Gradient estimates and blow-up analysis for stationary harmonic maps

Gradient estimates and blow-up analysis for stationary harmonic maps

By Fang-Hua Lin

Abstract

For stationary harmonic maps between Riemannian manifolds, we provide a necessary and sufficient condition for the uniform interior and boundary gradient estimates in terms of the total energy of maps. We also show that if analytic target manifolds do not carry any harmonic S², then the singular sets of stationary maps are m≤n−4 rectifiable. Both of these results follow from a general analysis on the defect measures and energy concentration sets associated with a weakly converging sequence of stationary harmonic maps.

Introduction

This paper studies some general properties of a sequence of weakly con- verging stationary harmonic maps between compact Riemannian manifolds.

In this part I of the paper we shall examine mainly two issues, the gradient estimates and the compactness of stationary maps in the H¹-norm. In Part II of this paper, we shall study asymptotic behavior at infinity of stationary harmonic maps from Rⁿ into a compact Riemannian manifold with bounded normalized energies. We shall also discuss there the analogous results as de- scribed in this paper for the heat flow case. The main results were announced in [Li].

Let u : M → N be a stationary harmonic map (cf. §1 below for the precise definition). Here M, N are compact, smooth Riemannian manifolds (with possible nonempty, smooth boundary ∂M). We are interested in the following question:

Under what conditions on the target manifold N is an estimate of the form

(0.1) k∇ukL^∞(M)≤C(M, N, E), where E= Z

M|∇u|²(x)dx, valid?

Naturally (0.1) contains both local interior and local near the boundary estimates. In the latter case, the right-hand side of (0.1) should also depend

786 FANG-HUA LIN

on a certain norm of u |∂M. Trivial examples, such as conformal maps be- tween spheres, or finite energy harmonic maps from R² into S², show there are obstructions for (0.1). One of the main results of the present paper is the following.

Theorem A. An interior gradient estimate of the form (0.1) is true for stationary maps provided that N does not carry any harmonic spheres, S^l, l= 2, . . . , n−1, n= dimM ≥3.

Here we say N does not carry harmonics S^l if there is no smooth, non- constant harmonic map from S^l intoN. We note that, forn= 2, the estimate (0.1) follows rather easily from the proof of the well-known theorem of Sacks- Uhlenbeck [SaU] provided thatN does not carry any harmonicS².

Theorem A generalizes the earlier results by Schoen-Uhlenbeck [SchU]

and, independently, by Giaquinta-Giusti [GG] for energy minimizing maps.

Note that, for energy minimizing maps, the boundary regularity is always true (see [SU2]) and this combined with the compactness of energy minimizing maps in H¹-norms imply the uniform boundary regularity for energy minimizing maps (cf. [M]). Such uniform boundary regularity can easily be seen to fail for smooth harmonic maps (cf.§4 below). Nevertheless, we have the following boundary regularity theorem.

Theorem B. Let M be a smooth, compact Riemannian manifold with smooth boundary∂M,and let φ:∂M →N be aC²-map. Supposeu:M →N is a smooth harmonic map with u |∂M= φ. Then there is a positive con- stant δ0 =δ0(M, N, φ, E) such that |∇u(x)| ≤C(M, N, φ, E), for all x ∈ M, dist(x, ∂M)≤δ0, provided that N does not carry any harmonic S². Here

E = Z

M|∇u|²dx.

As a consequence of Theorems A and B, we have:

Corollary. If the universal coverN^e of N supports a pointwise convex function,then under the same assumptions as Theorem B,

k∇ukL^∞(M)≤C(M, N, φ, E).

This latter result implies the well-known theorems of Eells and Sampson [ES] and of Hamilton [H] for nonpositively curved targetsN. It also generalizes results of [GH] and [Sch] (cf. [DL] for related discussions.)

To prove Theorem A and Theorem B, we have to consider a weakly con- verging sequence of stationary harmonic maps on a geodesic ball Br(p)⊂M.

We let {ui}^∞i=1 be a sequence of stationary maps from Br(p) intoN with Z

B_r(p)|∇ui|²(x)dx≤Λ.

Suppose ui * u weakly in H¹(Br(p), N). Then we define theenergy concen- tration set Σ (as in [Sch] for smooth maps) as follows:

(0.2) Σ =∩r>0

(

x∈Br(p) : lim inf

i→∞ r²⁻ⁿ Z

B_r(x)|∇ui|²(y)dy≥ε0

) . Here ε0 = ε0(M, n, N) is a suitable positive constant. We also introduce a nonnegative Radon measure ν such that µ = |∇u|²(x)dx+ν; here µ is the weak limit of Radon-measures |∇ui|²(x)dx onBr(p). We then show that

Σ = sptν∪singu;

(0.3)

ν(x) = Θ(x)Hⁿ⁻²bΣ, (0.4)

for anHⁿ⁻²-measurable function Θ(x) such thatε0≤Θ(x), and Θ(x) is locally uniformly bounded onBr(p);

(0.5) Hⁿ⁻²(Σ∩Bρ(p))≤C(ε0, M, N,Λ, ρ), for any 0< ρ < r.

Therefore ui → u strongly in H_loc¹ (Br(p), N) if and only if |∇ui|²dx *

|∇u|²dx if and only ifν = 0 if and only if Hⁿ⁻²(Σ) = 0.

Next we identify Br(p) (for r small, one can always do that) with a ball B1+δ0(0) inRⁿendowed with some nice metric. We letMbe the set of all such Radon measuresµ described above. That is, there is a sequence of stationary harmonic maps {ui} (with respect to suitable metrics on B1+δ0) from B1+δ0

intoN, such that|∇ui|²dx * µ. Note that|∇ui|²dxis the energy density with respect to a metric (may depend on i, but uniformly nice). We then show M has the following properties.

(0.6) µ∈ M, x∈B1,0< λ < δ0, then µx,λ∈ M.

Here µx,λ(A) =µ(x+λA), for Borel measurableA⊂B1+δ0; (0.7)

µ∈ M, x∈B1,{λk} &0 there is a subsequence{λk⁰} such thatµx,λ_k‘ * η ∈ M.

Moreover,η0,λ=η, for allλ >0.

The main result concerningMis the following.

TheoremC. For any µ∈ M,

µ = |∇u|²dx+ν, and π(µ) = Σ = (sptν∪singu), Σ is an Hⁿ⁻²-rectifiable set. Thus ν is also Hⁿ⁻²-rectifiable.

The above theorem follows from the arguments of D. Priess [P], and earlier contributions by Besicovitch, Federer, Marstrand and Mattila. See references in [P]. Here we present a self-contained direct proof.

788 FANG-HUA LIN

The next key step towards the proof of Theorem A and Theorem B is Lemma 3.1 (cf. also Lemma 4.11) that says: If Hⁿ⁻²(Σ)> 0, then there is a nonconstant, smooth harmonic map fromS² into N. We therefore obtain the following characterization:

Any sequence of weakly converging stationary harmonic maps converges strongly in the H¹-norm if and only if ν = 0 for all µ ∈ M, if and only if Hⁿ⁻²(Σ) = 0, π(µ) = Σ, for any µ ∈ M, if and only if there is no smooth, nonconstant harmonic map fromS² intoN.

The above statements lead to the following.

TheoremD. If there is no smooth,nonconstant harmonic map from S² into N, then the singular set of any stationary harmonic map has dimension m ≤ n−4. Moreover, if N is, in addition, analytic, then the singular set of any stationary harmonic map ism≤n−4 rectifiable.

The proof of Theorem D follows from the work of L. Simon [S3] and our characterization above.

The paper is organized as follows. In Section 1 we gather together various facts concerning stationary harmonic maps. In addition, we also establish a few preliminary results concerning the defect measures ν for µ∈ M and the concentration sets. In particular, we establish the properties ofµ∈ Mso that Federer and Almgren’s dimension-reducing principle can be applied.

The rectifiability of Σ andν are established by three key lemmas in Sec- tion 2. In Sections 3 and 4 we prove Theorem A and Theorem B, respectively.

The final section contains other discussions and describes some necessary modi- fications required in order to generalize all proofs in Sections 1 through 4, which are for the Euclidean domains, to the general Riemannian domains.

1. Preliminaries

Here we gather together some basic facts about stationary harmonic maps and related notions which are needed for the sequel. For a more detailed discussion of the facts reviewed here, we refer the reader to various articles cited below, and also monographs [Sim], and [J].

First, Ω will denote a bounded smooth domain of Rⁿ endowed with the standard Euclidean metric. We shall briefly discuss in Section 5 the exten- sion of the results here to the case where Ω is equipped with an arbitrary smooth Riemannian metric. This extension involves purely routine technical modifications of the arguments which we develop below for the Euclidean case.

Note that N denotes a smooth compact Riemannian manifold, which, by Nash’s isometric embedding theorem, we assume is isometrically embed- ded in some Euclidean space R^k. Also, H¹(Ω, N) denotes the set of maps

u ∈ H¹(Ω,R^k) such that u(x) ∈ N for a.e. x ∈ Ω. For a measurable subset A⊂Ω,

E(u, A) = Z

A|∇u|²dx.

Now u∈H¹(Ω, N) is said to beenergy-minimizing in Ω if E(u,Ω)≤E(v,Ω) whenever v∈H¹(Ω, N) withv=u on ∂Ω.

Ifu∈H⁽Ω, N) is energy-minimizing, thenuisstationary(cf. [Sch]) in the sense that

(1.1) d

dsE(Ω, us)^¯¯_¯¯

s=0

= 0

whenever the derivative on the left exists, provided that u0 = u and us ∈ H¹(Ω, N) withus(x) ≡u(x) for x ∈ ∂Ω and s ∈ (−ε, ε) for some ε > 0. In particular, by considering a familyus = Π(u+sξ), where Π denotes the nearest point projection of an R^k neighborhood ofN onto N, and ξ∈C₀^∞(Ω,R^k), we obtain the system of equations

(1.2) ∆u+A(u)(∇u,∇u) = 0 weakly in Ω.

Here ∆ is the usual Laplacian on Ω;A(u) denotes the second fundamental form of N at pointu. A mapu ∈H¹(Ω, N) which satisfies (1.2) is called aweakly harmonic map.

On the other hand if us(x) = u(x+sξ(x)), where ξ ∈ C₀^∞(Ω,Rⁿ), then (1.1) implies the integral identity

(1.3) Z

Ω

Xn i,j=1

³

δij|∇u|²−2DiuDju^´Diξ^jdx, ξ= (ξ¹,· · ·, ξ^k)∈C₀^∞(Ω,Rⁿ).

Notice that (1.3) implies (for a.e. ρ such thatBρ(z)⊂Ω) Z

B_ρ(z)

Xn i,j=1

³

δij|∇u|²−2DiuDju

´

Diξ^jdx (1.4)

= Z

∂Bρ(z)

Xn i,j=1

³

δij|Du|²−2DiuDju^´νiξ^j

for any ξ = (ξ¹,· · ·, ξⁿ) ∈ C^∞( ¯Bρ(z),Rⁿ), where ν = (x−z)/|x−z| is the outward pointing unit normal for∂Bρ(z). In particularξ(x) =x−z and then (1.4) implies

(1.5) (n−2)

Z

Bρ(z)|∇u|²dx=ρ Z

∂Bρ(z)

³|∇u|²−2|URz|^2´, a.e.ρ,

such that ¯Bρ(z)⊂Ω, where

UR_z = (x−z)

|x−z| · ∇u=uν.

790 FANG-HUA LIN

The latter can be written d

dρ Ã

ρ²⁻ⁿ Z

B_ρ(z)|∇u|²dx

!

= 2 Z

∂B_ρ(z)

|RzUR_z|² Rⁿ_z , whence by integration

(1.6) ρ²⁻ⁿ Z

B_ρ(z)|∇u|²dx−σ²⁻ⁿ Z

B_σ(z)|∇u|²dx= 2 Z

B_ρ(z)/B_σ(z)

|RzUR_z|² Rⁿ_z dx for any 0< σ < ρwith ¯Bρ(z)⊂Ω. HereRz =|x−z|. An obvious consequence of (1.6) is that

(1.7) ρ²⁻ⁿ

Z

B_ρ(z)|∇u|²dx is an increasing function ofρ so that the limit

(1.8) Θu(z) = lim

ρ→0ρ²⁻ⁿ Z

B_ρ(z)|∇u|²dx

exists at every point z ∈ Ω. Note that Θ_u(z) is an upper semicontinuous function of z∈Ω in the sense that

(1.9) Θ_u(z)≥lim sup

z_i→z

Θ_u(zi).

Letting σ→0 in (1.6) we obtain (1.10) ρ²⁻ⁿ

Z

Bρ(z)|∇u|²dx−Θ_u(z) = 2 Z

Bρ(z)

|RzURz|² Rⁿ_z dx.

By using (1.5) we have the alternative identity 2

Z

Bρ(z)

|RzUR_z|²

Rⁿ_z dx = ρ³⁻ⁿ n−2

Z

∂Bρ(z)

(|∇u|²−2|UR_z|²)−Θ_u(z) (1.11)

≤ (n−2)⁻¹ρ³⁻ⁿ Z

∂Bρ(z)|∇u|²−Θ_u(z).

For a map u ∈ H¹(Ω, N), we define the regular and singular sets, regu and singu, by

regu = {z∈Ω :u∈C^∞ in a neighborhood ofz}

singu = Ω\regu.

Notice that by definition reg u is open, and hence sing u is automatically relatively closed in Ω.

An important consequence of the small energy regularity theorem of Bethuel [B] (cf. also [E]) for stationary harmonic maps is that the regular set, regu, of u can be characterized in terms of density as follows:

z∈regu ⇐⇒ Θ_u(z)≤ε0(n, N)>0 ⇐⇒ Θ_u(z) = 0,

whereε0 =ε0(n, N)>0 is independent of u; equivalently, (1.12) z∈singu ⇐⇒ Θu(z)≥ε0 ⇐⇒ Θu(z)>0.

For energy minimizing maps, (1.12) was shown in the earlier work of Schoen- Uhlenbeck [SU] (cf. also [GG]). In fact, Schoen-Uhlenbeck proved a much stronger statement that can be described as follows.

Supposeu:Bρ(z)→N is an energy-minimizing map with ρ²⁻ⁿ

Z

Bρ(z)|∇u|²dx≤Λ and inf

λ∈R^kρ⁻ⁿ Z

Bρ(z)|u−λ|²dx≤ε.

Then if ε≤ε(n, N,Λ) thenBρ/2(z)⊂regu and

(1.13) ^X

B_ρ/2(z)

ρ^k|D^ku| ≤Ckε^1/2, k≥0.

One of the crucial consequences of the above theorem of Schoen-Uhlenbeck is that any weakly converging sequence of energy-minimizing mapsuj ∈H¹(Ω, N), uj * u weakly in H¹(Ω, N), converges strongly in H_loc¹ (Ω, N); cf. [SU]. The limituis also an energy-minimizing map. This latter fact was shown by Luck- haus [Lu] (cf. also [HL]). It is easy to see from examples below that the same statement cannot be true in general for stationary harmonic maps.

Example 1.1. Let v be a conformal map from S² into S². Then v gives rise to a finite energy harmonic mapu fromR² intoS² by composing with the inverse of the stereographic projection of S² onto R². Note that the converse is also true by Sacks-Uhlenbeck’s theorem [SaU]. Let uλ(x) = u(λx), x∈ R²; then uλ * constant = u(∞) weakly in H¹(R²,S²) as λ → ∞. Moreover,

|∇uλ|²dx *8πN δ0 as Radon measures. HereN =|degv|>0.

Now if we view u, uλ as smooth harmonic maps from Rⁿ into S² (thus u, uλ are independent of variables x3,· · ·, xn), thenuλ * constant as λ→ ∞ and |∇uλ|²dx *8πN Hⁿ⁻²b{0} ×Rⁿ⁻². Here Hⁿ⁻²b{0} ×Rⁿ⁻² denotes the (n−2) dimensional Hausdorff measure restricted to the (n−2)-dimensional plane {0} ×Rⁿ⁻² inRⁿ.

Example 1.2. In [HLP], we constructed examples of smooth stationary, axially symmetric harmonic mapsufromB³ intoS²with isolated singularities of degree zero. For suchu, we let the origin 0∈singu, and singu∩Bε(0) ={0} for some ε > 0. Then the degree u :∂Br(0) → S² is zero, for all r ∈ (0, ε).

Moreover, for uλ * constant, uλ(x) =u(λx), and |∇uλ|²dx *16πH¹b{0} × R¹, asλ→0⁺.

Example1.3. In [Po], Poon constructed examples of stationary harmonic maps u from B³ into S² such that U|∂B³(x) = x, and that u is smooth ev- erywhere except at one point on the boundary of B³. On the other hand, Riviere [R] constructed finite energy weakly harmonic mapsufromB³ intoS² such thatu is discontinuous everywhere on B³.

792 FANG-HUA LIN

From now on, we should assume B1 ⊂⊂ Ω, and let HΛ be the set of stationary harmonic maps u from Ω intoN such that E(u,Ω) ≤Λ, for some Λ > 0. The following result was shown in [Sch] for smooth harmonic maps (instead of stationary harmonic maps).

Proposition 1.4. Any map u in the weak H¹(Ω, N) closure of HΛ is smooth and harmonic outside a relatively closed subset of Ω with locally finite Hausdorff (n−2)-dimensional measure.

Proof. The proof given in [Sch] uses only the energy monotonicity and

“small energy regularity theorem.” Since both of these statements are true for stationary harmonic maps, the proof can be directly carried over here. In fact, the following lemma is essentially equivalent to Proposition 1.4. For the reader’s convenience we provide a proof below.

Lemma1.5. Let {ui} be a sequence of maps in HΛ,and suppose ui * u weakly in H¹(Ω, N). Let

Σ =∩r>0{x∈B1 : lim inf

i→∞ r²⁻ⁿ Z

Br(x)|∇ui|²dy≥ε0}.

Then Σis closed in B1 and

Hⁿ⁻²(Σ)≤C(ε0,Λ, N, δ0) where δ0= dist(B1, ∂Ω)>0.

Proof. Supposex0∈B1/Σ; then there isr0 >0 such that lim inf

i→∞ r₀²⁻ⁿ Z

Br0(x0)|∇ui|²dy < ε0. That is, there is a sequence ni→ ∞ such that

sup

n_i r²₀⁻ⁿ Z

B_r0(x0)|∇un_i|²dy < ε0.

Via the small energy regularity theorem of Bethuel [B] (cf. also [E]), one has sup

n_i

sup

x∈B_r

0/2(x0)|∇un_i(x)| ≤C0√ εor⁻₀¹, for some constantC0=C0(n, N). In particular,

sup

n_i sup

x∈B_r

0/4(x0)

r²⁻ⁿ Z

B_r(x)|∇un_i(x)|dy≤ ε0

2

wheneverr ≤r1(r0, ε0, N), for somer1 >0. ThereforeBr0/4(x0)⊂B1/Σ, and Σ is closed.

Next, for anyδ0> δ >0, we may find a finite collection of balls{Br_j(xj)} that cover Σ so thatrj < δ, that the collection{Br_j/2(xj)}is disjoint and that

xj ∈Σ. Forisufficiently large we then have µ1

2rj

¶_2−nZ

B_{rj /}2(x_j)|∇ui|²dy≥ε0 for all j.

Hence

X

j

rⁿ_j⁻² ≤ C(n) εo

E(ui,Ω)≤ C(n) ε0

Λ.

It follows that

Hⁿ⁻²(Σ)≤ C(n) ε0

Λ.

Letui ∈HΛ be such thatui * uinH¹(Ω, N), and let Σ be as in Lemma 1.5. Consider a sequence of Radon measure µi = |∇ui|²dx, i = 1,2, . . .; without loss of generality, we may assumeµi * µweakly as Radon measures.

By Fatou’s lemma, we may write

(1.14) µ=|∇u|²dx+ν

for some nonnegative Radon measure ν on Ω.

Lemma1.6. On the closed ballB1+δ0 ⊂Ω, (i) Σ = spt(ν)∪singu;

(ii) ν(x) = Θ(x)Hⁿ⁻²bΣ, x ∈B1 where ε0 ≤Θ(x) ≤δ²₀⁻ⁿΛ2ⁿ⁻², for Hⁿ⁻²- a.e. x∈Σ.

Proof. Supposex0∈B1/Σ; then the proof of Lemma 1.5 and higher order estimates (1.13) imply that there is a subsequence{un_i}such that un_i →u(x) inC^1,α(B_r0/2(x0)), for some 0< r0 < δ0. Thus

µn_i |B_r

0/2(x0)*|∇u|² |B_r

0/2(x0) asi→ ∞,

and u ∈ C^1,α(B_r0/2(x0)). The latter implies x0 6∈ singu and x0 6∈ sptν as ν ≡0 onBr0/2(x0)).

Suppose nowx0∈Σ, then for anyr∈(0, δ0), µi(Br(x0))

rⁿ⁻² ≥ ε0

2 for a sequence ofi→ ∞. Hence

µ(Br(x0)) rⁿ⁻² ≥ ε0

2 for a.e. r∈(0, δ0).

Ifx06∈ singu, thenu is smooth near x0, and hence r²⁻ⁿ

Z

B_r(x0)|∇u|²dx≤ ε0

4,

794 FANG-HUA LIN

for all r >0 sufficiently small. Thus, by the definition ofν, one has ν(Br(x0))

rⁿ⁻² ≥ ε0

4 ,

for all positive smallr. That isx0∈sptν. This completes the proof of (i).

To show (ii), we observe first the following facts.

(a) r²⁻ⁿµ(Br(x)) is a monotone increasing function of r ∈ (0,dist(x, ∂Ω)), forx∈Ω; thus the density

Θ(µ, x) = lim

r&0r²⁻ⁿµ(Br(x)) exists for everyx∈Ω.

(b) x∈Σ ⇐⇒ Θ(µ, x)≥ε0,x∈B1; (c) forHⁿ⁻² a.e. x∈Ω, Θu(x) = 0; here

Θu(x) = lim

r&0r²⁻ⁿ Z

B_r(x)|∇u|²dy.

Indeed, (a) follows from the energy monotonicity (1.7). For the statement (b), if x ∈ B1 and Θ(µ, x) ≥ ε0, then for any r ∈ (0, δ0), r²⁻ⁿµ(Br(x)) ≥ ε0 by (a); thusx ∈Σ by the definition of Σ. On the other hand, if x∈Σ, then, for any r ∈(0, δ0), r²⁻ⁿµ(Br(x))≥ε0; thus, by letting r &0, Θ(µ, x) ≥ε0. The statement (c) is a well-known fact proved by Federer-Ziemer (see [FZ]).

It is obvious, via the monotonicity of energy, that r²⁻ⁿµ(Br(x))≤δ²₀⁻ⁿµ(Ω)≤δ²₀⁻ⁿΛ,

for x ∈ B1. Thus µ |Σ is absolutely continuous with respect to Hⁿ⁻²bΣ. In other words, by the Radon-Nikodym theorem, one has

µ|Σ= Θ(x)Hⁿ⁻²bΣ,

for Hⁿ⁻²-a.e. x ∈Σ. Since Θ_u(x) = 0 for Hⁿ⁻²-a.e. x ∈Σ, we obtain (note that sptν ⊆Σ):

ν(x) = Θ(x)Hⁿ⁻²bΣ,

for Hⁿ⁻²- a.e. x ∈ Σ. The conclusion of Lemma 1.6 follows from the above density estimates, and also, forHⁿ⁻²- a.e. x∈Σ, that

(1.15) 2²⁻ⁿ≤lim inf

r&0

Hⁿ⁻²(Σ∩Br(x))

rⁿ⁻² ≤lim sup

r&0

Hⁿ⁻²(Σ∩Br(x)) rⁿ⁻² ≤1.

See [Sim2] for examples.

To explore further properties of Σ andµ, we assumeB1 ⊂B1+δ0 = Ω. Let Mdenote the set of all those Radon measures µon B1 such that µis a weak limit of Radon measures µi,µi =|∇ui|²dxdefined on B1, where ui ∈HΛ, for i = 1,2, . . . and Λ = Λ(µ) is a positive number. We also define F to be the set which consists of all whose compact subset E of B1 such thatE ⊂Σ for some Σ as defined in Lemma 1.5. We note that, forµ∈ M,µ=|∇u|²dx+ν, for some nonnegative ν as in Lemma 1.6, and for some u which is smooth and harmonic away from Σ ∈ F. For E ∈ F, y ∈ B1 with |y| < 1 and for 0< λ <1− |y|, we define

Ey,λ= E−y λ ∩B1.

Similarly, for µ∈ M we define a scaled Radon measureµy,λ by µy,λ(A) =µ(y+λA)λⁿ⁻²,

for|y|<1 and 0< λ <1− |y|.

Lemma 1.7. (i) If |y| < 1 and 0 < λ < 1− |y|, and if µ ∈ M, then µy,λ∈ M.

(ii) If {λk} &0 and if µ∈ M, then there is a subsequence {λ⁰_k} and η∈ M such that µy,λ_k * η;here |y|<1. Moreover,η0,λ=η for each λ >0.

(iii) M is closed with respect to weak-convergence of measures.

(iv) We define a map π :M → F as follows: If µ =|∇u|²dx+ν ∈ M so that ν(x) = Θ(x)Hⁿ⁻²bΣ (cf. Lemma 1.6), then π(µ) = Σ. If ν = 0, thenπ(µ) = singu. The mapπ has the following properties.

(a) If |y| ≤1−λ,0< λ <1, then

π(µy,λ) =λ⁻¹(π(µ)−y) for µ∈ M.

(b) If µ, µk ∈ M with µk * µ, then for each ε > 0 there is k(ε) such that

B1∩π(µk)⊂ {x∈B1+δ0 : dist(π(µ), x)< ε} for allk≥k(ε).

Similarly we have the following lemma concerningF.

Lemma1.8. 1. If E ∈ F,|y|<1,0< λ <1− |y|, thenEy,λ∈ F.

2. If {λk} &0≤ |y|<1, E ∈ F, then there is a subsequence {λk⁰} such that Ey,λ_k0 →F ∈ F in the Hausdorff metric as λk⁰ → 0. Moreover, F ⊂Σ_∗ for some Σ_∗ as defined in Lemma 1.5; and Σ_∗ is a cone.

796 FANG-HUA LIN

Remark 1.9. Letting µ ∈ M and Σ = π(µ), we consider FΣ, the clo- sure of

{E∈ F :E⊂Σ_y,λ for some|y|<1,0< λ <1− |y|}

under the Hausdorff metric. Note thatFΣ is a compact subset ofF. Indeed, if F ∈ FΣ then there are a sequence ofyk,|yk|<1, a sequence 0< λk<1− |yk| and a sequence Ek ⊂ Σ_yk,λk such that Ek → F in the Hausdorff metric.

Suppose µis the weak limit of µi; here µi =|∇ui|²dx such that E(ui, B1+δ0)

≤Λ, for some Λ>0. Define vi,k by

vi,k(x) =ui(yk+λkx), x∈B1+δ0. Note that

|yk+λkx| ≤ |yk|+λk|x|

< |yk|+ (1− |yk|)(1 +δ0)

≤ 1 +δ0, and thus thevi,k’s are well-defined. Moreover,

E(vi,k, B1+δ0) ≤ λ²_k⁻ⁿ Z

B_(1+δ0)λk(y_k)|∇ui|²dx

≤ (1 +δ0)ⁿ⁻²((1 +δ0)λk)²⁻ⁿ Z

B_(1+δ

0)λ_k(y_k)|∇ui|²dx

≤ (1 +δ0)ⁿ⁻²δ²₀⁻ⁿΛ by the energy monotonicity (1.7).

Thus for each fixed k,

µi,k =|∇vi,k|²dx * µy_k,λ_k

asi→ ∞. By taking subsequences as necessary, we may also assume µy_k,λ_k * µ_∗ ask→ ∞.

Then, by the diagonal sequence method, we may obtain a sequence{ik} → ∞ such that

µi_k,k* µ_∗ ask→ ∞.

As in Lemma 1.6, we may writeµ_∗=|∇u_∗|²dx+ν_∗. Moreover,F ⊂Σ_∗ by the definition. That isF ∈ F. We define a subset ofR+ by

(1.16) O={s∈R+:H^s(F) = 0 for everyF ∈ FΣ}. Then O is an open subset of R+ (cf. [W]).

Proof of Lemma 1.7. Part (i) of the lemma is obvious. Indeed, if {µi} ∈HΛ such that

µi=|∇ui|²dx * µ∈ M,

then ui,y,λ(x) = ui(y+λx), for every x ∈ B1+δ0 = Ω, i = 1,2, . . ., |y| < 1, 0< λ <1− |y|. Note that

|y+λx|<|y|+ (1− |y|)(1 +δ0)≤1 +δ0; thusui,y,λ is well-defined. Since

Z

B_1+δ0|∇ui,y,λ|²dx=λ²⁻ⁿ Z

B_λ(1+δ0)(y)|∇ui|²dx≤

µ1 +δ0

δ0

¶n−2

Λ, we have ui,y,λ∈HΛ˜,

Λ =˜

µ1 +δ0

δ0

¶_n−2 Λ, for each i= 1,2, . . . .Since

|∇ui,y,λ|²dx * µy,λ

by definition, we thus have µy,λ∈ M.

To prove part (ii), let {ui} ∈ HΛ be such that |∇ui|²dx * µ ∈ M. For any sequence {λk} &0, and for|y|<1, one has

klim→∞µy,λ_k(BR)≤Rⁿ⁻²Θ(µ, y)

(cf. the proof of Lemma 1.6), for everyR >0. Hence we obtain a subsequence {λk⁰}so that

µy,λ⁰_k * η

as Radon measures on Rⁿ. Note that if η is restricted to B1+δ0, thenη ∈ M. Indeed, since

|∇ui,y,λ⁰_k|²dx * µy,λ⁰_k asi→ ∞,

and µy,λ⁰_k * η ask→ ∞, we may obtain (by the diagonal sequence method) a sequence ik→ ∞ such that

|∇ui_k,y,λ_k|²dx * η.

By the monotonicity of r²⁻ⁿµ(Br(y)), for 0< r < δ0, we see that r²⁻ⁿη(Br(0))≡Θ(µ, y) for all r >0.

Letvk =ui_k,y,λ_k * v so that

η=|∇v|²dx+ν, ν(x) = Θ(x)Hⁿ⁻²bΣ.

Applying (1.6) tovk, we get for a.e. 0< r < R <∞, (1.17)

Z

B_R(0)/Br(0)

¯¯¯¯∂vk

∂ρ

¯¯¯¯²ρ²⁻ⁿdx→R²⁻ⁿη(BR)−r²⁻ⁿη(Br) = 0.

Thus, in particular,∂v/∂ρ= 0.

798 FANG-HUA LIN

Let φ:Sⁿ⁻¹ → R+ be a smooth function, and let ψ ∈C₀^∞(0,1) be such

that _Z

1 0

ψ(t)dx= 1, and ψ≥0.

We consider, for 0< a <∞, 0< ε¿a, the functions (1.18) E(vk, φ, a, ε) =

Z _∞

0

Z

Sⁿ⁻¹

"

(r+a)^2¯¯_¯¯∂vk

∂r

¯¯¯¯²+^¯¯_¯¯ ∂

∂θvk

¯¯¯¯²

#

(r+a, θ)·Odθdr;

here

O=φ(θ)·ψε(r), ψε(r) = 1 εψ

µr ε

¶ . Then a direct computation using the identity

Div[δi,j|∇vk|²−2DivkDjvk] = 0

in the sense of distributions (cf. (1.3) or more precisely the equivalent version of it in the polar coordinates system), we obtain

d

daE(vk, φ, a, ε) (1.19)

= 2 d da

Z _∞

0

Z

Sⁿ⁻¹(r+a)^2¯¯_¯¯∂vk

∂r

¯¯¯¯²(r+a, θ)·φ(θ)·ψε(r)dθdr

+2(n−2) Z _∞

0

Z

Sⁿ⁻¹(r+a)^¯¯_¯¯∂vk

∂r

¯¯¯¯²(r+a, θ)·φ(θ)ψε(r)dθdr

−^{Z ∞}

0

Z

Sⁿ⁻¹

2 ∂

∂rvk· ∂

∂θvk(r+a, θ) ∂

∂θφ(θ)ψε(r)dθdr.

Integrating both sides of (1.19) with respect to a∈(ρ, R), we then get (1.20)

E(vk, φ, R, ε)−E(vk, φ, ρ, ε)

= Z _∞

0

Z

Sⁿ⁻¹

2(r+a)^2¯¯_¯¯∂vk

∂r

¯¯¯¯²(r+a, θ)·φ(θ)·ψε(r)dθdr

¯¯¯¯

¯

a=R

a=ρ

+ Z _∞

0

Z _R

ρ

Z

Sⁿ⁻¹2(n−2)(r+a)^¯¯_¯¯∂vk

∂r

¯¯¯¯²(r+a, θ)·φ(θ)·ψε(r)dθdadr

−^{Z ∞}

0

Z _R

ρ

Z

Sⁿ⁻¹

2 ∂

∂rvk

∂

∂θvk(r+a, θ)φθ(r)ψε(r)dθdadr.

Now lettingε→0⁺, we obtain for a.e. 0< ρ < R <∞, that Z

Sⁿ⁻¹

"

R^2¯¯_¯¯∂vk

∂r

¯¯¯¯²+^¯¯_¯¯∂vk

∂θ

¯¯¯¯²(R, θ)

# φ(θ)dθ (1.21)

−^Z

Sⁿ⁻¹

"

ρ^2¯¯_¯¯∂vk

∂r

¯¯¯¯²(ρ, θ) +^¯¯_¯¯∂vk

∂θ

¯¯¯¯(ρ, θ)

# φ(θ)dθ

= 2 Z

Sⁿ⁻¹R^2¯¯_¯¯ ∂

∂rvk

¯¯¯¯²(R, θ)φ(θ)dθ+

−2 Z

Sⁿ⁻¹ρ^2¯¯_¯¯∂

∂rvk

¯¯¯¯²(ρ, θ)φ(θ)dθ

+ Z _R

ρ

Z

Sⁿ⁻¹

2(n−2)r^¯¯_¯¯∂

∂rvk

¯¯¯¯²(r, θ)φ(θ)dr

−^{Z R}

ρ

Z

Sⁿ⁻¹

2 ∂

∂rvk

∂vk

∂θ (r, θ)∂φ

∂θdθ)dr.

Note that

|∇vk|²dx= Ã

r^2¯¯_¯¯∂vk

∂r

¯¯¯¯²+^¯¯_¯¯∂vk

∂θ

¯¯¯¯²

!

(r, θ)rⁿ⁻³dθdr=rⁿ⁻³dσk(r, θ)dr;

then the above identity yields (1.22)

Z

Sⁿ⁻¹

φ(θ)dσk(R, θ)− Z

Sⁿ⁻¹

φ(θ)dσk(ρ, θ) is equal to the right-hand side of (1.21).

Now when k→ ∞, (1.22) goes to zero by (1.17) and (1.21), in the sense of distributions on (0,∞). Since |∇vk|²dx * dη, one has

r³⁻ⁿ|∇vk|²dx * r³⁻ⁿdη on (0,∞)×Sⁿ⁻¹. That is,

dσk(r, θ)dr * r³⁻ⁿdη(r, θ).

On the other hand, (1.22) yields also

dσk(r+a, θ)dr=dσk((r+a)θ)d(r+a)* r³⁻ⁿdη(r, θ)

for any a > 0. That is, r³⁻ⁿdη(r, θ) is translation invariant in r. We thus obtain

r³⁻ⁿdη(r, θ) =dσ(θ)dr, or equivalently

dη(r, θ) =r³⁻ⁿdrdσ(θ)

for some Radon measuredσ(θ) on Sⁿ⁻¹. Note thatdσ(θ) can also be obtained from the weak-limit of dσk(r, θ), for some suitabler⁰_ks, when k→ ∞.

Another way to see this is to integrate (1.22) again, and then let k→ ∞ to obtain

(1.23)

Z

B_R+δ/B_R−δ

φ²(θ)r³⁻ⁿdη(r, θ) = Z

B_ρ+δ/B_ρ−δ

φ²(θ)r³⁻ⁿdη(r, θ), for 0< ρ < R <∞, and for a.e. δ∈(0, R). Note that

η(Br)

rⁿ⁻² = Θ(µ, y),