of SX is � n − 1. Weremark that the singular set SX might be dense in X ([21]).

Some basic structures of Alexandrov spaces are in the following.

Proposition 2.2 Let k ∈ R and let X be an n-dim Alexandrov space with curvature� k.

(1) There exists a constant δn,k > 0 depending only on the dimension n and k suchthat for each δ ∈ (0, δn,k), the set X δ forms a Lipschitz manifold ([1]) and has aC∞-differentiable structure ([18]).

(2) There exists a BVloc-Riemannian metric g on X δ such that• the metric g is continuous in X\SX ([21,22]);• the distance function on X\SX induced from g coincides with the original one

of X ([21]);

123


• the Riemannian measure on X\SX induced from g coincides with the Hausdorffmeasure of X ([21]).

Definition 2.3 ([1]) The boundary of an Alexandrov space X is defined inductivelywith respect to dimension. If the dimension of X is one, then X is a completeRiemannian manifold and the boundary of X is defined as usual. Suppose that thedimension of X is n � 2. A point p is a boundary point of X if �p has non-emptyboundary.

From now on, we always consider Alexandrov spaces without boundary.

2.2 Sobolev Spaces and Laplacian

Several different notions of Sobolev spaces have been established on metric spaces;see [5,14,17–19,27].1 They coincide with each other on Alexandrov spaces.

From now on, we assume that X is an n-dim Alexandrov space with curvature � k,� ⊂ X is a domain of X . It is well known that the metric measure space (X, | · , · |, μ)

supports a local (weak) Poincaré inequality.We denote by Liploc(�) the set of locally Lipschitz continuous functions on�, and

by Lipc(�) the set of Lipschitz continuous functions on � with compact support.For a continuous function u : X → R, define

Lipu(x) = lim supy→x

|u(y) − u(x)||yx | .

Definition 2.4 For any 1 � p � +∞ and u ∈ Liploc(�), its W 1,p(�)-norm isdefined by

‖u‖W 1,p(�) := ‖u‖L p(�) + ‖Lipu‖L p(�).

Sobolev spaces W 1,p(�) are defined by the closure of the set

{u ∈ Liploc(�) : ‖u‖W 1,p(�) < +∞},

under W 1,p(�)-norm. Spaces W 1,p0 (�) are defined by the closure of Lipc(�) under

W 1,p(�)-norm. We denote by W 1,pc (�) = { f ∈ W 1,p

0 (�) : f has compact support}.We say that a function f ∈ W 1,p

loc (�) if f ∈ W 1,p(�′) for every open subset�′ ⊂⊂ �.

Cheeger [5, Theorem 4.48] proved that W 1,p(�) is reflexive when 1 < p < ∞. Wedenote by ∇u the weak gradient of u ∈ W 1,p(�).

1 In [5,14,27], Sobolev spaces are defined on metric measure spaces supporting a doubling condition anda Poincaré inequality.

123


We recall the chain derivation property of W 1,2(�). It is formulated as follows: Forf, g ∈ W 1,2(�)∩ L∞(�) and : R → R which is C1 on the range of f , then ( f )

belongs to W 1,2(�) and

〈∇ ( f ),∇g〉 = ′( f ) · 〈∇ f ,∇g〉 μ-a.e.

Given f ∈ W 1,2loc (�), the Laplacian of f is defined as a functional on Lipc(�) by

L f (ϕ) := −∫

�

〈∇ f ,∇ϕ〉 dμ, ∀ ϕ ∈ Lipc(�).

Given h ∈ L1loc(�). A function f ∈ W 1,2

loc (�) is said to satisfy the inequality

L f � hdμ

in the sense of distribution if the inequality

L f (ϕ) �∫

�

hϕdμ

holds for all 0 � ϕ ∈ Lipc(�). In this case, the functional L f is a signed Radonmeasure.

Remark 2.5 Moreover, the measure L f can be extended to a functional on ϕ ∈L∞(�) ∩ W 1,2

c (�).Indeed, let us fix arbitrarily a compact set K ⊂ �. Take a function �K ∈ Lipc(�)

with �K ≡ 1 on K . Given any function ϕ ∈ Lipc(K ), we have

0 � ‖ϕ‖L∞ · �K ± ϕ ∈ Lipc(�),

and then

L f

(‖ϕ‖L∞ · �K ± ϕ

)�

∫�

h ·(‖ϕ‖L∞�K ± ϕ

)dμ

� ‖ϕ‖L∞ ·∫supp�K

|h|(�K + 1)dμ.

This follows

±L f (ϕ) � ‖ϕ‖L∞ · (2‖h‖L1(supp�K ) + |L f (�K )|) := C · ‖ϕ‖L∞ .

Thus, since suppϕ ⊂ K , we have

∣∣∣∣∫

K〈∇ f,∇ϕ〉dμ

∣∣∣∣ = |L f (ϕ)| � C · ‖ϕ‖L∞ .

123


At last, any function φ ∈ W 1,2c (�) ∩ L∞(�) can be W 1,2(�)-approximated by a

sequence of functions ϕ j ∈ Lipc(�) with ‖ϕ j‖L∞ � ‖φ‖L∞ (see, for example, [5,Theorem 4.24]).

2.3 Energy Functional and Sobolev Spaces into Metric Space

From now on, we assume that � is a domain of an Alexandrov space and that (Y, d) isa complete metric space. Fix any p ∈ [1,∞). Fix a point P ∈ Y . A Borel measurablemap u : � → Y is said to be in the space L p(�, Y, P) if it has separable range and

∫�

d p(u(x), P)dμ(x) < ∞.

If μ(�) < ∞, the space L p(�, Y, P) does not depend on the choice of the pointP ∈ Y . In this case, we denote the space by L p(�, Y ). The space L p(�, Y ) is ametric space under the distance

dp( f, g) =(∫

�

d p( f (x), g(x))dμ(x)

) 1p

.

The space (L p(�, Y ), dp) forms a complete metric space.

Definition 2.6 For u ∈ L p(�, Y ), the approximating energy Eup,ε is defined by

Eup,ε(ϕ) =

∫�

ϕ(x)eup,εdμ(x),

for any ϕ ∈ Cc(�), where the approximating density of u is defined by

eup,ε(x) = n + p

cn,pεn

∫Bε(x)∩�

d p(u(x), u(y))

ε pdμ(y)

where the constant cn,p = ∫Sn−1 |x1|pσ(dx), and σ is the canonical volume on S

n−1.

Given any ϕ ∈ Cc(�), it is easy to check that, for any sufficiently small ε > 0, theapproximating energy Eu

p,ε(ϕ) coincides (up to a constant) with the one defined byKuwae and Shioya in [19], that is,

Eup,ε(ϕ) := n

2ωn−1εn

∫�

ϕ(x)

∫Bε(x)∩�

d p(u(x), u(y))

ε p· IQ(�)(x, y)dμ(y)dμ(x),

where

Q(�) := {(x, y) ∈ � × � : |xy| < |γxy, ∂�|, ∀geodesic γxy from x to y

},

123


and ωn−1 is the volume of Sn−1. It is proved in [19] that, for each ϕ ∈ Cc(�), the

limit

Eup(ϕ) := lim

ε→0+ Eup,ε(ϕ)

exists. We call Eup(ϕ) the energy functional of u, defined on Cc(�).

For u ∈ L p(�, Y ), the p-energy of u is defined by

Eup(�) := sup

φ∈Cc(�),0�ϕ�1Eu

p(ϕ).

Definition 2.7 We define the Sobolev space from � to Y by

W 1,p(�, Y ) ={

u ∈ L p(�, Y ) : Eup(�) < ∞

}.

Notice that the space of Lipschitz maps Lip(�, Y ) ⊂ W 1,p(�, Y ).

Proposition 2.8 ([19]) Let 1 < p < ∞ and u ∈ W 1,p(�, Y ). Then the followingassertions (1)–(5) hold.

(1) (Contraction property, [19, Lemma 3.3]) Consider another complete metric space(Z , dZ ) and a Lipschitz map ψ : Y → Z, we have ψ ◦ u ∈ W 1,p(�, Z) and

Eψ◦up (ϕ) � |ψ |p

Lip · Eup(ϕ)

for any 0 � ϕ ∈ Cc(�), where

|ψ |Lip := supy,y′∈Y, y �=y′

dZ (ψ(y), ψ(y′))d(y, y′)

.

In particular, for any point Q ∈ Y , we have d(Q, u(·)) ∈ W 1,p(�, R) and

Ed(Q,u(·))p (ϕ) � Eu

p(ϕ)

for any 0 � ϕ ∈ Cc(�).(2) (Lower semi-continuity, [19, Theorem 3.2]) For any sequence u j → u in

L p(�, Y ) as j → ∞, we have

Eup(ϕ) � lim inf

j→∞ Eu jp (ϕ)

for any 0 � ϕ ∈ Cc(�).(3) (Energy measure, [19, Theorem 4.1 and Proposition 5.1]) There exists a finite

Borel measure, denoted by Eup again, on �, which is called the energy measure

of u, such that for any 0 � ϕ ∈ Cc(�),

Eup(ϕ) =

∫�

ϕ(x)dEup(x).

123


Furthermore, the measure is strongly local. That is, for any nonempty open subsetO ⊂ �, we have u|O ∈ W 1,p(O, Y ), and moreover, if u is a constant map almosteverywhere on O, then Eu

p(O) = 0.(4) (Weak Poincaré inequality, [19, Theorem 4.2(ii)]) For any open set O = BR(Q)

with B6R(Q) ⊂⊂ �, there exists a positive constant C = C(n, k, R) such thatthe following holds: for any z ∈ O and any 0 < r < R/2, we have

∫Br (z)

∫Br (z)

d p(u(x), u(y))dμ(x)dμ(y) � Crn+2 ·

∫B6r (z)

dEup(x),

where the constant C given in [19, p. 61] depends only on the constants R, ϑ , and� in Definition 2.1 for WMCPBG condition in [19]. In particular, for the case ofAlexandrov spaces as shown in the proof of Theorem 2.1 in [19], one can chooseR > 0 arbitrarily, ϑ = 1 and

� = sup0<r<R

vol(Br (o) ⊂ Mnk )

vol(Br (o) ⊂ Rn)= C(n, k, R).

(5) (Equivalence for Y = R, [19, Theorem 6.2]) If Y = R, the above Sobolev spaceW 1,p(�, R) is equivalent to the Sobolev space W 1,p(�) given in Definition 2.4.Precisely, for any u ∈ W 1,p(�, R), the energy measure Eu

p is absolutely contin-uous with respect to μ and

dEup

dμ= |∇u|p.

For a subset A ⊂ X , denote by

Diam(A) := supx,y∈A

|xy|.

Lemma 2.9 Let � ⊂ X be a bounded domain. Then there exist positive constantsC1, C2 such that

C2 � μ(Br (x))

rn� C1

for any ball Br (x) ⊂ �.

Proof Set D := Diam(�). Using the Bishop inequality, we have, for any ball Br (x) ⊂�,

μ(Br (x)) � V kr � sup

0<r�D

V kr

rn· rn := C1(n, D) · rn,

where V kr is the volume of a ball of radius r in the space form M

nk .

123


For any ball Br (x) ⊂ �, by using the Bishop–Gromov inequality, we have, for all0 < r � D,

μ(Br (x))

V kr

� μ(BD(x))

V kD

.

Note that � ⊂ BD(x), thus we have

μ(Br (x))

rn� μ(�)

V kD

· V kr

rn� inf

0<r�D

V kr

rn· μ(�)

V kD

:= C2(n,�).

Now the proof is complete. ��To simplify, we use the following notation throughout this paper. Given a μ-

measurable function f and a μ-measurable subset A, we denote by

A

f dμ := 1

μ(A)

∫A

f dμ.

2.4 Absolute Continuity of the Energy Measures

We continue to assume that � is a domain of an Alexandrov space and that (Y, d) isa complete metric space.

In [24], Reshetnyak proposed a similar approach to define the Sobolev spaces formaps into a metric space. Given 1 � p < ∞, by the definition in [24], a mapu ∈ L p(�, Y ) belongs to Reshetnyak–Sobolev space, denoted by R1,p(�, Y ), if forevery Lipschitz function ψ : Y → R the composition ψ ◦ u ∈ W 1,p(�), and thereis a function w ∈ L p(�) (independent of ψ) such that, for every Lipschitz functionψ : Y → R, the inequality holds

|∇(ψ ◦ u)|(x) � |ψ |Lip · w(x) μ−a.e. x ∈ �.

Lemma 2.10 ([14]) Let 1 < p < ∞, we have W 1,p(�, Y ) ⊂ R1,p(�, Y ).

Proof The lemma is due to Heinonen–Koskela–Shanmugalingam–Tyson [14] essen-tially. For completeness, we include a proof here.

Let u ∈ W 1,p(�, Y ). Given any Lipschitz function ψ : Y → R, we have thecomposition ψ ◦ u ∈ W 1,p(�), because of Proposition 2.8(1) and (5). According toProposition 2.8(1), we know that the energy measure of u and ψ ◦ u satisfies

Eψ◦up � |ψ |p

Lip · Eup. (2.11)

Consider the Lebesgue decomposition of Eup with respect to μ on �,

Eup = |∇u|p · μ + (Eu

p)s,

123


where |∇u|p ∈ L1(�), because Eup(�) < ∞. By Proposition 2.8(5), the measure

Eψ◦up is absolutely continuous w.r.t. μ, and has density |∇(ψ ◦ u)|p. From (2.11), we

have

|∇(ψ ◦ u)|p · μ � |ψ |pLip · |∇u|p · μ + |ψ |p

Lip · (Eup)

s .

The singular part (Eup)

s is supported in a μ-zero measure set. Then we have

|∇(ψ ◦ u)|(x) � |ψ |Lip · |∇u|1/pp (x) μ−a.e. x ∈ �.

Therefore, u ∈ R1,p(�, Y ). ��Remark 2.12 If � is a bounded domain of a smooth manifold, it is shown in [14,25]that W 1,p(�, Y ) = R1,p(�, Y ). It is interesting to clarify whether the assertion isstill valid when � is a bounded domain of an Alexandrov space.

Proposition 2.13 (Absolute continuity for the energy measures) Let 1 < p < ∞.For each u ∈ W 1,p(�, Y ), the energy measure Eu

p is absolutely continuous w.r.t the

volume μ, i.e., there exists |∇u|p ∈ L1(�) such that

dEup

dμ= |∇u|p.

Proof By [14, Corollary 3.21] that R1,p(�, Y ) = N 1,p(�, Y ) ⊂ N 1,p(X, V :=�∞(Y )), where N 1,p(�, Y ) is the Newtonian–Sobolev space in [14, Definition 3.9]by the upper gradient. From Lemma 2.10, we have u ∈ N 1,p(X, V ). By [14, Propo-sition 4.6], we have

d(u(x), u(y)) � C(n,�)(Mg(x) + Mg(y)) · |xy|,

for μ-a.e. x, y ∈ �, where Mg(x) is the maximal function of g defined by

Mg(x) = supr>0

Br (x)

g(y)dμ(y),

and g ∈ L p(�) is a weak upper gradient in the sense of [14, Definition 3.9]. Denoteby h = Mg for convenience. Without loss of generality, for any 0 � ϕ ∈ Lipc(�),we have

cn,p

n + pEu

p,ε(ϕ) =∫

�

ϕ(x)dμ(x)

∫Bε(x)

d p(u(x), u(y))

εn+pdμ(y)

� C(n, p,�)

∫�

ϕ(x)dμ(x)

∫Bε(x)

h p(x) + h p(y)

εndμ(y) (2.14)

123


� C(n, p,�)

∫�

ϕ(x)h p(x)dμ(x)

+ C(n, p,�)

∫�

ϕ(x)dμ(x)

∫Bε(x)

h p(y)

εndμ(y),

where we have used that μ(Bε(x)) � C2εn . Set

|ϕ|Lip = supx,y∈�,x �=y

|ϕ(y) − ϕ(x)|d(x, y)

.

Define a function Iε(x, y) on � × � by

Iε(x, y) ={1, if |xy| < ε,

0, if |xy| � ε.

Note that Iε(x, y) = Iε(y, x). On the other hand,

∫�

ϕ(x)dμ(x)

∫Bε(x)

h p(y)

εndμ(y)

=∫

�×�

ϕ(x)h p(y)

εnIε(x, y)dμ(y)dμ(x)

�∫

�×�

(ϕ(y) + |ϕ|Lip · |xy|)h p(y)

εnIε(x, y)dμ(y)dμ(x)

=∫

�×�

(ϕ(y) + |ϕ|Lip · |xy|)h p(y)

εnIε(x, y)dμ(x)dμ(y)

=∫

�

h p(y)dμ(y)

∫Bε(y)

ϕ(y) + |ϕ|Lip · |xy|εn

dμ(x)

�∫

�

h p(y)dμ(y)

∫Bε(y)

ϕ(y) + |ϕ|Lip · ε

εndμ(x)

� C∫

�

(ϕ(y) + |ϕ|Lip · ε)h p(y)dμ(y)

= C∫

�

ϕ(y)h p(y)dμ(y) + Cε

∫�

|ϕ|Lip · h p(y)dμ(y).

(2.15)

Note that by the classical Hardy–Littlewood estimate, h = Mg ∈ L p(�). Then, bycombining (2.14) and (2.15), and taking ε → 0, we obtain that

Eup(ϕ) � C

∫�

ϕ(x)h p(x)dμ(x).

Hence, the proof is complete. ��Recall that a result about the point-wise convergence of approximating density is

given in [29, Corollary 4.6].

123


Lemma 2.16 ([29]) Let 1 < p < ∞ and u ∈ W 1,p(�, Y ). Then, for any sequenceof positive numbers {εi } converging to 0, there is a subsequence {εi } ⊂ {εi } such that

limi→∞ eu

p,εi(x) = |∇u|p(x) μ−a.e. x ∈ �.

Combining the absolute continuity of energy measure and the above point-wise con-vergence result, we can obtain a locally L1-convergence of the approximating density.

Proposition 2.17 Let 1 < p < ∞ and u ∈ W 1,p(�, Y ). Then we have

eup,ε → |∇u|p in L1

loc(�),

as ε → 0.

Proof It suffices to show that for any ball B ⊂⊂ � we have eup,ε → |∇u|p in L1(B).

We argue it by contradiction. Suppose that there exists some ball B ⊂⊂ �, such that

eup,ε � |∇u|p in L1(B),

as ε → 0. Then there exist δ0 > 0 and a sequence of positive numbers {εi }, going to0, such that ∫

B|eu

p,εi− |∇u|p| � δ0, (2.18)

for all large enough i � 1.Notice that μ(∂ B) = 0. By Proposition 2.13, we have Eu

p(∂ B) = 0. Recall thatEu

p,εi= eu

p,εidμ ⇀ Eu

p = |∇u|pdμ weakly as Radon measures on �. Thus, we have

limi→∞ Eu

p,εi(B) = Eu

p(B).

That is,

limi→∞

∫B

eup,εi

dμ =∫

B|∇u|pdμ. (2.19)

By Lemma 2.16, there is a subsequence {εi } ⊂ {εi } such that

limi→∞ eu

p,εi(x) = |∇u|p μ−a.e. x ∈ �. (2.20)

Thus, by combination of (2.19) and (2.20) implies that

eup,εi

→ |∇u|p in L1(B).

This contradicts with (2.18). Hence, the proof is complete. ��

123


3 Harmonic Maps into Space of Curvature Bounded from Above

We continue to assume that � is a domain of an Alexandrov space and that (Y, d) is acomplete metric space. In this section, we will prove the existence of harmonic maps,Theorem 1.1.

3.1 Some Representations of Energy Functionals

We will give some technical lemmas about the representation of energy functionals.

Lemma 3.1 For any fixed α > 0 and u ∈ W 1,2(�, Y ), we have for any v ∈W 1,2(�, Y ) and ϕ ∈ Cc(�)

cn,2

n + 2Ev2 (ϕ) = lim

ε→0

∫�

ϕ(x)dμ(x)

∫{y∈Bε(x):d(u(x),u(y))<α}

d2(v(x), v(y))

εn+2 dμ(y).

(3.2)

Proof Step 1. We firstly show that (3.2) holds in the case where � is a bounded ofthe Euclidean space R

n .Let ω ∈ S

n−1 be an unit vector. The corresponding directional energy ω Ev2 of v is

defined by (see [17, Theorem 1.8.1])

ω Ev2 (ϕ) := lim

ε→0

∫�

d2(v(x), v(x + εω))

ε2ϕ(x)dμ(x).

According to [26, Corollary 1.5], we have

d(v(x), v(x + εω))

ε→ |v∗(ω)|(x) in L2

loc(�), (3.3)

as ε → 0, for some |v∗(ω)|(x) ∈ L2(�).We denote by �ε := {x ∈ � : |x, ∂�| > ε} and Aε

u = {x ∈ �ε : d(u(x), u(x +εω)) � α}.Sublemma 3.4 For any ϕ ∈ Cc(�), we have

ω Ev2 (ϕ) = lim

ε→0

∫�\Aε

u

d2(v(x), v(x + εω))

ε2ϕ(x)dμ(x). (3.5)

Proof of Sublemma By definition, we have

ω Ev2 (ϕ) = lim

ε→0

∫�\Aε

u

d2(v(x), v(x + εω))

ε2ϕ(x)dμ(x)

+ limε→0

∫Aε

u

d2(v(x), v(x + εω))

ε2ϕ(x)dμ(x).

123


However, we have∣∣∣∣∣∫

Aεu

d2(v(x), v(x + εω))

ε2ϕ(x)dμ(x)

∣∣∣∣∣ � |ϕ|∞∫

Aεu∩suppϕ

d2(v(x), v(x + εω))

ε2dμ(x)

converges to 0, because of (3.3) and thatμ(Aεu ∩ suppϕ) → 0. Hence, we have proved

this sublemma. ��We now continue the proof of Lemma 3.1. Fix any ϕ ∈ Cc(�). Without loss of

generality, we may assume that ϕ � 0. Define a function I εu (x, ω) on �ε × S

n−1 by

I εu (x, ω) =

{1, if d(u(x), u(x + εω)) < α,

0, if d(u(x), u(x + εω)) � α.

It follows from (3.5) that (by [17, Theorem 1.8.1])

cn,2 · Ev2 (ϕ) =

∫Sn−1

ω Ev2 (ϕ)dσ(ω)

=∫Sn−1

dσ(ω) limε→0

∫�

d2(v(x), v(x + εω))

ε2I εu (x, ω)ϕ(x)dμ(x).

(3.6)

By [17, Lemma 1.8.2], we have

∫�

d2(v(x), v(x + εω))

ε2ϕ(x)dμ(x) � C |ϕ|∞Ev

2 (�), (3.7)

for some constant C > 0, independent of ω. Hence, by (3.6), (3.7) and the boundedconvergence theorem, we have

cn,2 · Ev2 (ϕ) = lim

ε→0

∫Sn−1

dσ(ω)

∫�

d2(v(x), v(x + εω))

ε2I εu (x, ω)ϕ(x)dμ(x).

(3.8)

Applying Fubini’s theorem to (3.8), we obtain that


ε→0

∫�

ϕ(x)dμ(x)

∫Sn−1

d2(v(x), v(x + εω))

ε2I εu (x, ω)dσ(ω).

(3.9)

Let S(x, ε) be the sphere centered at x with radius ε, i.e., S(x, ε) := {y : |xy| = ε},and dσx,ε be the area volume on S(x, ε). Define K ε

u,x (y) be the characteristic functionof {y ∈ S(x, ε) : d(u(x), u(y)) < α}. We obtain from (3.9) that


ε→0

∫�

ϕ(x)dμ(x)

∫S(x,ε)

d2(v(x), v(y))

εn+1 K εu,x (y)dσx,ε(y). (3.10)

123


Divding both sides of (3.10) by n + 2, we have

cn,2

n + 2Ev2 (ϕ)

=∫ 1

0λn+1dλ lim

ε→0

∫�

ϕ(x)dμ(x)

∫S(x,λε)

d2(v(x), v(y))

(λε)n+1 K λεu,x (y)dσx,λε(y).

(3.11)

According to [17, Theorem 1.5.1], there exists ε0 > 0, such that

∫�

ϕ(x)dμ(x)

∫S(x,δ)

d2(v(x), v(y))

δn+1 dσx,δ(y) � 1 + Ev2 (ϕ)

for any 0 < δ < ε0. Thus, by applying the bounded convergence theorem to (3.11),we have

cn,2

n + 2Ev2 (ϕ)

= limε→0

∫ 1

0λn+1dλ

∫�

ϕ(x)dμ(x)

∫S(x,λε)

d2(v(x), v(y))

(λε)n+1 K λεu,x (y)dσx,λε(y)

= limε→0

∫ 1

0dλ

∫�

ϕ(x)dμ(x)

∫S(x,λε)

d2(v(x), v(y))

εn+1 K λεu,x (y)dσx,λε(y)

= limε→0

∫�

ϕ(x)dμ(x)

∫ 1

0dλ

∫S(x,λε)

d2(v(x), v(y))

εn+1 K λεu,x (y)dσx,λε(y).

(3.12)

Note that dμ = εdλdσx,λε. Thus, we obtain from (3.12) that

cn,2

n + 2Ev2 (ϕ) = lim

ε→0

∫�

ϕ(x)dμ(x)


d2(v(x), v(y))

εn+2 dμ(y).

Equivalently,

limε→0

∫�

ϕ(x)dμ(x)

∫{y∈Bε(x):d(u(x),u(y))�α}

d2(v(x), v(y))

εn+2 dμ(y) = 0. (3.13)

Hence, we have proved this lemma for the case where the domain � ⊂ Rn .

Step 2. We secondly show that (3.2) holds in the case where� is a bounded domainof aLipschitzmanifoldwith an L∞-Riemannianmetric. Equivalently,wewant to show

limε→0

∫�

ϕ(x)dμ(x)


d2(v(x), v(y))

εn+2 dμ(y) = 0. (3.14)

123


By compactness of � and the partition of unit, we may assume that � is contained insome local chart (U, ψ). Notice that ψ is bi-Lipschitz, i.e.,

m · |xy| � |ψ(x) − ψ(y)|Rn � M · |xy|

for some constants m, M > 0. Denote by Ber (x) the ball centered at x ∈ R

n withradius r , and by g = det(gi j )n×n . Without loss of generality, for any 0 � ϕ ∈ Cc(�),

∫�

ϕ(x)dμ(x)


d2(v(x), v(y))

εn+2 dμ(y)

�∫

ψ(�)

ϕ ◦ ψ−1(x ′)√

g(x ′)dx ′

∫{y′∈Be

Mε(x ′):d(u◦ψ−1(x ′), u◦ψ−1(y′))�α}d2(v ◦ ψ−1(x ′), v ◦ ψ−1(y′))

εn+2

√g(y′)dy′

= Mn+2∫

ψ(�)

ϕ ◦ ψ−1(x ′)√

g(x ′)dx ′

∫{y′∈Be

Mε(x ′):d(u◦ψ−1(x ′), u◦ψ−1(y′))�α}d2(v ◦ ψ−1(x ′), v ◦ ψ−1(y′))

(Mε)n+2

√g(y′)dy′

→ 0, as ε → 0 (by (3.13)).

Hence, (3.14) holds true.Step 3. At last, we shall finish the lemma by proving (3.2) in the case where � is

a bounded domain of an Alexandrov space.Fix δ ∈ (0, δn,k), where δn,k is defined in Proposition 2.2. Let �δ = {x ∈ � :

vol(�x ) > (1 − δ)vol(Sn−1)}. �δ is an open subset of �. Let K δ = �\�δ . Thenμ(K δ) = 0. Furthermore, by Proposition 2.2, �δ forms a Lipschitz manifold.

Hence we have for each ϕ ∈ Cc(�δ),

limε→0

∫�

ϕ(x)dμ(x)


d2(v(x), v(y))

εn+2 dμ(y) = 0. (3.15)

We now prove that (3.15) still holds for each ϕ ∈ Cc(�). Fix any ϕ ∈ Cc(�).Without loss of generality, we may assume that ϕ � 0. For any γ > 0 sufficientlysmall, we can choose f ∈ Cc(�

δ), such that 0 � f � ϕ and

Ev2 (ϕ) − Ev

2 ( f ) � κ2(γ ), (3.16)

where κ2(γ ) is a positive function with

limγ→0

κ2(γ ) = 0. (3.17)

This can be easily done by the following arguments. Recall that K δ = �\�δ andμ(K δ) = 0. Let K δ

γ = {x ∈ � : |x, K δ| < γ } and h : � �→ [0, 1] be a continuous

123


function such that h∣∣�\K δ

γ= 1 and h

∣∣K δ

γ /2= 0. Choose f = h◦ϕ, then by Proposition

2.13, we have

Ev2 (ϕ) − Ev

2 ( f ) �∫

K δγ

ϕ|∇v|2dμ := κ2(γ ).

Hence, (3.16) holds true. We compute

∫�

ϕ(x)dμ(x)


d2(v(x), v(y))

εn+2 dμ(y)

�∫

�

ϕ(x) − f (x)dμ(x)

∫Bε(x)

d2(v(x), v(y))

εn+2 dμ(y)

+∫

�

f (x)dμ(x)


d2(v(x), v(y))

εn+2 dμ(y)

:= I1 + I2.

(3.18)

It follows from (3.16) thatlimε→0

I1 � κ2(γ ). (3.19)

We obtain from (3.15) thatlimε→0

I2 = 0. (3.20)

Hence, the combination of (3.17)–(3.20) implies that

limε→0

∫�

ϕ(x)dμ(x)


d2(v(x), v(y))

εn+2 dμ(y) = 0.

Equivalently, (3.2) holds true. ��

Corollary 3.21 Fix any α, β > 0 and u, v ∈ W 1,2(�, Y ). Let Iε(x) := {y ∈ Bε(x) :d(u(x), u(y)) � α and d(v(x), v(y)) � β}. Then we have, for any w ∈ W 1,2(�, Y )

and any ϕ ∈ Cc(�),

Ew2 (ϕ) =

∫�

|∇w|2ϕdμ = n + 2

cn,2limε→0

∫�

ϕ(x)dμ(x)

∫Iε(x)

d2(w(x), w(y))

εn+2 dμ(y).

Corollary 3.22 Fix any α, β > 0 and u, v ∈ W 1,2(�, Y ). Let Iε(x) := {y ∈ Bε(x) :d(u(x), u(y)) � α and d(v(x), v(y)) � β}. Then, for any w ∈ W 1,2(�, Y ) and anyϕ ∈ L∞(�) with compact support

Ew2 (ϕ) =

∫�

|∇w|2ϕdμ = n + 2

cn,2limε→0

∫�

ϕ(x)dμ(x)

∫Iε(x)

d2(w(x), w(y))

εn+2 dμ(y)

123


Proof By Proposition 2.17, we obtain that

ew2,ε → |∇w|2 in L1

loc(�).

Thus, for any 0 � ϕ ∈ L∞(�) with compact support, we have

∫�

|∇w|2ϕdμ = n + 2

cn,2limε→0

∫�

ϕ(x)dμ(x)

∫Bε(x)

d2(w(x), w(y))

εn+2 dμ(y) (3.23)

On the other hand, choose f ∈ Cc(�) such that ϕ � f . Then, by Corollary 3.21, wehave

lim supε→0

∫�

ϕ(x)dμ(x)

∫Bε(x)\Iε(x)

d2(w(x), w(y))

εn+2 dμ(y)

� lim supε→0

∫�

f (x)dμ(x)

∫Bε(x)\Iε(x)

d2(w(x), w(y))

εn+2 dμ(y) = 0. (3.24)

Thus, this corollary follows from the combination of (3.23) and (3.24). ��Corollary 3.25 For any δ > 0 and u ∈ W 1,2(�, Y ) with Diam(u(�)) < ∞, we have

limε→0

∫�

ϕ(x)dμ(x)

∫Bε(x)

d2+δ(u(x), u(y))

εn+2 dμ(y) = 0

for any ϕ ∈ Cc(�).

Proof Let D = Diam(u(�)). Fix any α > 0. Without loss of generality, for any0 � ϕ ∈ Cc(�),

∫�

ϕ(x)dμ(x)

∫Bε(x)

d2+δ(u(x), u(y))

εn+2 dμ(y)

=∫

�

ϕ(x)dμ(x)


d2+δ(u(x), u(y))

εn+2 dμ(y)

+∫

�

ϕ(x)dμ(x)

∫{y∈Bε(x):d(u(x),u(y))>α}

d2+δ(u(x), u(y))

εn+2 dμ(y)

�αδ

∫�

ϕ(x)dμ(x)

∫Bε(x)

d2(u(x), u(y))

εn+2 dμ(y)

+ Dδ

∫�

ϕ(x)dμ(x)


d2(u(x), u(y))

εn+2 dμ(y)

:=I1 + I2.

One haslimε→0

I1 = αδ Eu2 (ϕ)

123


By Lemma 3.1, we obtain thatlimε→0

I2 = 0.

Thus, we have

lim supε→0

∫�

ϕ(x)dμ(x)

∫Bε(x)

d2+δ(u(x), u(y))

εn+2 dμ(y) � αδ Eu2 (ϕ).

By taking α → 0, we conclude this corollary. ��Corollary 3.26 Let h : [0, β) → [0,∞) be a continuous function which is differ-entiable at 0 with h(0) = 0, h′(0) = 1, and satisfies h(t) � ct for some constantc > 0 and all t ∈ (0, β). Fix any α > 0 and u ∈ W 1,2(�, Y ), we have for anyv ∈ W 1,2(�, Y ) and ϕ ∈ Cc(�),

cn,2

n + 2Ev2 (ϕ) = lim

ε→0

∫�

ϕ(x)dμ(x)


h2(d(v(x), v(y)))

εn+2 dμ(y).

In particular, we have

cn,2

n+2Ev2 (ϕ)= lim

ε→0

∫�

ϕ(x)dμ(x)


sin2(d(v(x), v(y)))

εn+2 dμ(y).

Proof For any given γ ∈ (0, 1), choose δ > 0 such that

|h(t) − t | � γ t when 0 � t < δ. (3.27)

Fix any ϕ ∈ Cc(�). Without loss of generality, we may assume that ϕ � 0. Then wehave,

∫�

ϕ(x)dμ(x)


h2(d(v(x), v(y)))

εn+2 dμ(y)

=∫

�

ϕ(x)dμ(x)

∫{y∈Bε(x):d(u(x),u(y))<α and d(v(x),v(y))<δ}

h2(d(v(x), v(y)))

εn+2 dμ(y)

+∫

�

ϕ(x)dμ(x)

∫{y∈Bε(x):d(u(x),u(y))<α and d(v(x),v(y))�δ}

h2(d(v(x), v(y)))

εn+2 dμ(y)

:= I1 + I2.

By Corollary 3.21 and (3.27), we have

(1 − γ )2cn,2

n + 2Ev2 (ϕ) � lim inf

ε→0I1 � lim sup

ε→0I1 � cn,2

n + 2(1 + γ )2Ev

2 (ϕ).

Recall that h(t) � ct for some constant c > 0. Applying Lemma 3.1, we have

limε→0

I2 = 0.

123


Finally, by taking γ → 0, we conclude this corollary. ��

3.2 The Existence of Harmonic Maps to a Space with Curvature BoundedAbove

Let us begin from an introduction of the space with curvature �1.

Definition 3.28 A geodesic space (Y, d) is said to be with curvature � 1 (in thesense of Alexandrov), if the following comparison property holds: Given any tripleof points P, Q, R ∈ Y with d(P, Q) + d(Q, R) + d(R, Q) < 2π , and a pointS ∈ geodesic Q R with

d(Q, S) = d(R, S) = 1

2d(Q, R),

there exists a comparison triangle P Q R in S2 and a point S ∈ geodesic Q R with

dS2(Q, S) = dS2(R, S) = 1

2dS2(Q, R)

such that

d(P, S) � dS2(P, S).

As a simple consequence of Definition 3.28, we have, for any pair of points P, Q ∈ Ywith d(P, Q) < π , there exists a unique geodesic joining them.

Definition 3.29 Given any pair of points P, Q ∈ Y with d(P, Q) < π and λ ∈ [0, 1],one defines (1 − λ)P + λQ to be a point on a unique geodesic joining P and Q thatis a fraction λ away from P , that is

d((1 − λ)P + λQ, P) = λd(P, Q).

From now on, we always assume that Y is a complete geodesic space with curvature� 1.

Definition 3.30 We say that a quadruple of points P, Q, R, S ∈ Y determines aquadrilateral �P Q RS in Y if d(P, Q) + d(Q, R) + d(R, S) + d(S, P) < 2π .

Definition 3.31 We say that a triple of points P, Q, R ∈ Y determines a triangleP Q R in Y if d(P, Q) + d(Q, R) + d(R, P) < 2π .

123


Lemma 3.32 ([26], Estimate I) Let �P Q RS be a quadrilateral in Y . Let Q1/2 =Q+R2 and P1/2 = P+S

2 . Then

2d2(P1/2, Q1/2) � 1

cos2(d(P, S)/2)(d2(P, Q) + d2(R, S))

− 1

2 cos2(d(P, S)/2)(d(Q, R) − d(P, S))2

+ Cub(d(P, Q), d(R, S), |d(Q, R) − d(P, S)|).

Here Cub(·, ·, ·) represents terms that are cubic in the indicated variables.

Lemma 3.33 ([26], Estimate II) Let P QS be a triangle in Y . For a pair of numbers0 ≤ η, η′ ≤ 1, define Pη′ = (1 − η′)P + η′Q and Sη = (1 − η)S + ηQ. Then

d2(Pη′ , Sη) � sin2((1 − η)d(Q, S))

sin2(d(Q, S))(d2(P, S) − (d(Q, P) − d(Q, S))2)

+ ((1 − η′)d(Q, P) − (1 − η)d(Q, S))2 + Cub(|η′ − η|, d(P, S)).

Here Cub(·, ·) represents terms that are cubic in the indicated variables.

Given a ball Bρ(Q) ⊂ Y with ρ < π/2, we denote by

W 1,2(�, Bρ(Q)) = {u ∈ W 1,2(�, Y ) : u(�) ⊂ Bρ(Q)}

and

W 1,2ϕ (�, Bρ(Q)) = {u ∈ W 1,2(�, Bρ(Q)) : d(u, ϕ) ∈ W 1,2

0 (�))}

for ϕ ∈ W 1,2(�, Bρ(Q)).

Given u0, u1 ∈ W 1,2(�, Bρ(Q)). For any x, y ∈ �, the sum

d(u0(x), u0(y)) + d(u0(y), u1(y)) + d(u1(y), u1(x)) + d(u1(x), u0(x))

may exceed 2π , the four points u0(x), u0(y), u1(y), u1(x) do not determine a quadri-lateral in this case. Thanks to Corollary 3.22, when computing the energies, we onlyneed to consider the points x, y such that

d(u0(x), u0(y)) � π − 2ρ

and

d(u1(x), u1(y)) � π − 2ρ.

In this case, we have

d(u0(x), u0(y)) + d(u0(y), u1(y)) + d(u1(y), u1(x)) + d(u1(x), u0(x)) < 2π.

123


Hence, the four points u0(x), u0(y), u1(y), u1(x) determine a quadrilateral in thiscase.

A convexity of energy functional from a smooth Riemannian domain has beenestablished by Serbinowski in [26]. In the following, we will extend the convexity toa domain of an Alexandrov space.

Theorem 3.34 (Convexity of the energy functional) Given a ball Bρ(Q) ⊂ Y withρ < π/2, and u0, u1 ∈ W 1,2(�, Bρ(Q)). Let u1/2(x) = 1

2 (u0(x) + u1(x)), λ(x) =d(u0(x), u1(x)), R(x) = d(u1/2(x), Q), w(x) = (1− η(x))u1/2(x) + η(x)Q, whereη : � → [0, 1] is defined by

{sin((1−η)R)

sin R = cos λ2 when R > 0

1 − η = cos λ2 when R = 0.

(3.35)

Then we have, for any 0 � φ ∈ Cc(�),

cos8 ρ

∫�

∣∣∣∣∇ tan(λ/2)

cos R

∣∣∣∣2

φdμ � 1

2Eu02 (φ) + 1

2Eu12 (φ) − Ew

2 (φ).

In particular,

cos8 ρ

∫�

∣∣∣∣∇ tan(λ/2)

cos R

∣∣∣∣2

dμ � 1

2Eu02 (�) + 1

2Eu12 (�) − Ew

2 (�).

Proof For any x, y ∈ � with d(u0(x), u0(y)) � π − 2ρ and d(u1(x), u1(y)) �π − 2ρ, by applying Lemma 3.32 to the quadrilateral �u0(x)u0(y)u1(y)u1(x), weget

d2(u1/2(x), u1/2(y)) � 1

2 cos2(λ(x)/2)(d2(u0(x), u0(y)) + d2(u1(x), u1(y)))

− 1

4 cos2(λ(x)/2)(λ(y) − λ(x))2

+Cub(d(u0(x), u0(y)), d(u1(x), u1(y)), |λ(y) − λ(x)|).(3.36)

Applying Lemma 3.33 to the triangle Qu1/2(x)u1/2(y), we have

d2(w(y), w(x)) � sin2((1 − η(x))R(x))

sin2 R(x)d2(u1/2(x), u1/2(y))

− sin2((1 − η(x))R(x))

sin2 R(x)(R(y) − R(x))2

+ ((1 − η(y))R(y) − (1 − η(x))R(x))2

+ Cub(|η(y) − η(x)|, |R(y) − R(x)|).

(3.37)

123


Recall (3.35), the definition of η. The combination of (3.36) and (3.37) implies that

− d2(w(y), w(x)) + d2(u0(x), u0(y))

2+ d2(u1(x), u1(y))

2

� (λ(y) − λ(x))2

4+ cos2

(λ(x)

2

)(R(y) − R(x))2

− ((1 − η(y))R(y) − (1 − η(x))R(x))2

+ Cub(|η(y) − η(x)|, |R(y) − R(x)|)+ Cub(d(u0(x), u0(y)), d(u1(x), u1(y)), |λ(y) − λ(x)|).

(3.38)

For ε > 0 sufficiently small, we let Iε(x) := {y ∈ Bε(x) : d(u0(x), u0(y)) �π − 2ρ and d(u1(x), u1(y)) � π − 2ρ}. For any 0 � φ ∈ Cc(�), we obtain from(3.38) that

−∫

�

φ(x)dμ(x)

∫Iε(x)

d2(w(y), w(x))

εn+2 dμ(y)

+ 1

2

∫�

φ(x)dμ(x)

∫Iε(x)

d2(u0(x), u0(y))

εn+2 dμ(y)

+ 1

2

∫�

φ(x)dμ(x)

∫Iε(x)

d2(u1(x), u1(y))

εn+2 dμ(y)

�1

4

∫�

φ(x)dμ(x)

∫Iε(x)

(λ(y) − λ(x))2

εn+2 dμ(y)

+∫

�

φ(x) cos2(

λ(x)

2

)dμ(x)

∫Iε(x)

(R(y) − R(x))2

εn+2 dμ(y)

−∫

�

φ(x)dμ(x)

∫Iε(x)

((1 − η(y))R(y) − (1 − η(x))R(x))2

εn+2 dμ(y)

+∫

�

φ(x)dμ(x)

∫Iε(x)

Cub(|η(y) − η(x)|, |R(y) − R(x)|)εn+2 dμ(y)

+∫

�

φ(x)dμ(x)

∫Iε(x)

Cub(d(u0(x), u0(y)), d(u1(x), u1(y)), |λ(y)−λ(x)|)εn+2 dμ(y).

Note that φ(x) cos2(

λ(x)2

)∈ L∞(�) has compact support. Applying Corollary 3.25

and Corollary 3.22 to the above inequality, we have

−Ew2 (φ) + 1

2Eu02 (φ) + 1

2Eu12 (φ) � 1

4Eλ2 (φ) + E R

2

(cos2

(λ

2

)φ

)− E (1−η)R

2 (φ),

Recall that the energy measures are absolutely continuous w.r.t μ. Hence, we obtainthat

123


−Ew2 (φ) + 1

2Eu02 (φ) + 1

2Eu12 (φ)

�∫

�

(1

4|∇λ|2 + cos2

(λ

2

)|∇ R|2 − |∇((1 − η)R)|2

)φdμ

=∫

�

(1

4|∇λ|2 + cos2

(λ

2

)|∇ R|2 − |∇(cos λ

2 sin R)|21 − cos2 λ

2 sin2 R

)φdμ

=∫

�

cos4 R cos4 λ2

1 − sin2 R cos2 λ2

∣∣∣∣∣∇tan λ

2

cos R

∣∣∣∣∣2

φdμ

� cos8 ρ

∫�

∣∣∣∣∣∇tan λ

2

cos R

∣∣∣∣∣2

φdμ

Thus, by letting φ ↗ 1, we have

cos8 ρ

∫�

∣∣∣∣∇ tan(λ/2)

cos R

∣∣∣∣2

dμ(x) � 1

2Eu02 (�) + 1

2Eu12 (�) − Ew

2 (�).

��Now, we are in position to prove the existence result Theorem 1.1.

Proof of Theorem 1.1 Denote by

E0 = inf{Ev2 (�) : v ∈ W 1,2

ϕ (�, Bρ(Q))}.

Choose a sequence {un} ⊂ W 1,2ϕ (�, Bρ(Q)) such that Eun

2 (�) ↓ E0. Let λn,m =d(un, um), Rn,m = d

( un+um2 , Q

), wn,m(x) = (1−ηn,m(x))( un+um

2 )(x)+ηn,m(x)Q,where ηn,m : � → [0, 1] is defined by

{sin((1−ηn,m )Rn,m)

sin Rn,m= cos λn,m

2 when Rn,m > 0

1 − ηn,m = cos λn,m2 when Rn,m = 0.

Notice that wn,m ∈ W 1,2ϕ (�, Bρ(Q)) and hence E0 � E

wn,m2 (�). Thus, we have

1

2Eun2 (�) + 1

2Eum2 (�) − E

wn,m2 (�) � 1

2Eun2 (�) + 1

2Eum2 (�) − E0.

By Poincaré’s inequality and Theorem 3.34, we obtain that

limn,m→0

∫�

λ2n,m(x)dμ(x) � C limn,m→0

∫�

∣∣∣∣∇ tan(λn,m/2)

cos Rn,m

∣∣∣∣2

dμ = 0.

Hence, {un} is a Cauchy sequence in L2(�, Y ). By completeness of L2(�, Y ), {un}converges to some u ∈ L2(�, Y ), in the sense of L2(�, Y ). By semi-continuity of the

123


energy functional, Eu2 (�) � E0 and u ∈ W 1,2(�, Bρ(Q)). Notice that d(un, ϕ) ∈

W 1,20 (�), and d(un, ϕ) → d(u, ϕ) in L2(�), so d(u, ϕ) ∈ W 1,2

0 (�). Hence, wehave u ∈ W 1,2

ϕ (�, Bρ(Q)) and Eu2 (�) = E0. The uniqueness follows directly from

Theorem 3.34. ��

4 Global Hölder Regularity for Harmonic Maps

Let � be a bounded domain of an Alexandrov space and let (Y, dY ) be a geodesicspace with curvature � 1. Suppose that u : � → Y is a harmonic map such that theimage u(�) is contained in a ball Bρ(Q) with radius ρ < π/2.

4.1 Interior Hölder Regularity

Proposition 4.1 Assume that u : � → Bρ(Q) ⊂ Y is a harmonic map with 0 < ρ <

π/4. Let R(x) := d(u(x), P) for P ∈ Bρ(Q) fixed. Then we have

(1) Lcos R � − cos R · |∇u|2dμ in the sense of distribution;(2) if ρ < π/4, then LR � 0 in the sense of distribution;(3) if ρ < π/4, then LR2 � 2 cos(2ρ) · |∇u|2dμ in the sense of distribution.

Proof (1) Fix any 0 � φ ∈ Lipc(�). Without loss of generality, we may assumethat |φ|∞ � 1. Let w = (1 − φ)u + φP . Applying Lemma 3.33 to the trianglePu(x)u(y), we have

d2(w(y), w(x)) � sin2((1 − φ(x))R(x))

sin2 R(x)d2(u(x), u(y))

− sin2((1 − φ(x))R(x))

sin2 R(x)(R(y) − R(x))2

+ ((1 − φ(y))R(y) − (1 − φ(x))R(x))2

+ Cub(|φ(y) − φ(x)|, d(u(y), u(x))).

(4.2)

Fix any 0 � η ∈ Cc(�). For ε > 0 sufficiently small, we obtain from (4.2) that

∫�

η(x)

∫Bε(x)

d2(w(y), w(x))

εn+2 dμ(y)

�∫

�

η(x)sin2((1 − φ(x))R(x))

sin2 R(x)

∫Bε(x)

d2(u(x), u(y))

εn+2 dμ(y)

−∫

�

η(x)sin2((1 − φ(x))R(x))

sin2 R(x)

∫Bε(x)

(R(y) − R(x))2

εn+2 dμ(y)

+∫

�

η(x)

∫Bε(x)

((1 − φ(y))R(y) − (1 − φ(x))R(x))2

εn+2 dμ(y)

+∫

�

η(x)

∫Bε(x)

Cub(|φ(y) − φ(x)|, d(u(y), u(x)))

εn+2 dμ(y).

(4.3)

123


Note that η(x)sin2((1−φ(x))R(x))

sin2 R(x)∈ L∞(�) has compact support. Applying Corol-

lary 3.25 and Proposition 2.17 to (4.3), we obtain that

Ew2 (η) ≤

∫�

ηsin2((1 − φ)R)

sin2 R|∇u|2dμ −

∫�

ηsin2((1 − φ)R)

sin2 R|∇ R|2dμ

+∫

�

η|∇((1 − φ)R)|2dμ.

Thus, by letting η ↗ 1, we get

Ew2 (�) ≤

∫�

sin2((1 − φ)R)


∫�

sin2((1 − φ)R)

sin2 R|∇ R|2dμ

+∫

�

|∇((1 − φ)R)|2dμ.

Recall that u is a harmonic map, then Ew2 (�) ≥ Eu

2 (�). Thus, we have

Eu2 (�) ≤

∫�

sin2((1 − φ)R)


∫�

sin2((1 − φ)R)

sin2 R|∇ R|2dμ

+∫

�

|∇((1 − φ)R)|2dμ,

and hence

0 ≤ ∫�

sin2((1−φ)R)−sin2 Rsin2 R

|∇u|2dμ − ∫�

sin2((1−φ)R)

sin2 R|∇ R|2dμ

+ ∫�

|∇ R − R∇φ − φ∇ R|2dμ. (4.4)

For any t ∈ (0, 1), by replacing φ with tφ in (4.4), and dividing by t , we get

0 ≤∫

�

sin2((1 − tφ)R) − sin2 R

t sin2 R|∇u|2dμ −

∫�

sin2((1 − tφ)R)

t sin2 R|∇ R|2dμ

+∫

�

|∇ R − t R∇φ − tφ∇ R|2t

dμ.

Letting t ↘ 0, we obtain that

0 � −2∫

�

φR cot R|∇u|2dμ + 2∫

�

(−R〈∇φ,∇ R〉 − φ|∇ R|2)dμ

+ 2∫

�

φR|∇ R|2 cot Rdμ,

i.e.,

0 ≤∫

�

〈∇(

φR

sin R

),∇ cos R〉 −

∫�

φR

sin Rcos R · |∇u|2dμ. (4.5)

123


Note that 2ρsin 2ρ � R

sin R � 1, and hence φ/(R/ sin R) ∈ L∞(�)∩ W 1,2c (�). From

Remark 2.5, we know that (4.5) continues to hold for any nonnegative function inL∞(�) ∩ W 1,2

c (�). Thus, by replacing φ with φ/(R/ sin R) in (4.5), we have

Lcos R(φ) � −∫

�

φ cos R · |∇u|2dμ (4.6)

for any 0 � φ ∈ Lipc(�). This proves the assertion (1).(2) Fix any ε > 0 sufficiently small. Note that 0 � R � 2ρ < π . Let Rε :=

arccos(cos R − ε). HenceLcos R = Lcos Rε . (4.7)

For any 0 � φ ∈ Lipc(�), by (4.6) and (4.7), we have

0 � −∫

�

〈∇ cos Rε,∇φ〉 dμ +∫

�

cos R · |∇u|2φdμ

=∫

�

〈sin Rε∇ Rε,∇φ〉 dμ +∫

�


=∫

�

〈∇ Rε, sin Rε∇φ〉 dμ +∫

�


=∫

�

〈∇ Rε,∇(φ sin Rε) − φ∇ sin Rε〉 dμ +∫

�


=∫

�

〈∇ Rε,∇(φ sin Rε)〉 dμ −∫

�

〈φ cos Rε∇ Rε,∇ Rε〉 dμ

+∫

�


= −LRε (φ sin Rε) −∫

�

〈φ cos Rε∇ Rε,∇ Rε〉 dμ +∫

�


Hence, we get

LRε (φ sin Rε) �∫

�

φ cos R · |∇u|2dμ −∫

�

φ cos Rε|∇ Rε|2dμ. (4.8)

From Remark 2.5, we know that (4.8) continues to hold for any nonnegativefunction in L∞(�) ∩ W 1,2

c (�).On the other hand, it is easy to check that

0 < κ(ε) � sin Rε � 1

for some positive function κ(ε) with limε→0 κ(ε) = 0. Hence, φ/ sin Rε ∈ L∞(�) ∩W 1,2

c (�). Thus, by replacing φ with φ/ sin Rε in (4.8), we obtain that

LRε (φ) �∫

�

φcos R

sin Rε

|∇u|2dμ −∫

�

φ cot Rε · |∇ Rε|2dμ (4.9)

123


for any 0 � φ ∈ Lipc(�). If ρ < π/4, then we have 0 � R < π/2 and hencecos R > 0. Note that |∇u|2 � |∇ R|2 and ∇ Rε = sin R

sin Rε∇ R. Thus, we obtain from

(4.9) that

LRε (φ) �∫

�

φcos R

sin Rε

|∇u|2dμ −∫

�

φ cot Rε · |∇ Rε|2dμ

�∫

�

φcos R

sin Rε

|∇ R|2dμ −∫

�

φ cot Rε · sin2 R

sin2 Rε

|∇ R|2dμ

=∫

�

φcos R sin2 Rε − cos Rε sin2 R

sin3 Rε

|∇ R|2dμ

=∫

�

φε(1 − ε cos R + cos2 R)

sin3 Rε

|∇ R|2dμ� 0.

On the other hand, it is easy to check that Rε is uniformly bounded in W 1,2(�) and

Rε → R in L2(�) as ε → 0.

Recall that W 1,2(�) is reflexive. Hence, by taking a subsequence of Rε, we concludethat

LR � 0.

This is the assertion (2).(3) Fix any ε > 0 sufficiently small. Let Rε := arccos(cos R − ε). For any 0 �

φ ∈ Lipc(�), we have

LR2ε(φ) = −

∫�

⟨∇ R2

ε ,∇φ⟩dμ

= −∫

�

〈2Rε∇ Rε,∇φ〉 dμ

= −∫

�

〈∇ Rε, 2Rε∇φ〉 dμ

= −∫

�

〈∇ Rε,∇(2Rεφ) − φ∇(2Rε)〉 dμ

= LRε (2Rεφ) + 2∫

�

φ|∇ Rε|2dμ.

(4.10)

We obtain from (4.9) and (4.10) that

LR2ε(φ) � 2

∫�

φRε

cos R

sin Rε

|∇u|2dμ − 2∫

�

φRε cot Rε|∇ Rε|2dμ

+ 2∫

�

φ|∇ Rε|2dμ

= 2∫

�

φRε

cos R

sin Rε

|∇u|2dμ + 2∫

�

φ(1 − Rε cot Rε)|∇ Rε|2dμ.

123


If ρ < π/4, then R < 2ρ < π/2 and hence Rε < π/2. Notice that 1− Rε cot Rε �0. Thus, we have

LR2ε(φ) � 2

∫�

φRε

cos R

sin Rε

|∇u|2dμ.

Note that Rε � sin Rε and cos R � cos(2ρ). Hence,

LR2ε

� 2 cos(2ρ) · |∇u|2dμ.

On the other hand, it is easy to check that R2ε is uniformly bounded in W 1,2(�) and

R2ε → R2 in L2(�) as ε → 0.

Recall that W 1,2(�) is reflexive. Hence, by taking a subsequence of R2ε , we conclude

thatLR2 � 2 cos(2ρ) · |∇u|2dμ. (4.11)

This is the assertion (3), and hence the proof is finished. ��Along the same arguments as in the proof of [11, Theorem 2], the inequality (4.11)

(as same as [11, Eq. (5.2)]) implies the following interior Hölder estimate:

Theorem 4.12 Let X be an (n-dim) Alexandrov space with curvature � k, � ⊂ Xbe a bounded domain, and let Y be a complete geodesic space with curvature � 1.Suppose that u : � → Bρ(Q) ⊂ Y is a harmonic map with ρ < π/2. Then u islocally Hölder continuous on �.

4.2 Boundary Hölder Regularity

We will prove the boundary Hölder regularity by constructing a barrier via harmonicfunctions. Let us recall the concept of the relative capacity.

Definition 4.13 ([2]) Let B ⊂ X be a ball and E ⊂ B. We define

Capp(E, 2B) := infu

∫2B

|∇u|pdμ,

where the infimum is taken over all u ∈ W 1,p0 (2B) such that u � 1 on E .

To obtain the boundary regularity of the harmonic maps, we need the followingtheorem for harmonic functions:

Theorem 4.14 ([2], Theorem 2.12) Let X be a doubling metric measure space admit-ting a weak (1,p)-Poincaré inequality for some p < 2, and let � ⊂ X be a boundeddomain. Then there exists a constant C > 0 such that if w ∈ W 1,2(X) is Höldercontinuous at x0 ∈ ∂�,

lim infθ→0

1

| log θ |∫ 1

θ

exp(−Cγ (p, θ)2/(2−p)

) dθ

θ> 0, (4.15)

123


and u ∈ W 1,2(X) is a harmonic function on � with boundary data w, then u is Höldercontinuous at x0. Here

γ (p, θ) := θ−pμ(Bθ (x0))

Capp(Bθ (x0)\�, B2θ (x0)). (4.16)

Remark 4.17 From [2, Remark 2.15], the inequality (4.15) in Theorem4.14 is satisfiedif the measure density condition is satisfied for the domain X\� at x0, namely, thereexists a constant C > 0 such that

μ(Br (x0)\�) � C · μ

(Br (x0)

)

for all 0 < r < min{1,Diam(X\�)}.Consider the modified distance function on Bρ(Q) ⊂ Y with ρ < π/2 as follows

f (x) := 1 − cos(d(P, x))

cos(d(Q, x)), (4.18)

here P ∈ Bρ(Q) is fixed. Note that f is Lipschitz continuous on Bρ(Q).

Definition 4.19 We say that a function f is λ-convex on a domain E ⊂ Y , if f ◦ γ

is λ-convex for all geodesics γ ⊂ E .

Lemma 4.20 ([4], Proposition 9.2.18) Let (Z , dZ ) be a complete geodesic space. Acontinuous function f : Z → R is λ-convex if and only if, for any x, y ∈ Z and za midpoint (z lies on some geodesic γ connecting x and y, and satisfies dZ (z, x) =dZ (z, y) = dZ (x, y)/2) of x and y,

f (z) � f (x) + f (y)

2− λ

4d2

Z (x, y).

Lemma 4.21 f is λ-convex on Bρ(Q), for some positive constant λ.

Proof For any geodesic xy ⊂ Bρ(Q), choose x ′, y′ ∈ Bρ(Q′) ⊂ S2 such that

d(Q, x) = dS2(Q′, x ′), d(x, y) = dS2(x ′, y′), d(Q, y) = dS2(Q′, y′),

then choose P ′ ∈ S2 such that P ′ and Q′ are on the same side of the geodesic x ′y′

and

d(P, x) = dS2(P ′, x ′), d(P, y) = dS2(P ′, y′).

Let z = x+y2 and z′ = x ′+y′

2 . Define a function g on Bρ(Q′) ⊂ S2 by

g(x) := 1 − cos(dS2(P ′, x))

cos(dS2(Q′, x)).

123


According to [30, Lemma 3], the function g is λ-convex for some λ > 0. By curvaturecomparison, we have

d(P, z) � dS2(P ′, z′) and d(Q, z) � dS2(Q′, z′).

Hence, f (z) � g(z′). By Lemma 4.20, we obtain that

f (z) � g(z′) � g(x ′) + g(y′)2

− λ

4d2S2

(x ′, y′)

= f (x) + f (y)

2− λ

4d2(x, y).

By Lemma 4.20 again, we conclude that f is λ-convex. ��

Lemma 4.22 Let f be defined in (4.18), and let u : � → Bρ(Q) ⊂ Y (0 < ρ < π/2)be a harmonic map. Then

L f ◦u � 0

in the sense of distribution.

Proof Fix any ε ∈ (0, 1). Denote by Rp = cos d(u(x), P), Rq = cos d(u(x), Q) and

fε = 1 + ε − Rp

Rq.

Thus, we have1 + ε − Rp = Rq · fε. (4.23)

Then we obtain from (4.23) that

⟨∇ Rp,∇ Rq⟩ = − ⟨∇(Rq fε),∇ Rq

⟩ = −|∇ Rq |2 fε − Rq⟨∇ fε,∇ Rq

⟩,

and hence ⟨∇ fε,∇ Rq⟩ = 1

Rq

(− ⟨∇ Rp,∇ Rq

⟩ − |∇ Rq |2 fε)

. (4.24)

123


For any 0 � φ ∈ Lipc(�), by (4.23), we have

−LRp (φ) = LRq fε (φ)

= −∫

�

⟨∇φ,∇(Rq fε)⟩dμ

= −∫

�

⟨∇φ, Rq∇ fε + fε∇ Rq⟩dμ

= −∫

�

⟨∇ fε, Rq∇φ⟩dμ −

∫�

⟨fε∇φ,∇ Rq

⟩dμ

= −∫

�

⟨∇ fε,∇(φRq) − φ∇ Rq⟩dμ −

∫�

⟨∇ Rq ,∇( fεφ) − φ∇ fε⟩dμ

= 2∫

�

⟨φ∇ fε,∇ Rq

⟩dμ + L fε (φRq) + LRq ( fεφ).

(4.25)

The combination of (4.24) and (4.25) implies that

− LRp (φ) = 2∫

�

φ−|∇ Rq |2 fε − ⟨∇ Rp,∇ Rp

⟩Rq

dμ + LRq ( fεφ) + L fε (φRq).

(4.26)Note that fε > 0, by Young’s inequality, we have

− ⟨∇ Rp,∇ Rq⟩� |∇ Rq |2 fε + |∇ Rp|2

4 fε.

Hence, we obtain from the above inequality and (4.26) that

L fε (φRq) = −LRp (φ) + 2∫

�

φ|∇ Rq |2 fε + ⟨∇ Rp,∇ Rp

⟩Rq

dμ − LRq ( fεφ)

� −LRp (φ) −∫

�

|∇ Rp|22Rq fε

φdμ − LRq ( fεφ)

By the above inequality and Proposition 4.1(1), we have

L fε (φRq) �∫

�

Rq fε|∇u|2φdμ +∫

�

Rp|∇u|2φdμ −∫

�

|∇ Rp|22Rq fε

φdμ

=∫

�

(1+ε−Rp)|∇u|2φdμ+∫

�

Rp|∇u|2φdμ−∫

�

|∇ Rp|22Rq fε

φdμ (4.27)

=∫

�

(1 + ε)|∇u|2φdμ −∫

�

|∇ Rp|22(1 + ε − Rp)

φdμ

123


Notice that |∇ Rp|2 = sin2 d(u(x), P) · |∇d(u(x), P)|2 = (1 − R2p)|∇d(u(x), P)|2,

thus we obtain from (4.27) that

L fε (φRq) �∫

�

(1 + ε)|∇u|2φdμ −∫

�

(1 − R2p)|∇d(u(x), P)|2

2(1 + ε − Rp)φdμ

�∫

�

(1 + ε)|∇u|2φdμ −∫

�

(1 − R2p)|∇d(u(x), P)|22(1 − Rp)

φdμ

=∫

�

(1 + ε)|∇u|2φdμ −∫

�

(1 + Rp)|∇d(u(x), P)|22

φdμ

Note that 1+Rp2 < 1 + ε and |∇u|2 � |∇d(u(x), P)|2. Hence, we conclude that

L fε (φRq) � 0. (4.28)

From Remark 2.5, we know that (4.28) continues to hold for any nonnegative functionin L∞(�) ∩ W 1,2

c (�).Note that 1 � Rq � cos ρ > 0, hence 0 � φ/Rq ∈ W 1,2

c (�) ∩ L∞(�). Thus, byreplacing φ with φ/Rq in (4.28), we have

L fε (φ) � 0

for any 0 � φ ∈ W 1,2c (�) ∩ L∞(�). Finally, it is easy to check that

fε → f ◦ u as ε → 0, in W 1,2(�).

Thus, we conclude that

L f ◦u � 0.

This completes the proof. ��Theorem 4.29 Let X be an (n-dim) Alexandrov space with curvature � k, and let� ⊂ X be a bounded domain satisfying the measure density condition. Let Y be acomplete geodesic space with curvature � 1 (in the sense of Alexandrov). Given aball Bρ(Q) ⊂ Y with radius 0 < ρ < π/2. Then there exists a constant C0 > 0 suchthat if w ∈ W 1,2(X, Y ) with w(X) ⊂ Bρ(Q) is Hölder continuous at x0 ∈ ∂�,

lim infθ→0

1

| log θ |∫ 1

θ

exp(−C0γ (p, θ)2/(2−p)

) dθ

θ> 0 (4.30)

holds for some 1 < p < 2, and u ∈ W 1,2(�, Bρ(Q)) is a harmonic map on � whichagrees w on ∂�, then u is Hölder continuous at x0. Here γ (p, θ) is defined in (4.16).

123


Proof Let f be the modified distance function given in (4.18) with P = w(x0). Letg ∈ W 1,2(�) be the solution of the Dirichlet problem

{{Lg = 0 on �

g − f ◦ w ∈ W 1,20 (�).

Because that � satisfies the measure density condition, according to [3, Proposi-tion 5.1], there exists a W 1,2(X)-extension of the function (g − f ◦ w) by takingvalue = 0 outside of �. Note that f ◦ w ∈ W 1,2(X), by Proposition 2.8(1), one hasg ∈ W 1,2(X). Using Theorem 4.14 and the assumption (4.30), we have g is Höldercontinuous at x0.

By Lemma 4.22, we have

L f ◦u−g � 0,

then by applying the weak maximum principle (see, for example, [5, Theorem 7.17]),we get

f ◦ u − g � 0 a.e. x ∈ �.

Note that f (w(x0)) = 0, g − f ◦ w ∈ W 1,20 (�) and that both g and f ◦ w are Hölder

continuous at x0, we have g(x0) = 0. Hence, we have

f (u(x)) − f (w(x0)) � g(x) − g(x0) a.e. x ∈ �.

Choose a geodesic γ connecting w(x0) and u(x), by the λ-convexity of f for someλ > 0, Lemma 4.21, we have, for any x ∈ �,

f (u(x))− f (w(x0)) � d+

dt

∣∣∣∣t=0

( f ◦ γ (t))+ λ

2d2(u(x), u(x0)) � λ

2d2(u(x), u(x0)),

where we have used d+dt |t=0( f ◦ γ (t)) � 0, since γ (0) = w(x0) is a minimum point

of f . Thus, we have

d2(u(x), w(x0)) � 2

λ(g(x) − g(x0)) a.e. x ∈ �.

By using the triangle inequality, we get

d2(u(x), w(x))

� 2d2(w(x), w(x0)) + 4

λ(g(x) − g(x0)) a.e. x ∈ �. (4.31)

By using that � satisfies the measure density condition again, according to [3,Proposition 5.1], the zero-extension of d

(u(x), w(x)

)is in W 1,2(X). Since w(X) is

contained in Bρ(Q) with ρ < π/2, we have f (w(x)) � 0. Hence, by g − f ◦ w � 0,

123


we have g(x) � 0. Thus, the right-hand side of (4.31) is nonnegative. It follows, fora fixed constant ε > 0, that

d2(u(x), w(x))

� 2d2(w(x), w(x0)) + 4

λ(g(x) − g(x0)) a.e. x ∈ Bε(x0).

Since both w and g are Hölder continuous at x0, we get

esssupBδ(x0)

d2(u(x), w(x))

� C(δα + δβ)

for some C > 0 and 0 < α, β < 1 and all 0 < δ < ε. After a redefinition on a set ofmeasure zero, d

(u(x), w(x)

)is Hölder continuous at x0.

At last, by using the facts that both w and d(u(x), w(x)

)are Hölder continuous at

x0 and the triangle inequality, we have u is Hölder continuous at x0. Now the proof isfinished. ��Theorem 1.3 is a consequence of the combination of Theorem 4.12, Remark 4.17 andTheorem 4.29.

Acknowledgements The second author is partially supported by NSFC 11201492.

References

1. Burago, Y., Gromov, M., Perelman, G.: A. D. Alexandrov spaces with curvatures bounded below.Russian Math. Surv. 47, 1–58 (1992)

2. Björn, J.: Boundary continuity for quasiminimizers on metric spaces. Ill. J. Math. 46(2), 383–403(2002)

3. Björn, J., Shanmugalingam, N.: Poincaré inequalities, uniform domains and extension properties forNewton-Sobolev functions in metric spaces. J. Math. Anal. Appl. 332(1), 190–208 (2007)

4. Burago, D., Burago Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics,vol. 33. AMS, New York (2001)

5. Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9,428–517 (1999)

6. Chen, J.: On energy minimizing mappings between and into singular spaces. Duke Math. J. 79, 77–99(1995)

7. Daskalopoulos, G., Mese, C.: Harmonic maps between singular spaces I. Commun. Anal. Geom. 18,257–337 (2010)

8. Eells, J., Fuglede, B.: Harmonic Maps Between Riemannian Polyhedra. Cambridge Tracts Mathse-matics, vol. 142. Cambridge University Press, Cambridge (2001)

9. Fuglede, B.: Hölder continuity of harmonic maps from Riemannian polyhedra to spaces of upperbounded curvature. Calc. Var. PDE 16(4), 375–403 (2003)

10. Fuglede, B.: The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upperbounded curvature. Trans. Am. Math. Soc. 357(2), 757–792 (2005)

11. Fuglede, B.: Harmonic maps from Riemannian polyhedra to geodesic spaces with curvature boundedfrom above. Calc. Var. PDE 31(1), 99–136 (2008)

12. Gregori, G.: Sobolev spaces and harmonic maps between singular spaces. Calc. Var. PDE 7, 1–18(1998)

13. Gromov, M., Schoen, R.: Harmonic maps into singular spaces and p-adic superrigidity for lattices ingroups of rank one. Publ. Math. IHES 76, 165–246 (1992)

14. Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.: Sobolev classes of Banach space-valuedfunctions and quasiconformal mappings. J. Anal. Math. 85, 87–139 (2001)

15. Jost, J.: Equilibrium maps between metric spaces. Calc. Var. PDE 2(2), 173–204 (1994)

123


16. Jost, J.: Generalized Dirichlet forms and harmonic maps. Calc. Var. PDE 5, 1–19 (1997)17. Korevaar, N., Schoen, R.: Sobolev spaces and harmonic maps for metric space targets. Commun. Anal.

Geom. 1, 561–659 (1993)18. Kuwae, K., Machigashira, Y., Shioya, T.: Sobolev spaces, Laplacian, and heat kernel on Alexandrov

spaces. Math. Z. 238(2), 269–316 (2001)19. Kuwae, K., Shioya, T.: Sobolev andDirichlet spaces overmaps betweenmetric spaces. J. ReineAngew.

Math. 555, 39–75 (2003)20. Lin, F.H.: Analysis on Singular Spaces. Collection of Papers on Geometry, Analysis andMathematical

Physics, pp. 114–126. World Science Publications, River Edge (1997)21. Otsu, Y., Shioya, T.: The Riemannian structure of Alexandrov spaces. J. Differ. Geom. 39, 629–658

(1994)22. Perelman, G.: DC structure on Alexandrov spaces. Preprint, preliminary version available online at

www.math.psu.edu/petrunin/23. Petrunin, A.: Semiconcave Functions in Alexandrov’s Geometry. Surveys in Differential Geometry

XI: Metric and Comparison Geometry, pp. 137–201. International Press, Somerville (2007)24. Reshetnyak, Y.G.: Sobolev classes of functions with values in a metric space. Sibirsk. Mat. Zh. 38(3),

657–675 (1997)25. Reshetnyak, Y.G.: Sobolev classes of functions with values in ametric space II. Sibirsk.Mat. Zh. 45(4),

709–721 (2004)26. Serbinowski, T.: Harmonic Maps into Metric Spaces with Curvature Bounded from above, Ph.D.

Thesis: University of Utah (1995)27. Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces.

Rev. Mat. Iberoamericana 16, 243–279 (2000)28. Zhang, H.C., Zhu, X.P.: Yau’s gradient estimates on Alexandrov spaces. J. Diff. Geom. 91, 445–522

(2012)29. Zhang H.C., Zhu, X.P.: Lipschitz continuity of harmonic maps between Alexandrov spaces, available

at arXiv:1311.133130. Zhou, Z.-R.: A note on boundary regularity of subelliptic harmonic maps. Kodai Math. J. 28(3), 525–

533 (2005)

123

www.math.psu.edu/petrunin/

http://arxiv.org/abs/1311.1331

harmonic maps between alexandrov spaces -...

Documents