Harmonic Maps Between Alexandrov Spaces

DOI 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 of harmonic maps from an Alexandrov space into a geodesic space with curvature1 in 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 singular spaces has been studied extensively. In [17], Korevaar and Schoen introduced a con- cept of energy and Sobolev maps for a metric space target, and developed a satisfactory existence 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 curvature of the target, one does not have either existence or regularity of the minimizers.

Let(X,|·,· |),(Y,d)be two metric spaces and letbe a bounded domain (con- nected open subset) ofX.μis a Radon measure onX. Givenp 1,ε >0 and a Borel measurable mapu:→Y, the approximating energy functionalE^u_p,εofuis given as follows. For each compactly supported continuous functionϕ∈Cc(), we set

B

[email protected]

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

E^u_p,ε(ϕ):=Cn,p

ϕ(x)dμ(x)

B_ε(x)∩

d^p(u(x),u(y)) ε^n+p dμ(y),

whereCn,pis a normalized constant. Thep-th energy functional ofuis defined by E^u_p(ϕ):=lim sup

ε→0

E^u_p,ε(ϕ), ∀ϕ∈Cc().

We say thatu ∈W^1,p(,Y)ifu ∈L^p(,Y)and it has finitep-energy sup

ϕ∈Cc(),0ϕ1

E^u_p(ϕ) <∞.

Whenis 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 lim_ε→0E^u_p,ε(ϕ) exists. This is extended to the case whereis a domain of a Lipschitz manifold [12], or a domain of a polyhedra [8], oris a domain of an Alexandrov space with curvature bounded below [19].

Given a domainof an Alexandrov space with curvature bounded below and a metric space(Y,d). A map u : → Y is called a harmonic map if it is a local energy minimizer of E₂^u. Our purpose in this paper is to study the Dirichlet problem of harmonic maps, including the existence and regularity, from a bounded domain of an Alexandrov space with curvature bounded below into a complete geodesic space of curvature1 in the sense of Alexandrov.

When the target(Y,d)is a non-positively curved space, by utilizing the convexity of distance functiond(·,·)on the product space Y ×Y, the Dirichlet problem of harmonic 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 harmonic maps fromtoY has been solved by Serbinowski [26] for the case where the images of the maps are contained in a geodesic ball ofY with radius < π/2. In [17,26], the essential tool is a concept of directional energy, which is a generalization of the directional derivative of functions, defined by aC¹-vector field on. In fact, the energy of a Sobolev map in [17] is able to be represented as an integration of its directional energy. J. Eells and B. Fuglede [8,10,11] extended this method to the case where is a domain of a Riemannian polyhedra.

It is well known that the set of singular points might be dense in a general Alexandrov space with curvature bounded from below [21]. Whenis a domain of an Alexandrov space, since theC¹(even Lipschitz continuous) vector fields do not make sense, it seems difficult to employ directly the method of directional energy in the setting of Alexandrov spaces. We need more arguments to conclude the following existence result when the domain is in an Alexandrov space. For a subset A ⊂ X and a map u :A→Y, we denote byu(A):= {u(x):x∈ A}.

Theorem 1.1 Let X be an Alexandrov space with curvaturek,⊂X be a bounded domain, and let Y be a complete geodesic space with curvature1(in the sense of

Alexandrov). Given a ball B_ρ(Q) ⊂ Y with ρ < π/2, andϕ ∈ W^1,2(,Y)with ϕ()⊂B_ρ(Q), we write

W_ϕ^1,2(,B_ρ(Q)):= {v∈W^1,2(,Y):d(v, ϕ)∈W₀^1,2(), v()⊂B_ρ(Q)}.

Then W_ϕ^1,2(,B_ρ(Q))has a unique element u of least2-energy. It is called the har- monic map onwhich agreesϕon∂.

Next, we will consider the regularity of harmonic maps given in the above Theorem 1.1. See, for example, [6,8,9,16,17,29] for the relational results for the case that the target has non-positive curvature. Letu :→Y be a harmonic map from a smooth Riemannian domainto a metric space with curvature1, it is proved in [26] thatuis locally Lipschitz continuous and globally Hölder continuous. J. Eells and B. Fuglede [8–11] proved the global Hölder continuity for harmonic maps from a domain with regular boundary of a Riemannian polyhedra to a metric space with curvature1.

Here, we shall prove the global Hölder continuity for harmonic maps from a domain of an Alexandrov space to a metric space with curvature1. Letbe a domain of an Alexandrov spaceX. It is said to satisfy themeasure density condition, if there exists a constantC>0 such that

μ(Br(x)∩)Cμ(Br(x)) (1.2)

for allx∈and all 0<r<min{1,Diam()}. Here Diam():=sup_x,y∈|x y|.

Theorem 1.3 Let X be an Alexandrov space with curvaturek and let⊂ X be a bounded domain such that both domainsand X\satisfy the measure density condition, and let Y be a complete geodesic space with curvature1(in the sense of Alexandrov). Assume thatw∈W^1,2(X,Y)is Hölder continuous onand the image w(X)is contained in a geodesic ball B_ρ(Q)⊂Y with radiusρ < π/2. Suppose that u is the harmonic map onwhich agreeswon∂. Then u is Hölder continuous on .

Remark 1.4(1) If⊂Mis a domain of a smooth manifold with Lipschitz continuous boundary∂, then bothandM\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), both domains 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 Lipschitz continuous in one of the following cases:

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

(b) X is a smooth manifold,Y is a metric space with curvature1, and the image u()is contained in a geodesic ball with radius< π/2 (see [26]).

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

Organization of the paper. In Sect.2, we will provide some necessary materials on Alexandrov 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) must be absolutely continuous with respect to the Hausdorff measure. In Sect.3, we will discuss the existence of harmonic maps and prove Theorem1.1. The regularity of harmonic maps is included in the last section.

2 Preliminaries

2.1 Alexandrov Spaces

Letk∈Randl ∈N. Denote byM^l_kthe simply connected,l-dimensional space form of constant sectional curvaturek. The spaceM²_kis calledk-plane.

Let(X,| ·,· |)be a complete metric space. A rectifiable curveγ connecting two points p,q is called ageodesicif its length is equal to|pq|and it has unit speed. A metric space Xis called ageodesic spaceif, for every pair of pointsp,q ∈ X, there exists some geodesic connecting them.

Fix anyk ∈ R. Given three points p,q,r in a geodesic space X, we can take a triangle ¯pq¯r¯ink-planeM²_k such that| ¯pq¯| = |pq|,| ¯qr| = |qr¯ |and|¯rp| = |r p|.¯ Ifk > 0, we add the assumption that|pq| + |qr| + |r p| < 2π/√

k. The triangle ¯pq¯r¯ ⊂ M²_k is unique up to a rigid motion. We call it comparison triangle. We let kpqr denote the angle at the vertexq¯ of the triangle ¯pq¯r¯, and we call it a k-comparison angle.

Definition 2.1 Letk ∈ R. A geodesic space X is called anAlexandrov space with curvaturekif it satisfies the following properties:

(1) it is locally compact;

(2) for any pointx ∈ X, there exists a neighborhoodUofxsuch that the following condition is satisfied: for any two geodesics γ (t) ⊂ U and σ (s) ⊂ U with γ (0)=σ(0):=p, thek-comparison angle

κγ (t)pσ(s)

is non-increasing with respect to each of the variablestands.

It is well known that the Hausdorff dimension of an Alexandrov space with curvature k, for some constantk∈ R, is always an integer (see, for example, [4] or [1]). Let X be ann-dimensional Alexandrov space with curvaturek. We denote byμthen- dimensional Hausdorff measure onX. The Bishop inequality and the Bishop–Gromov inequality are satisfied onX, i.e., for everyx∈ X, the ratio

μ(Br(x))

k 1

and

μ(Br(x)) V_r^k

is non-increasing inr, whereV_r^k is the volume of a ball of radiusrin the space form Mⁿ_k, cf. [1]. In particular, the doubling condition is satisfied onX.

Letp ∈X, given two geodesicsγ (t)andσ(s)withγ (0)=σ(0)= p, the angle

γ(0)σ(0):= lim

s,t→0

κγ (t)pσ (s)

is well defined. We denote bypthe set of equivalence classes of geodesicγ (t)with γ (0)= p, whereγ (t)is equivalent toσ(s)if γ(0)σ(0)=0. The completion of metric space(p, )is called the space of directions atp, denoted byp. The tangent cone atp,Tp, is the Euclidean cone overp. For two tangent vectorsu, v∈Tp, their

“scalar product” is defined by (see Sect. 1 in [23]) u, v := 1

2(|u|²+ |v|²− |uv|²).

For anyδ >0, we denote X^δ :=

x∈ X : vol(x) > (1−δ)·vol(Sⁿ⁻¹) ,

whereSⁿ⁻¹is the standard(n−1)-sphere, and vol denotes the(n−1)-dimensional Hausdorff measure. X^δ an open set (see [1]). The set S_δ := X\X^δ is called theδ- singular set. Each point p∈S_δis called aδ-singular point. The set

SX :=

δ>0

S_δ

is calledsingular set. A pointp ∈Xis called asingular pointifp∈SX. Otherwise it is called aregular point. Equivalently, a pointpis regular if and only ifTpis isometric toRⁿ ([1]). It is proved in [1] that the Hausdorff dimension ofSX is n−1. We remark that the singular setSX might be dense inX([21]).

Some basic structures of Alexandrov spaces are in the following.

Proposition 2.2 Let k ∈Rand let X be an n-dim Alexandrov space with curvature

k.

(1) There exists a constantδn,k >0depending only on the dimension n and k such that for eachδ∈(0, δn,k), the set X^δforms a Lipschitz manifold([1])and has a C^∞-differentiable structure([18]).

(2) There exists a B Vloc-Riemannian metric g on X^δsuch that

• the metric g is continuous in X\SX ([21,22]);

• the distance function on X\SXinduced from g coincides with the original one of X([21]);

• the Riemannian measure on X\SXinduced from g coincides with the Hausdorff measure of X ([21]).

Definition 2.3 ([1]) The boundary of an Alexandrov spaceX is defined inductively with respect to dimension. If the dimension of X is one, then X is a complete Riemannian manifold and theboundary of X is defined as usual. Suppose that the dimension of X isn 2. A point pis aboundary point of X ifphas non-empty boundary.

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].¹They coincide with each other on Alexandrov spaces.

From now on, we assume thatXis ann-dim Alexandrov space with curvaturek, ⊂Xis a domain ofX. It is well known that the metric measure space(X,| ·,· |, μ) supports a local (weak) Poincaré inequality.

We denote by Lip_loc()the set of locally Lipschitz continuous functions on, and by Lip_c()the set of Lipschitz continuous functions onwith compact support.

For a continuous functionu:X →R, define Lipu(x)=lim sup

y→x

|u(y)−u(x)|

|yx| .

Definition 2.4 For any 1 p +∞ andu ∈ Lip_loc(), its W^1,p()-norm is defined by

uW^1,p():= uL^p()+ LipuL^p(). Sobolev spacesW^1,p()are defined by the closure of the set

{u ∈Lip_loc(): uW^1,p()<+∞},

underW^1,p()-norm. SpacesW₀^1,p()are defined by the closure of Lip_c()under W^1,p()-norm. We denote byWc^1,p()= {f ∈W₀^1,p(): f has compact support}.

We say that a function f ∈W_loc^1,p()if f ∈W^1,p()for every open subset⊂⊂. Cheeger [5, Theorem 4.48] proved thatW^1,p()is reflexive when 1< p <∞. We denote by∇uthe weak gradient ofu∈W^1,p().

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

We recall the chain derivation property ofW^1,2(). It is formulated as follows: For f,g∈W^1,2()∩L^∞()and : R→Rwhich isC¹on the range of f, then(f) belongs toW^1,2()and

∇(f),∇g =(f)· ∇f,∇g μ-a.e.

Given f ∈W_loc^1,2(), the Laplacian of f is defined as a functional on Lip_c()by Lf(ϕ):= −

∇f,∇ϕdμ, ∀ϕ∈Lip_c().

Givenh∈ L¹_loc(). A function f ∈W_loc^1,2()is said to satisfy the inequality Lf hdμ

in the sense of distribution if the inequality Lf(ϕ)

hϕdμ

holds for all 0 ϕ ∈ Lip_c(). In this case, the functionalLf is a signed Radon measure.

Remark 2.5 Moreover, the measure Lf can be extended to a functional on ϕ ∈ L^∞()∩Wc^1,2().

Indeed, let us fix arbitrarily a compact setK ⊂. Take a functionK ∈Lip_c() withK ≡1 onK. Given any functionϕ∈Lip_c(K), we have

0ϕL^∞·K±ϕ ∈Lip_c(), and then

Lf

ϕL^∞·K±ϕ

h·

ϕL^∞K ±ϕ dμ

ϕL∞·

suppK

|h|(K+1)dμ.

This follows

±Lf(ϕ)ϕL^∞· 2hL¹(suppK)+ |Lf(K)|

:=C· ϕL^∞. Thus, since suppϕ⊂K, we have

K

∇f,∇ϕdμ

= |Lf(ϕ)|C· ϕL^∞.

At last, any functionφ ∈ Wc^1,2()∩L^∞()can be W^1,2()-approximated by a sequence of functionsϕj ∈ Lip_c()withϕjL^∞ φL^∞ (see, for example, [5, Theorem 4.24]).

2.3 Energy Functional and Sobolev Spaces into Metric Space

From now on, we assume thatis a domain of an Alexandrov space and that(Y,d)is a complete metric space. Fix anyp∈ [1,∞). Fix a pointP∈Y. A Borel measurable mapu: →Y is said to be in the spaceL^p(,Y,P)if it has separable range and

d^p u(x),P

dμ(x) <∞.

Ifμ() < ∞, the space L^p(,Y,P)does not depend on the choice of the point P ∈ Y. In this case, we denote the space by L^p(,Y). The space L^p(,Y)is a metric space under the distance

dp(f,g)=

d^p(f(x),g(x))dμ(x) ¹_p

.

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

Definition 2.6 Foru ∈L^p(,Y), the approximating energyE^u_p,εis defined by E^u_p,ε(ϕ)=

ϕ(x)e^up,εdμ(x),

for anyϕ ∈Cc(), where the approximating density ofuis defined by e^u_p,ε(x)= n+p

cn,pεⁿ

B_ε(x)∩

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

ε^p dμ(y)

where the constantcn,p=

Sⁿ⁻¹|x¹|^pσ (dx), andσis the canonical volume onSⁿ⁻¹. Given anyϕ∈Cc(), it is easy to check that, for any sufficiently smallε >0, the approximating energy E^u_p,ε(ϕ)coincides (up to a constant) with the one defined by Kuwae and Shioya in [19], that is,

E^u_p,ε(ϕ):= n 2ωn−1εⁿ

ϕ(x)

B_ε(x)∩

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

ε^p ·IQ()(x,y)dμ(y)dμ(x), where

Q():=

(x,y)∈×: |x y|<|γ , ∂|, ∀geodesicγ fromxtoy ,

andωn−1is the volume ofSⁿ⁻¹. It is proved in [19] that, for eachϕ ∈ Cc(), the limit

E^u_p(ϕ):= lim

ε→0⁺E^u_p,ε(ϕ)

exists. We callE^u_p(ϕ)theenergy functionalofu, defined onCc().

Foru ∈L^p(,Y), thep-energy ofuis defined by E^u_p():= sup

φ∈C_c(),0ϕ1

E^u_p(ϕ).

Definition 2.7 We define the Sobolev space fromtoY by W^1,p(,Y)=

u∈L^p(,Y):E^u_p() <∞ . Notice that the space of Lipschitz maps Lip(,Y)⊂W^1,p(,Y).

Proposition 2.8 ([19])Let1 < p < ∞and u ∈ W^1,p(,Y). Then the following assertions (1)–(5) hold.

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

E^ψ◦_p ^u(ϕ)|ψ|_{Li p}^p ·E^u_p(ϕ) for any0ϕ ∈Cc(), where

|ψ|Li p := sup

y,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 E^d_p^(Q,u(·))(ϕ)E^u_p(ϕ)

for any0ϕ ∈Cc().

(2) (Lower semi-continuity, [19, Theorem 3.2]) For any sequence uj → u in L^p(,Y)as j→ ∞, we have

E^u_p(ϕ)lim inf

j→∞ E^up^j(ϕ) for any0ϕ ∈Cc().

(3) (Energy measure, [19, Theorem 4.1 and Proposition 5.1])There exists a finite Borel measure, denoted by E^u_pagain, on, which is called theenergy measure of u, such that for any0ϕ ∈Cc(),

E^u_p(ϕ)=

ϕ(x)dE^up(x).

Furthermore, the measure is strongly local. That is, for any nonempty open subset O ⊂, we have u|O ∈W^1,p(O,Y), and moreover, if u is a constant map almost everywhere on O, then E^u_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 that the following holds: for any z∈ O and any0<r <R/2, we have

Br(z)

Br(z)d^p u(x),u(y)

dμ(x)dμ(y)Crⁿ⁺²·

B6r(z)dE^u_p(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 of Alexandrov spaces as shown in the proof of Theorem 2.1 in[19], one can choose R>0arbitrarily,ϑ =1and

= sup

0<r<R

vol(Br(o)⊂Mⁿ_k)

vol(Br(o)⊂Rⁿ) =C(n,k,R).

(5) (Equivalence forY =R, [19, Theorem 6.2])If Y =R, the above Sobolev space W^1,p(,R)is equivalent to the Sobolev space W^1,p()given in Definition2.4.

Precisely, for any u∈W^1,p(,R), the energy measure E^u_pis absolutely contin- uous with respect toμand

dE^u_p

dμ = |∇u|^p. For a subsetA⊂X, denote by

Diam(A):= sup

x,y∈A

|x y|.

Lemma 2.9 Let ⊂ X be a bounded domain. Then there exist positive constants C1,C2such that

C2 μ(Br(x)) rⁿ C1

for any ball Br(x)⊂.

Proof SetD:=Diam(). Using the Bishop inequality, we have, for any ballBr(x)⊂ ,

μ(Br(x))V_r^k sup

0<rD

V_r^k

rⁿ ·rⁿ:=C1(n,D)·rⁿ, whereV^kis the volume of a ball of radiusrin the space formMⁿ.

For any ballBr(x)⊂, by using the Bishop–Gromov inequality, we have, for all 0<r D,

μ(Br(x))

V_r^k μ(BD(x)) V_D^k . Note that⊂BD(x), thus we have

μ(Br(x))

rⁿ μ()

V_D^k ·V_r^k

rⁿ inf

0<rD

V_r^k rⁿ ·μ()

V_D^k :=C2(n, ).

Now the proof is complete.

To simplify, we use the following notation throughout this paper. Given a μ- measurable function f and aμ-measurable subsetA, we denote by

A

fdμ:= 1 μ(A)

A

fdμ.

2.4 Absolute Continuity of the Energy Measures

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

In [24], Reshetnyak proposed a similar approach to define the Sobolev spaces for maps into a metric space. Given 1 p < ∞, by the definition in [24], a map u ∈ L^p(,Y)belongs to Reshetnyak–Sobolev space, denoted byR^1,p(,Y), if for every Lipschitz functionψ :Y → Rthe compositionψ◦u ∈ W^1,p(), and there is a functionw ∈ L^p()(independent ofψ) such that, for every Lipschitz function ψ:Y →R, the inequality holds

|∇(ψ◦u)|(x)|ψ|Li p·w(x) μ−a.e. x∈.

Lemma 2.10 ([14])Let1< p<∞, we have W^1,p(,Y)⊂R^1,p(,Y).

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

Letu ∈ W^1,p(,Y). Given any Lipschitz functionψ : Y → R, we have the compositionψ◦u ∈W^1,p(), because of Proposition2.8(1) and (5). According to Proposition2.8(1), we know that the energy measure ofuandψ◦usatisfies

E^ψ◦_p ^u|ψ|_{Li p}^p ·E^u_p. (2.11) Consider the Lebesgue decomposition ofE^u_pwith respect toμon,

E^u_p= |∇u|p·μ+(E^up)^s,

where |∇u|p ∈ L¹(), because E^u_p() < ∞. By Proposition2.8(5), the measure E^ψ◦p ^uis absolutely continuous w.r.t.μ, and has density|∇(ψ◦u)|^p. From (2.11), we have

|∇(ψ◦u)|^p·μ|ψ|^p_{Li p}· |∇u|p·μ+ |ψ|_{Li p}^p ·(E^u_p)^s. The singular part(E^up)^s is supported in aμ-zero measure set. Then we have

|∇(ψ◦u)|(x)|ψ|Li p· |∇u|¹p^/p(x) μ−a.e. x∈.

Therefore,u ∈ R^1,p(,Y).

Remark 2.12 Ifis a bounded domain of a smooth manifold, it is shown in [14,25]

that W^1,p(,Y)= R^1,p(,Y). It is interesting to clarify whether the assertion is still valid whenis a bounded domain of an Alexandrov space.

Proposition 2.13 (Absolute continuity for the energy measures)Let1 < p < ∞.

For each u∈ W^1,p(,Y), the energy measure E^upis absolutely continuous w.r.t the volumeμ, i.e., there exists|∇u|p∈ L¹()such that

dE^u_p

dμ = |∇u|p.

Proof By [14, Corollary 3.21] that R^1,p(,Y) = N^1,p(,Y) ⊂ N^1,p(X,V :=

^∞(Y)), whereN^1,p(,Y)is the Newtonian–Sobolev space in [14, Definition 3.9]

by the upper gradient. From Lemma2.10, we haveu ∈ N^1,p(X,V). By [14, Propo- sition 4.6], we have

d(u(x),u(y))C(n, )(Mg(x)+Mg(y))· |x y|, forμ-a.e.x,y∈, whereMg(x)is the maximal function ofgdefined by

Mg(x)=sup

r>0 Br(x)g(y)dμ(y),

andg ∈L^p()is a weak upper gradient in the sense of [14, Definition 3.9]. Denote byh = Mgfor convenience. Without loss of generality, for any 0ϕ ∈ Lip_c(), we have

cn,p

n+pE^u_p,ε(ϕ)=

ϕ(x)dμ(x)

B_ε(x)

d^p(u(x),u(y)) ε^n+p dμ(y)

C(n,p, )

ϕ(x)dμ(x)

B_ε(x)

h^p(x)+h^p(y)

εⁿ dμ(y) (2.14)

C(n,p, )

ϕ(x)h^p(x)dμ(x) +C(n,p, )

ϕ(x)dμ(x)

B_ε(x)

h^p(y) εⁿ dμ(y), where we have used thatμ(Bε(x))C2εⁿ. Set

|ϕ|Li p = sup

x,y∈,x=y

|ϕ(y)−ϕ(x)|

d(x,y) . Define a functionI_ε(x,y)on×by

I_ε(x,y)=

1, if|x y|< ε, 0, if|x y|ε.

Note thatI_ε(x,y)=I_ε(y,x). On the other hand,

ϕ(x)dμ(x)

B_ε(x)

h^p(y) εⁿ dμ(y)

=

×ϕ(x)h^p(y)

εⁿ I_ε(x,y)dμ(y)dμ(x)

×(ϕ(y)+ |ϕ|Li p· |x y|)h^p(y)

εⁿ I_ε(x,y)dμ(y)dμ(x)

=

×(ϕ(y)+ |ϕ|Li p· |x y|)h^p(y)

εⁿ I_ε(x,y)dμ(x)dμ(y)

=

h^p(y)dμ(y)

B_ε(y)

ϕ(y)+ |ϕ|Li p· |x y|

εⁿ dμ(x)

h^p(y)dμ(y)

B_ε(y)

ϕ(y)+ |ϕ|Li p·ε

εⁿ dμ(x)

C

(ϕ(y)+ |ϕ|Li p·ε)h^p(y)dμ(y)

=C

ϕ(y)h^p(y)dμ(y)+Cε

|ϕ|Li p·h^p(y)dμ(y).

(2.15)

Note that by the classical Hardy–Littlewood estimate,h = Mg ∈ L^p(). Then, by combining (2.14) and (2.15), and takingε→0, we obtain that

E^u_p(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].

Lemma 2.16 ([29])Let1 < p <∞and u ∈W^1,p(,Y). Then, for any sequence of positive numbers{i}converging to0, there is a subsequence{εi} ⊂ {i}such that

ilim→∞e^u_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 locallyL¹-convergence of the approximating density.

Proposition 2.17 Let1<p<∞and u∈W^1,p(,Y). Then we have e^u_p,ε→ |∇u|p in L¹_loc(),

asε→0.

Proof It suffices to show that for any ballB⊂⊂we havee^u_p,ε→ |∇u|pinL¹(B). We argue it by contradiction. Suppose that there exists some ballB⊂⊂, such that

e^u_p,|∇u|p inL¹(B),

asε→0. Then there existδ0>0 and a sequence of positive numbers{i}, going to

0, such that

B

|e^u_p,i − |∇u|p|δ0, (2.18) for all large enoughi 1.

Notice thatμ(∂B)=0. By Proposition2.13, we have E^u_p(∂B) =0. Recall that E^u_p,i =e^u_p,idμ E^u_p= |∇u|pdμweakly as Radon measures on. Thus, we have

ilim→∞E^u_p,i(B)=E^u_p(B).

That is,

ilim→∞

B

e^u_p,idμ=

B

|∇u|pdμ. (2.19)

By Lemma2.16, there is a subsequence{εi} ⊂ {i}such that

ilim→∞e^u_p,εi(x)= |∇u|p μ−a.e. x∈. (2.20) Thus, by combination of (2.19) and (2.20) implies that

e^u_p,εi → |∇u|p inL¹(B).

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

3 Harmonic Maps into Space of Curvature Bounded from Above

We continue to assume thatis a domain of an Alexandrov space and that(Y,d)is a complete metric space. In this section, we will prove the existence of harmonic maps, Theorem1.1.

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)andϕ∈Cc()

cn,2

n+2E^v₂(ϕ)=lim

ε→0

ϕ(x)dμ(x)

{y∈B_ε(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 whereis a bounded of the Euclidean spaceRⁿ.

Letω∈Sⁿ⁻¹be an unit vector. The corresponding directional energy^ωE₂^vofvis defined by (see [17, Theorem 1.8.1])

ωE₂^v(ϕ):= lim

ε→0

d²(v(x), v(x+εω))

ε² ϕ(x)dμ(x).

According to [26, Corollary 1.5], we have d(v(x), v(x+εω))

ε → |v_∗(ω)|(x) inL²_loc(), (3.3) asε→0, for some|v∗(ω)|(x)∈L²().

We denote byε := {x ∈ : |x, ∂|> ε}andA^ε_u = {x ∈ε :d(u(x),u(x+ εω))α}.

Sublemma 3.4 For anyϕ∈Cc(), we have

ωE₂^v(ϕ)=lim

ε→0

\A^ε_u

d²(v(x), v(x+εω))

ε² ϕ(x)dμ(x). (3.5)

Proof of Sublemma By definition, we have

ωE₂^v(ϕ)= lim

ε→0

\Aεu

d²(v(x), v(x+εω))

ε² ϕ(x)dμ(x) + lim

ε→0

Aεu

d²(v(x), v(x+εω))

ε² ϕ(x)dμ(x).

However, we have

A^ε_u

d²(v(x), v(x+εω))

ε² ϕ(x)dμ(x) |ϕ|_∞

A^ε_u∩suppϕ

d²(v(x), v(x+εω))

ε² dμ(x)

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

this sublemma.

We now continue the proof of Lemma3.1. Fix anyϕ ∈ Cc(). Without loss of generality, we may assume thatϕ0. Define a functionI_u^ε(x, ω)on_ε×Sⁿ⁻¹by

I_u^ε(x, ω)=

1, ifd(u(x),u(x+εω)) < α, 0, ifd(u(x),u(x+εω))α.

It follows from (3.5) that (by [17, Theorem 1.8.1]) cn,2·E^v₂(ϕ)=

Sⁿ⁻¹

ωE₂^v(ϕ)dσ(ω)

=

Sⁿ⁻¹dσ (ω)lim

ε→0

d²(v(x), v(x+εω))

ε² I_u^ε(x, ω)ϕ(x)dμ(x).

(3.6) By [17, Lemma 1.8.2], we have

d²(v(x), v(x+εω))

ε² ϕ(x)dμ(x)C|ϕ|_∞E^v₂(), (3.7) for some constantC >0, independent ofω. Hence, by (3.6), (3.7) and the bounded convergence theorem, we have

cn,2·E₂^v(ϕ)= lim

ε→0

Sⁿ⁻¹dσ (ω)

d²(v(x), v(x+εω))

ε² I_u^ε(x, ω)ϕ(x)dμ(x).

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

cn,2·E₂^v(ϕ)= lim

ε→0

ϕ(x)dμ(x)

Sⁿ⁻¹

d²(v(x), v(x+εω))

ε² I_u^ε(x, ω)dσ(ω).

(3.9) LetS(x, ε)be the sphere centered atxwith radiusε, i.e.,S(x, ε):= {y: |x y| =ε}, and dσx,εbe the area volume onS(x, ε). DefineK_u^ε_,x(y)be the characteristic function of{y∈ S(x, ε):d(u(x),u(y)) < α}. We obtain from (3.9) that

cn,2·E₂^v(ϕ)=lim

ε→

ϕ(x)dμ(x)

d²(v(x), v(y))

εⁿ⁺¹ K_u^ε_,x(y)dσx,ε(y). (3.10)

Divding both sides of (3.10) byn+2, we have cn,2

n+2E₂^v(ϕ)

= ₁

0

λⁿ⁺¹dλlim

ε→0

ϕ(x)dμ(x)

S(x,λε)

d²(v(x), v(y))

(λε)ⁿ⁺¹ K_u^λε_,x(y)dσx,λε(y).

(3.11) According to [17, Theorem 1.5.1], there existsε0>0, such that

ϕ(x)dμ(x)

S(x,δ)

d²(v(x), v(y))

δⁿ⁺¹ dσx,δ(y)1+E^v₂(ϕ)

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

cn,2

n+2E^v₂(ϕ)

= lim

ε→0

₁

0 λⁿ⁺¹dλ

ϕ(x)dμ(x)

S(x,λε)

d²(v(x), v(y))

(λε)ⁿ⁺¹ K_u^λε_,x(y)dσx,λε(y)

= lim

ε→0

₁

0

dλ

ϕ(x)dμ(x)

S(x,λε)

d²(v(x), v(y))

εⁿ⁺¹ K_u^λε_,x(y)dσx,λε(y)

= lim

ε→0

ϕ(x)dμ(x) ₁

0

dλ

S(x,λε)

d²(v(x), v(y))

εⁿ⁺¹ K_u^λε_,x(y)dσx,λε(y).

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

cn,2

n+2E₂^v(ϕ)= lim

ε→0

ϕ(x)dμ(x)

{y∈B_ε(x):d(u(x),u(y))<α}

d²(v(x), v(y)) εⁿ⁺² dμ(y).

Equivalently,

ε→lim0

ϕ(x)dμ(x)

{y∈B_ε(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 whereis a bounded domain of a Lipschitz manifold with anL^∞-Riemannian metric. Equivalently, we want to show

ε→lim0

ϕ(x)dμ(x)

{y∈B_ε(x):d(u(x),u(y))α}

d²(v(x), v(y))

εⁿ⁺² dμ(y)=0. (3.14)