We shall consider harmonic maps from n-dimensional compact connected Riemannian manifold with boundary to the unit sphere under the Dirichlet boundary condition

Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 34, pp. 1–7.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

ANOTHER PROOF OF THE REGULARITY OF HARMONIC MAPS FROM A RIEMANNIAN MANIFOLD TO

THE UNIT SPHERE

JUNICHI ARAMAKI

Abstract. We shall consider harmonic maps from n-dimensional compact connected Riemannian manifold with boundary to the unit sphere under the Dirichlet boundary condition. We claim that if the Dirichlet data is smooth and so-called “small”, all minimizers of the energy functional are also smooth and “small”.

1. Introduction

Let (M, g) be a n-dimensional Riemannian manifold with boundary ∂M en- dowed with a smooth Riemannian metricg. For any p∈M, let (x₁, . . . , x_n) be a coordinate system nearp. Thengcan be represented by

g=

n

X

α,β=1

g_αβdx_α⊗dx_β

where (g_αβ) is a positive definite symmetric n×n matrix. We write the inverse matrix of (gαβ) by (g^αβ) and the volume element of (M, g) bydvg =√

gdx where g = det(gαβ), and we use the notations that for any vector fields u,v, hu,vig = g(u,v) and|u|²_g=hu,uig. We view maps fromM into ak-dimensional unit sphere S^k ⊂R^k+1, extrinsically. The Sobolev space W^1,2(M,R^k+1) is standardly defined and the spaceW^1,2(M,S^k) is defined by

W^1,2(M,S^k) =

u= (u¹, . . . , u^k+1)∈W^1,2(M,R^k+1); u(x)∈S^k a.e. x∈M . For anyu∈W^1,2(M,S^k), the Dirichlet energy density is defined by

e(u) = 1

2|∇u|²_g (1.1)

where |∇u|²_g =Pk+1

i=1 |∇uⁱ|²_g. In any local coordinate system x= (x1, . . . , xn), we see that

e(u) = 1 2

n

X

α,β=1 k+1

X

i=1

g^αβ∂uⁱ

∂x_α

∂uⁱ

∂x_β,

2000Mathematics Subject Classification. 58E20, 53C43, 58E30.

Key words and phrases. Harmonic maps; minimizing harmonic maps; weak Harnack inequality.

c

2014 Texas State University - San Marcos.

Submitted September 16, 2013. Published January 27, 2014.

1

and the Dirichlet energy is defined by E(u, M) =

Z

M

e(u)dv_g. (1.2)

We sayu∈W^1,2(M,S^k) is weakly harmonic map, if Z

M n

X

α,β=1

g^αβ∂u

∂xα

· ∂φ

∂xβ

+ ∂u

∂xα

· ∂u

∂xβ

u·φ

dvg= 0 (1.3) for any φ∈C₀^∞(M,R^k+1) where· denotes the Euclidean inner product in R^k+1. Thenusatisfies the harmonic map equation in the sense of distribution

∆_gu+

n

X

α,β=1

g^αβ ∂u

∂x_α· ∂u

∂x_βu=0 inM (1.4)

where ∆_g is the Laplace-Beltrami operator on (M, g) given by

∆g= 1

√g

n

X

α,β=1

∂

∂xα

√ gg^αβ ∂

∂xβ

.

Next we sayu∈W^1,2(M,S^k) is a minimizing harmonic map, if for any Ω⊂M, E(u,Ω) :=

Z

Ω

e(u)dvg≤E(v,Ω) (1.5)

for allv∈W^1,2(Ω,S^k) withv|_∂M =u|_∂M.

The regularity of minimizing harmonic maps has been studied by many authors for a general target Riemannian manifold N instead of S^k. For the case where dimM = 2, Morrey [13] showed that ifu∈W^1,2(M, N) is a minimizing harmonic map, thenu∈C^∞(M, N). Forn≥3, Schoen and Uhlenbeck [14] have shown that if we define the singular set of any minimizing mapu∈W^1,2(M, N) by

sing(u) ={x∈M;uis discontinuous atx}, then sing(u) is a closed set, and it is discrete forn= 3, and

dim_H(sing(u))≤n−3

forn≥4 where dim_H(sing(u)) is the Hausdorff dimension of sing(u). Moreover, it is well known thatuis analytic inM\sing(u) (cf. Borchers and Garber [5]).

For p∈N, r > 0, let Br(p) = {q ∈ N; distN(q, p)≤r} be the closed geodesic ball with centerpand radius r, and let C(p) be the cut locus ofp. We callBr(p) is a regular ball if the following two conditions hold.

(i) √

κr < π/2 whereκ= max{0,sup_B

r(p)K^N},K^N is the sectional curvature ofN.

(ii) C(p)∩B_r(p) =∅.

Hildebrandt et al. [9] have established the following existence theorem of smooth harmonic maps with given boundary data contained in a regular ball. (see also Lin and Wang [12, Theorem 3.1.7]).

Theorem 1.1 ([9]). Suppose that Br(p) ⊂ N is a regular ball and Ω ⊂ M is a bounded domain andg: Ω→Br(p)is continuous map and has finite energy. Then there exists a harmonic map u∈C^2+α(Ω, N)∩C⁰(Ω, N) withu|∂M =g.

As the first step of their proof, they considered the following variational problem.

Find a minimizer of

inf

u∈V

Z

Ω

e(u)dv_g

where the admissible space V is as follows. Choose r1 ∈ (r, π/2√

κ) such that Br1(p)⊂N is also regular ball, and define

V ={u∈W^1,2(Ω, Br₁(p));u|∂M =g}.

This admissible space seems to be restrictive. Thus in the present paper, we report that in order to get the same result for the target manifold N =S^k, we can take the admissible spaceV =W^1,2(M,S^k,g) :={u∈W^1,2(M,S^k);u|∂M =g}.

We note that in the case where N = S^k, since K^N = 1 and C(p) = {−p}, if 0< r < π/2, then the ballBr(p) is regular.

2. Preliminaries

LetM be an-dimensional connected compact Riemannian manifold with smooth boundary ∂M and S^k ⊂R^k+1 the unit sphere inR^k+1 (k≥2). For every p∈S^k and r > 0, we denote the closed geodesic ball in S^k with center p and radius r byBr(p). Throughout this paper we treat theBr(p) which is an closed ball with 0< r < π/2, soBr(p) is a regular ball in this case. We denote the standard Sobolev space byW^1,2(Ω,R^k+1), and define

W^1,2(Ω,S^k) ={u∈W^1,2(Ω,R^k+1);u(x)∈S^k a.e. x∈M}.

Lete:∂M →S^k be a smooth given vector field, for instance,e∈C^2+α(∂M,S^k), and define

W^1,2(M,S^k,e) ={u∈W^1,2(M,S^k);u|∂M =e}.

Here we assume the hypotheses

(H1) e∈C^2+α(∂M,S^k) has a finite energy extensionee∈W^1,2(M,S^k) such that ee|∂M =e.

Remark 2.1. It is not trivial that W^1,2(M,S^k,e)6=∅. However ifM = Ω ⊂Rⁿ is a boundedC² domain, Hardt and Lin [8, Theorem 6.2] (cf. [12, Lemma 2.2.10]) have proved the fact in the case where the target space is a more general simply connected Riemannian manifold N (i.e., Π0(N) = Π1(N) = 0) that any map e∈W^1/2,2(∂M, N) admits a finite energy extension ee∈W^1,2(Ω, N). Recall that N =S^k has Π₀(S^k) = Π₁(S^k) = 0, unlessk= 1.

u ∈ W^1,2(M,S^k) is called weakly harmonic map in the sense of Introduction with boundary dataeif for anyv∈W₀^1,2(M,R^k+1),

Z

M

(h∇u,∇vig− |∇u|²_gu·v)dx= 0, (2.1) andu|_∂M =ewhere

h∇u,∇vig=

k+1

X

i=1

h∇uⁱ,∇vⁱig

foru= (u¹, . . . , u^k+1),v= (v¹, . . . , v^k+1) andu·vis the standard Euclidean inner product. Thenusatisfies the following equations, in the sense of distributions,

∆gu+|∇u|²_gu=0 inM,

u=e on∂M. (2.2)

We also say that u∈ W^1,2(M,S^k) is a minimizing harmonic map with boundary dataeifuis a minimizer of

inf

u∈W^1,2(M,S^k,e)

Z

M

|∇u|²_gdvg. (2.3)

Lemma 2.2. Any minimizing harmonic map u ∈ W^1,2(M,S^k) is a weakly har- monic map.

The proof is well known. For example, see [12, Proposition 2.1.5].

We state the main theorem.

Theorem 2.3. Assume thatM is aC^2+αconnected compact Riemannian manifold with boundary ∂M for some 0 < α < 1 and assume that a boundary data e ∈ C^2+α(∂M,S^k)satisfying(H1)is given, and satisfies that e(∂M)⊂Br(p)for some point p∈S^k and0< r < π/2. Then ifuis any minimizer of

u∈Vinf Z

M

|∇u|²_gdvg

where V =W^1,2(M,S^k,e), then u(M)⊂Br(p)and uis a unique harmonic map inC^2+α(M,S^k).

Remark 2.4. In [9] and[12, Theorem 3.17], they took the admissible space V as V ={u∈H¹(M,S^k,e);u(M)⊂Br1(p)}for somer < r1< π/2, and they call such solution a “small solution”. However, we can remove the rather stronger condition.

We emphasize that even if we takeW^1,2(M,S^k,e)as the admissible space, we can get the same result as[9], and we seem to make more natural. To do so, we shall use the weak Harnack inequality (cf. Gilbarg and Trudinger[7, Theorem 8.18]or Chen and Wu [6, Chapter 4, Lemma 1.3]) and the maximum principle for minimizing harmonic maps (cf. Jost [11, Lemma 4.10.1]). Such strategy also appear in the author’s papers Aramaki [1, 2, 3]and Aramaki, Chinen, Ito and Ono[4].

3. Proof of Theorem 2.3

For the proof we need the following lemma which is can be found for example in [12, Proposition 2.1.5].

Lemma 3.1. Let V =W^1,2(M,S^k,e). Then

u∈Vinf Z

M

|∇u|²_gdvg

is achieved inV.

Let u ∈ W^1,2(M,S^k,e) be a minimizer of (2.3). Then u satisfies the Euler- Lagrange equation in the sense of distribution

−∆gu=|∇u|²_gu inM,

u=e on∂M . (3.1)

Proposition 3.2. Let e ∈ C^2+α(∂M,S^k) for some 0 < α < 1 and assume that e(∂M)⊂Br(p) for some p∈S^k and 0< r < π/2. Then for any minimizeru of (2.3)satisfiesu(Ω)⊂Br(p).

Proof. After the rotation of coordinate axis ofR^k+1, we can choose the centerp of Br(p) so that p = (1,0, . . . ,0). We write e(x) = (e¹(x), . . . , e^k+1(x)). The hypothesis means that e¹(x)≥ cosr for x∈∂M. Let u= (u¹, . . . , u^k+1) be any minimizer of (2.3). Since u¹ ∈ W^1,2(M), it is well known that |u¹| ∈ W^1,2(M) and|∇|u¹||=|∇u¹|a.e. inM. Definew= (w₁, . . . , w^k+1) = (|u¹|, u², . . . , u^k+1)∈ W^1,2(M,R^k+1). Since u¹ =e¹ >0 on∂M, we can see that w ∈ W^1,2(M,S^k,e), andw is also a minimizer of (2.3). Thereforewalso satisfies (3.1), andw∈C^2+α near the boundary (cf. Schoen and Uhlenbeck [15, Proposition 3.1]. In particular, w¹ satisfiesw¹≥0 and

−∆_gw¹=|∇w|²_gw¹ inM,

w¹=e¹ on∂M (3.2)

For anyq∈M, choose a local coordinate neighborhoodUq and a local coordinate system (x1, . . . , xn). Thenw¹ is a bounded non-negative weak supersolution of

∆_g= 1

√g

n

X

α,β=1

∂

∂xα

√ gg^αβ ∂

∂xβ

;

that is to say, ∆_gw¹≤0 inU_q. We can apply the weak Harnack inequality (cf. [7, Theorem 8.18] or [6, Chapter 4, Lemma 1.3]). Thus for any 1 ≤p < n/(n−2), B2R⊂Uq

ess inf_BRw¹≥c 1

|B2R| Z

B_2R

(w¹)^pdx^1/p

where c > 0 depends on n, p. Since w¹ ∈ C^2+α near the boundary and w¹ = e¹ ≥ cosr > 0 on ∂M, there exists δ > 0 such that if we define Mδ = {x ∈ M; dist(x, ∂M)≤δ}, thenw¹≥c0:= cosr/2 inMδ. Since dimHsing(w¹)≤n−3 (in the case where n = 3, sing(w¹) is discrete), for any x0 ∈ M \sing(w¹), we can choose x1 ∈ Mδ and a continuous curve l in M joining x0 and x1 such that l∩sing(w¹) =∅. For everyx∈l, there existsR >0 such thatB_2R(x) is contained in a local coordinate neighborhood and

ess inf_BR(x)w¹≥c 1

|B2R(x)|

Z

B_2R(x)

(w¹)^pdx^1/p

. (3.3)

Sincel is compact, there exist finitely many R_j >0 andx_(j)∈l (j= 1,2, . . . , N) such that∪^N_j=1BR_j(x_(j))⊃landx₍₁₎=x0, x_(N)=x1. Since ess inf_BR(x(N))w¹>0, it follows from (3.3) that ess inf_BR(x(N−1))w¹ > 0. Repeating this procedure, we have ess inf_BR(x0)w¹ >0. In particular, w¹(x₀)>0. Thus we see thatw¹ >0 in M\sing(w¹). Hence we see thatu¹>0 inM\sing(u¹) oru¹<0 inM\sing(u¹).

Sinceu¹=e1>0 on∂M, we haveu¹>0 inM \sing(u¹). Since u¹ is continuous near∂M, there existδ >0 andc0>0 such thatu¹≥c0onMδ. DefineM^δ ={x∈ M; dist(x, ∂M)≥δ}. ChooseR >0 so that 2R < δand fix 1≤p < n/(n−2). For anyy ∈M^δ, there existsc⁰ =c⁰(n, p)>0 such that for any B2R(y) contained in a local coordinate neighborhood,

ess inf_BR(y)u¹≥c⁰ 1

|B2R(y)|

Z

B2R(y)

(u¹)^pdv_g1/p

.

Since M^δ is compact, there exists finitely many points yi and positive numbers Ri(i= 1,2, . . . , L) such that∪^L_i=1BR_i(yi)⊃M^δ. If we define

ci=c⁰_i 1

|B2R_i(yi)|

Z

B_2Ri(y_i)

(u¹)^pdx^1/p

(i= 1,2, . . . , L),

and c= min{c0, c1, . . . , cL}, we haveu¹≥c a.e. onM. Therefore we can find r⁰ withr < r⁰ < π/2 such thatn(M)⊂Br⁰(p).

Next, we use the following maximum principle by Jost.

Lemma 3.3 ([11]). Let B0 and B1 be closed subsets ofS^k andB0⊂B1. Suppose that there exists a C¹ retraction mapΠ :B1→B0 satisfying the condition

|∇Π(x)(v)|<|v| for allx∈B1\B0, and allv∈TxS^k.

For any boundary datae:∂M →B₀, ifu∈W^1,2(M,S^k,e) :M →B₁is an energy minimizing map of (2.3) with the boundary datae, thenu(x)∈B0 a.e. x∈M.

We apply this lemma withB0=Br(p), B1=Br⁰(p), we see thatu(M)⊂Br(p).

Then we can see thatu∈C^2+α(M,R^k+1) by the regularity theory in [14, 15] and [9]. The uniqueness of the solution follows from J¨ager and Kaul [10]. This completes the proof.

Acknowledgments. We would like to thank the anonymous referee who indicates some errors and gave us some advice for a previous version of this article.

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Junichi Aramaki

Division of Science, Faculty of Science and Engineering, Tokyo Denki University, Hatoyama-machi, Saitama 350-0394, Japan

E-mail address:[email protected]