The boundary value problem for the super-Liouville equation

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Available online atwww.sciencedirect.com

Ann. I. H. Poincaré – AN 31 (2014) 685–706

www.elsevier.com/locate/anihpc

The boundary value problem for the super-Liouville equation

^✩

Jürgen Jost

^a

, Guofang Wang

^b

, Chunqin Zhou

^c

, Miaomiao Zhu

^d,^∗

aMax Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany bMathematisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany

cDepartment of Mathematics, Shanghai Jiaotong University, Shanghai, 200240, China dMathematics Institute, University of Warwick, CV4 7AL, Coventry, UK

Received 18 April 2012; accepted 22 June 2013 Available online 17 July 2013

Abstract

We study the boundary value problem for the — conformally invariant — super-Liouville functional E(u, ψ )=

M

1

2|∇u|²+K_gu+

/

D+e^u ψ, ψ

−e^2u

dz

that couples a functionuand a spinorψon a Riemann surface. The boundary condition that we identify (motivated by quantum field theory) couples a Neumann condition foruwith a chirality condition forψ. Associated to any solution of the super-Liouville system is a holomorphic quadratic differentialT (z), and when our boundary condition is satisfied,Tbecomes real on the boundary.

We provide a complete regularity and blow-up analysis for solutions of this boundary value problem.

1. Introduction

In[12], we have introduced thesuper-Liouville functional, a conformally invariant functional that couples a real- valued functionuand a spinorψon a Riemann surfaceMwith conformal metricgand a spin structure,

E(u, ψ )=

M

1

2|∇u|²+K_gu+

/

D+e^u ψ, ψ

−e^2u

dz. (1)

HereK_gis the Gaussian curvature ofM. The Dirac operatorD

/

is defined byDψ

/

:= ²_α₌₁e_α· ∇eαψ, where{e₁, e₂} is an orthonormal basis onT M and∇ is the connection on the spinor bundleΣ MofM, which is induced from the Levi-Civita connection onM with respect togand·denotes the Clifford multiplication in the spinor bundle Σ M.

✩ The first author is supported by the ERC Advanced Grant FP7-267087. The second is supported partly by SFB/TR71 “Geometric partial differential equations” of DFG. The third author is supported partially by NSFC of China (No. 10871126). The fourth author is supported in part by The Leverhulme Trust.

* Corresponding author.

E-mail addresses:jost@mis.mpg.de(J. Jost),Guofang.Wang@math.uni-freiburg.de(G. Wang),cqzhou@sjtu.edu.cn(C. Zhou), Miaomiao.Zhu@warwick.ac.uk(M. Zhu).

http://dx.doi.org/10.1016/j.anihpc.2013.06.002

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Finally,·,·is the natural Hermitian metric onΣ Minduced byg. The system of Euler–Lagrange equations associated to(1)is called the super-Liouville equation. For the geometric background, see[15]or[10].

In this paper, we wish to address the boundary value problem for the super-Liouville functional. We therefore first of all need to identify the appropriate boundary condition for the spinor fieldψ. Since the super-Liouville functional is inspired by quantum field theory, we likewise turn to the physics literature [2,6,16,17]to get some clue about a natural boundary condition. This boundary condition then will be of chirality type. The main point of the paper then is an analytical investigation of solutions of the boundary value problem. In particular, we shall show the regularity of solutions and identify the blow-up behavior for limits of sequences of solutions. In other words, we analytically understand the non-compactness of the solution space.

The key property of the functional is of course its conformal invariance. Therefore, the boundary conditions to be imposed likewise need to be conformally invariant. Conformal invariance on one hand makes the solution space non-compact, but on the other hand allows for a control of limits of solutions via a blow-up analysis. This is, of course, a well-known scheme, but the details are technically somewhat tricky and interesting.

Conformal invariance, like any invariance, by Noether’s theorem leads to some conserved current. For two- dimensional conformally invariant variational problems, this conserved quantity can be identified with a holomorphic quadratic differential associated to a solution. From this perspective, our boundary condition is the natural one, because it renders that holomorphic quadratic differential real on the boundary. At a more technical level, this is important for the study of the asymptotic behavior of an entire solution on the upper half-plane with finite energy. Also, our boundary condition allows for the reflection of solutions across the boundary, which, at least heuristically, reduces boundary to interior regularity and which therefore, technically, is a useful device.

In[5], we have investigated the chirality boundary condition for Dirac-harmonic maps. Since in the present case, the coupling between the two fields is different, so then necessarily is the boundary analysis. Since the Liouville fieldu is scalar valued, in particular, here we can achieve a more precise blow-up analysis at the boundary. Since the chirality boundary condition is of physical interest, its general mathematical understanding should be useful.

2. The boundary value problem for the super-Liouville equation

In this section, we shall derive the boundary condition to be imposed on solutions of the super-Liouville equation.

Thus, letMbe a compact Riemann surface with smooth boundary∂Mand with a fixed spin structure. When∂M= ∅, we know that the Laplacian operatoris in general not formally self-adjoint, and neither is the Dirac operatorD. In

/

fact, we have

M

ψ,Dϕ

/

dv=

M

Dψ, ϕ

/

dv−

∂M

n·ψ, ϕdv

for allψ, ϕ∈C^∞(Σ M). Herenis the outward unit normal vector field on∂M.

As is well known, the natural boundary condition for the functionuis of Neumann type. This condition is clearly conformally invariant. We now shall derive a boundary conditions for the spinor fieldψthat is likewise conformally invariant.

We recall the chirality boundary conditions for the Dirac operatorD

/

first introduced in[7]. See also[9]. LetMbe a compact Riemann surface with∂M= ∅and with a fixed spin structure, admitting a chirality operatorG, which is an endomorphism of the spinor bundleΣ Msatisfying:

G²=I, Gψ, Gϕ = ψ, ϕ, and

∇X(Gψ )=G∇Xψ, X·Gψ= −G(X·ψ ),

for anyX∈T M, ψ, ϕ∈Γ (Σ M). HereIdenotes the identity endomorphism ofΣ M.

We usually takeG=γ (ω₂), the Clifford multiplication by the complex volume formω₂=ie₁e₂, wheree₁, e₂is a local orthonormal frame onM.

Let

S:=Σ M|∂M

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denote the restricted spinor bundle with induced Hermitian product. The outward unit normal vector fieldninduces an operatornG :Γ (S)→Γ (S), which is a self-adjoint endomorphism satisfying

(nG) ²=I, nGψ, ϕ = ψ,nGϕ .

Hence, we can decomposeS=V⁺⊕V⁻, whereV^±is the eigensubbundle corresponding to the eigenvalue±1 ofnG.

One can check that the orthogonal projection onto the eigensubbundleV^±: B^±:L²(S)→L²

V^± ψ→1

2(I± nG)ψ,

defines a local elliptic boundary condition for the Dirac operatorD, see

/

[9]. We say that a spinorψ∈W^1,⁴³(Γ (Σ M)) satisfies the chirality boundary conditionsB^±if

B^±ψ

∂M =0.

It is shown in[9]that ifψ, ϕ∈W^1,⁴³(Γ (Σ M))satisfy the chirality boundary conditions, resp., then n·ψ, ϕ =0, on∂M.

In particular,

∂M

n·ψ, ϕ =0. (2)

It follows that the Dirac operatorD

/

is self-adjoint when we impose the chirality boundary conditions.

Let us note that on a surface the (usual) Dirac operatorD

/

can be seen as the (doubled) Cauchy–Riemann operator.

Consider R² with the Euclidean metric ds²+dt². Let e₁=_∂s^∂ and e₂= _∂t^∂ be the standard orthonormal frame.

A spinor field is simply a mapΨ :R²→₂=C², and the Cliford multiplication ofe₁ande₂acting on spinor fields can be identified by the multiplication with matrices

e₁= 0 i

i 0

, e₂=

0 1

−1 0

.

Here, without loss of generality, we keepe₁ande₂consistence with that in[5]. If exchanginge₁ande₂, thene₁and e₂are consistent with that in[12]and this case can be handled analogously. IfΨ :=

f g

:R²→C²is a spinor field, then the Dirac operator is

/

DΨ= 0 i

i 0

∂f

∂s

∂g

∂s

+

0 1

−1 0

∂f

∂t

∂g

∂t

=2i ∂g

∂z

∂f

∂z¯

, where

∂

∂z=1 2

∂

∂s −i∂

∂t

, ∂

∂z¯=1 2

∂

∂s+i∂

∂t

.

Therefore, the elliptic estimates developed for (anti-)holomorphic functions can be used to study the Dirac equation.

IfMis the upper half Euclidean spaceR²₊, then the chirality operator is simplyG=ie1e2=

1 0 0−1

. Note that

n= −e2, we get that B^±=1

2(I± n·G)=1 2

1 ±1

±1 1

.

By the standard chirality decomposition, we can writeψ=

ψ₊ ψ₋

, then the boundary condition becomes ψ₊= ∓ψ₋ on∂R²₊.

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In this paper, we will consider the functional EB(u, ψ )=

M

1

2|∇u|²+Kgu+

/

D+e^u ψ, ψ

−e^2u

dv+

∂M

hgu−ce^u

dσ, (3)

whereh_gis geodesic curvature of∂Mandcis a given constant.

Proposition 2.1.The Euler–Lagrange system forEB(u, ψ )with Neumann/chirality boundary conditions is

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

−u=2e^2u−e^uψ, ψ −K_g, inM^o,

/

Dψ= −e^uψ, inM^o,

∂u

∂n=ce^u−hg, on∂M,

B^±ψ=0, on∂M.

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Hereis the Laplacian with respect tog, andK_gis the Gaussian curvature inM, andh_gis the geodesic curvature of∂M.

Proof. Letu_t be a family of function with ^∂u_∂t^t|t=0=η, and letψ_t be a family of spinor with^∂ψ_∂t^t|t=0=ξ. Since dE_B(u, ψ_t)

dt

t=0

=

M

Dξ, ψ

/

+ Dψ, ξ

/

+e^uξ, ψ +e^uψ, ξdv

=2

M

Reξ,Dψ

/

+2e^uReξ, ψdv−

∂M

ξ,n·ψdσ,

and

dE_B(u_t, ψ ) dt

t=0

=

M

∇u· ∇η+Kgη+ηe^uψ, ψ −2e^2uη dv+

∂M

hgη−ce^uη dσ

= −

M

ηu dv+

∂M

η∂u

∂ndσ+

M

K_gη+ηe^uψ, ψ −2e^2uη dv+

∂M

h_gη−ce^uη dσ,

one can easily obtain(4). 2

For simplicity, we shall call(4)the Neumann boundary problem in the sequel.

Now we come to an important property of the Neumann boundary problem(4).

Proposition 2.2.Assume that(u, ψ )is a solution of (4). For any conformal diffeomorphismϕ:M→M, if we set

˜

u=u◦ϕ−φ,

ψ˜ =e⁻^φ²ψ◦ϕ (5)

wheree^φ is the conformal factor of the conformal mapϕ, i.e.,ϕ^∗(g)=e^2φg, then(u,˜ ψ )˜ is also a solution of (4).

Moreover, the functionalEB(u, ψ )is conformally invariant.

Proof. Letg˜=ϕ^∗g, wheregis the metric onM. LetD,

/

Bbe the Dirac operator and the chirality boundary operator with respect to the new metricg˜respectively. We identify the new and old spin bundles as in[8]. Since the relation between the two Dirac operatorsD

/

andD

/

is

/

Dψ˜ =λ⁻³²D

/

λ¹²ψ˜

=λ⁻³²Dψ

/

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forλ=e^φ, and the relation between the two Gaussian curvatures and between the two geodesic curvatures are respectively

−_gφ=K_g_˜e^2φ−K_g,

∂φ

∂n =h_g_˜e^φ−hg.

We can show by a direct computation that(u,˜ ψ )˜ satisfies

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

−_g_˜u˜=2e²û^˜−eû^˜ ˜ψ,ψ˜ −K_g_˜, inMô,

/

Dψ˜ = −e^u^˜ψ ,˜ inM^o,

∂u˜

∂n=ce^u^˜−h_g_˜, on∂M,

B^±ψ˜ =0, on∂M.

Similarly, one can also show that the functional is conformally invariant. The proof of the proposition is complete. 2 In the sequel, we will only consider the case ofB⁺and omit the symbol “+”. The case ofB⁻can, of course, be handled analogously.

Let us recall that aKilling spinoris a spinorψsatisfying

∇Xψ=λX·ψ, for any vector fieldX

for some constantλ. On the standard sphere, there are Killing spinors with the Killing constantλ=¹₂, see for in- stance[3]. A Killing spinor is an eigenspinor, i.e.,

/

Dψ= −ψ, (6)

with constant|ψ|². Choose a Killing spinorψ with|ψ|²=1. If we identifyS²\{north pole}by the stereographic projection with the Euclidean planeR²with the metric

4

(|1+ |x|²)²|dx|²,

then any Killing spinor has the form v+x·v

1+ |x|²,

for a constantv∈C², up to a translation or a dilation. See[3].

We can now construct some special solutions of(4).

Proposition 2.3.LetM=R²₊. Then

log

√2 1+ |x−x0|²,0

is a solution of (4), wherex₀=(s₀, t₀)fors₀∈Randt₀= −^√₂^2c for any constantc. Furthermore, ifc=0, then

log 2

1+ |x−x₁|²,√

2v+(x−x₁)·v 1+ |x−x₁|²

is also a solution of (4), wherex₁=(s₁,0)fors₁∈R, andv=v₁ v2

∈ {v∈C²| |v| =1}andv₁= −v₂.

Proof. Set ψ=√

2^v₁⁺_+|^(x⁻^x¹⁾^·^v

x−x1|² . SinceDψ

/

= −ψ, according to the argument in [12], it is sufficient to show that Bψ|_∂_R²

+=0. Since

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v+(x−x₁)·v=v+(s−s₁)e₁·v+t e₂·v

= v₁

v₂

+(s−s₁) 0 i

i 0 v₁ v₂

+t

0 1

−1 0 v₁ v₂

=

v₁+i(s−s₁)v₂+t v₂ v₂+i(s−s₁)v₁−t v₁

, we have

ψ=

√2 1+ |x−x₁|²

v1+i(s−s1)v2+t v2

v2+i(s−s1)v1−t v1

. Hence we have by usingv1= −v2on∂R²₊,

ψ=

√2 1+ |x−x1|²

v₁−i(s−s₁)v₁

−v₁+i(s−s₁)v₁

on∂R²₊. This means thatBψ|_∂_R²

+=0. 2

3. Regularity of solutions for the Neumann boundary problem

In this section, we consider the regularity of solutions for the Neumann boundary problem(4)under the condition that

M

e^2u+ |ψ|⁴dv+

∂M

e^u

dσ <∞.

First, we define weak solutions of (4). We say that (u, ψ ) is a weak solution of(4), if u∈W^1,2(M)and ψ∈ W^1,

4 3

B (Γ (Σ M))satisfy

M

∇u∇φ dv=

M

2e^2u−e^u|ψ|²−K_g φ dv+

∂M

ce^u−h_g φ dσ,

M

ψ,Dξ

/

dv= −

M

e^uψ, ξdv

forφ∈C^∞(M)and any smooth spinorξ ∈C^∞∩W^1,

4 3

B (Γ (Σ M)). Here W^1,

4 3

B

Γ (Σ M)

=

ψψ∈W^1,⁴³

Γ (Σ M)

, Bψ|∂M=0 . It is clear that (u, ψ )∈W^1,2(M)×W^1,

4 3

B (Γ (Σ M))is a weak solution if and only if (u, ψ ) is a critical point of E_B(u, ψ )inW^1,2(M)×W^1,

4 3

B (Γ (Σ M)). A weak solution is a classical solution by the following.

Proposition 3.1.Let(u, ψ )be a weak solution of(4)with

Me^2u+ |ψ|⁴dv+

∂Meûdσ <∞. Thenu∈C^2,α(Mô)∩ C^1,α(M)andψ∈C^2,α(Γ (Σ Mô))∩C^1,α(Γ (Σ M))for someα∈(0,1).

To prove this proposition, we need several lemmas.

Lemma 3.2.(See[4].) AssumeΩ⊂R²is a bounded domain and letube a solution of −u=f (x) inΩ,

u=0 on∂Ω

withf ∈L¹(Ω). Then for everyδ∈(0,4π )we have

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Ω

exp

(4π−δ)|u(x)| f1

dx4π²

δ (diamΩ)², (7)

wheref1=

Ω|f (x)|dx.

Lemma 3.3.(See[13].) Assume thatuis a solution of

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−u=0, inB_R⁺,

∂u

∂t =f (x), on{t=0} ∩∂B_R⁺, u=0, on∂B_R⁺∩B_R⁺,

withf∈L¹({t=0} ∩∂B_R⁺)for anyR >0. Then for everyδ1∈(0,4π )we have

B_R⁺

exp

(4π−δ₁)|u(x)| f1

dx16π²R² δ₁ and for everyδ₂∈(0,2π )

∂B_R⁺∩{t=0}

exp

(2π−δ₂)|u(x)| f1

ds4π R δ₂

wheref1=

{t=0}∩∂B_R⁺|f|ds.

ByLemma 3.2andLemma 3.3, we obtain the following Lemma 3.4.If(u, ψ )is a weak solution to(4)with

Me^2u+ |ψ|⁴dv+

∂Me^udσ <∞, then we have for0< α <1 u⁺∈L^∞(M), ψ∈C^α

Γ (Σ M) .

Proof. By the conformal invariance of(4)and by the interior regularity Lemma 4.3 in[12], it suffices to show that, for anyx₀∈∂M,uis bounded from above inB_r^M(x₀)∩Mandψis continuous inΓ (Σ (B_r^M(x₀)∩M)), whereB_r^M(x₀) is a geodesic ball atx₀ of M. Without loss of generality, we assume thatx₀=0 and B_r^M(x₀)∩M= {x =(s, t)| s²+t²< r², t0} ⊂R²₊. Set B_r⁺= {x=(s, t)|s²+t²< r², t >0},B_r⁻= {(s, t)|s²+t²< r², t <0}and Γ1=∂B_r⁺∩∂R²₊,Γ2=∂B_r⁺∩R²₊. By using the conformality again, we may assume that

M

e^2u+ |ψ|⁴dv+

∂M

e^udσ <1 4π.

First, we show the boundedness from above ofu. Set f =2e^2u−e^u|ψ|² and g=ce^u.

Then we consider

⎧⎨

⎩

−u=f, inB_r⁺,

∂u

∂n=g, onΓ1.

It is clear thatg∈L¹(Γ₁). Setg=g₁+g₂withg₁L¹(Γ1)πandg₂∈L^∞(Γ₁). Defineu₁,u₂andu₃by

⎧⎪

⎪⎨

⎪⎪

⎩

−u₁=f, inB_r⁺,

∂u₁

∂n =0, onΓ1, u1=0, onΓ2,

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⎧⎪

⎪⎨

⎪⎪

⎩

−u₂=0, inB_r⁺,

∂u₂

∂n =g1, onΓ1, u2=0, onΓ2,

⎧⎪

⎪⎨

⎪⎪

⎩

−u₃=0, inB_r⁺,

∂u₃

∂n =g₂, onΓ₁, u₃=0, onΓ₂. Extendingu1andf evenly we have

−u₁=f, inB_r, u₁=0, on∂B_r. Since

B_r⁺e^2u+ |ψ|⁴dx <∞, we know thatf ∈L¹(B_r⁺)withfL¹π. By applyingLemma 3.2we have e⁴^|^u¹^|∈L¹(Br).

Foru2, byLemma 3.3, we have

B⁺_r

exp 4|u₂|

dxC,

Γ1

exp 2|u₂|

dsC.

Foru₃, it is obvious that u₃_L∞(B⁺r

2

)C.

Letu₄=u−u₁−u₂−u₃. Then we have

⎧⎨

⎩

−u₄=0, inB_r⁺,

∂u₄

∂n =0, onΓ1.

Extending u₄evenly,u₄becomes a harmonic function inB_r. Then the mean value theorem for harmonic functions implies that

u⁺

4

L∞(B⁺_r

2

)Cu⁺

4

L¹(B_r⁺). Notice that

u⁺₄ u⁺+ |u1| + |u2| + |u3|,

and

B⁺_r

u⁺dx1 2

B_r⁺

e^2udx <∞.

We get u⁺₄

L^∞(B⁺_r

2

)C.

Altogether, we find thatf∈L²(B_r⁺)andg∈L²(Γ1).

The standard elliptic estimates imply that u⁺

L^∞(B⁺r 4

)C.

Next we show the continuity of the spinor fieldψ. For this purpose, we extend(u, ψ )to the lower half diskB_r⁻. Assumex¯is the reflection point ofx about∂R²₊, and define

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u(x)¯ :=u(x), x¯∈B_r⁻, ψ (x)¯ :=ie₁·ψ (x), x¯∈B_r⁻. Since we have for a.e.x∈Γ1

ψ (x)= −nGψ (x)=ie₁·ψ (x),

it is clear that the extension forψis well defined.

Now assume that(u, ψ )is a weak solution of(4) andξ is inW^1,⁴³(Γ (Σ B_r)) with compact support. Then we obtain

Br

ψ,Dξ

/

=

Br⁺

ψ,Dξ

/

+

B⁻r

ψ,Dξ

/

=

B_r⁺

ψ,Dξ

/

+

x∈B_r⁺

ψ (x),¯ Dξ(

/

x)¯

=

Br⁺

ψ,Dξ

/

+

x∈Br⁺

ie1·ψ (x),Dξ(

/

x)¯

=

B_r⁺

ψ,Dξ

/

+

x∈B_r⁺

ψ (x),D

/

ie₁·ξ(x)¯

=

B_r⁺

ψ (x),D

/

ξ(x)+ie₁·ξ(x)¯ .

By the definition of the chirality operatorB, we have for a.e.x∈Γ₁ B

ξ(x)+ie₁·ξ(x)¯

=1

2(I−ie₁)·

ξ(x)+ie₁·ξ(x)

=0.

Then by the definition of a weak solution we obtain

B_r⁺

ψ (x),D

/

ξ(x)+ie₁·ξ(x)¯

=

B_r⁺

Dψ (x), ξ(x)

/

+ie₁·ξ(x)¯

= −

B⁺r

e^u

ψ (x), ξ(x)+ie1·ξ(x)¯

= −

B⁺_r

e^u

ψ (x), ξ(x)

−

x∈B_r⁺

e^u

ψ (x), ie₁·ξ(x)¯

= −

B⁺_r

e^u

ψ (x), ξ(x)

−

x∈B_r⁻

e^u(^x)^¯

ψ (x), ie¯ 1·ξ(x)

= −

B⁺r

e^u

ψ (x), ξ(x)

−

x∈Br⁻

e^u(^x)^¯

ψ (x), ξ(x) .

Therefore we obtain that

Br

ψ,Dξ

/

= −

Br⁺

e^u

ψ (x), ξ(x)

−

x∈B⁻r

e^u(^x)^¯

ψ (x), ξ(x) .

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Set

A(x)=

e^u(x), x∈B_r⁺, e^u(^x)^¯ , x∈B_r⁻. It follows thatψsatisfies

/

Dψ= −A(x)ψ, inB_r. SinceA(x)∈L^∞(B^r

4)and

B_r|ψ|⁴dx <∞, we haveψ∈W^1,4(Γ (Σ B^r

8))and in particularψ∈C^α(Γ (Σ B⁺r 8

))for some 0< α <1. 2

Proof of Proposition 3.1. Assume that(u, ψ )is a weak solution of(4). For anyq >2, let 2> p=_q^q₋₁>1. Then we have

∇uL^q(M)sup

M

∇u∇ϕ dv

ϕ∈W^1,p(M),

M

ϕ dv=0, ϕW^1,p(M)=1

.

Since fromLemma 3.4

M

∇u∇ϕ dv =

M

−uϕ dv+

∂M

∂u

∂nϕ dσ

=

M

2e^2u−e^u|ψ|²−K_g ϕ dv+

∂M

ce^u−h_g ϕ dσ

C

M

|ϕ|dv+

∂M

|ϕ|dσ

C,

we have∇uL^q(M)Cfor anyq >2. Therefore we haveu∈W^1,q(M)for anyq >2. ByW²⁺^k,qestimates for the Neumann boundary problem (see[1], see also[14])

uW²⁺^k,q(M)C

uW^k,q(M)+ ∂u

∂n

W¹⁺^k,q(∂M)

+ uW¹⁺^k,q(M)

,

we haveu∈W^2,q(M)for anyq >2. By the Sobolev embedding theorem we knowu∈C^1,α(M)for someα∈(0,1).

Similarly we obtain thatu∈C^2,α(M^o)for someα∈(0,1).

Forψ, since(u, ψ )satisfies

/

Dψ= −A(x)ψ

in the neighborhood ofx0∈∂Mafter the reflection, seeLemma 3.4. By the well-know Lichnerowitz formulaD

/

²ψ=

−ψ+¹₄Kgψ(see e.g.[10]), we know

−ψ= −dA(x)·ψ+A²(x)ψ−1 4K_gψ.

It follows that ψ∈W^2,q for any q >1 in the neighborhood of x₀∈∂M. Hence we haveψ∈C^2,α(Γ (Σ M^o))∩ C^1,α(Γ (Σ M))for someα∈(0,1). This concludes the proof. 2

We call(u, ψ )aregularsolution to(4)ifu∈C^2,α(M^o)∩C^1,α(M)andψ∈C^2,α(Γ (Σ M^o))∩C^1,α(Γ (Σ M))for someα∈(0,1).

Next we discuss the convergence of a sequence of regular solutions to (4), under a smallness condition for the energy.

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Lemma 3.5.Forε₁< π, andε₂< π. If a sequence of regular solutions(u_n, ψ_n)satisfy

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

−un=2e^2uⁿ−e^uⁿψn, ψn, inB_r⁺,

/

Dψ_n= −e^uⁿψ_n, inB_r⁺,

∂u_n

∂n =ce^uⁿ, on∂B_r⁺∩ {t=0}, Bψ_n=0, on∂B_r⁺∩ {t=0}

and

B_r⁺

e^2uⁿdx < ε₁, |c|

∂B_r⁺∩{t=0}

e^uⁿdσ < ε₂,

B⁺_r

|ψ_n|⁴dx < C.

Thenu⁺_n_L∞(B⁺r 4

)andψ_n_L∞(B⁺r 8

)are uniformly bounded.

Proof. Ifc=0, by extending(u_n, ψ_n)to the lower half diskB_r⁻asLemma 3.4, we have −un=2e^2uⁿ−e^uⁿψn, ψn, inBr,

/

Dψ_n= −e^uⁿψ_n, inB_r. From Lemma 4.4 of[12], we obtain the conclusions.

Next we assume thatc=0. LetΓ₁=∂B_r⁺∩ {t=0}andΓ₂=∂B_r⁺∩ {t >0}. Defineu_1,n, u_2,nby

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−u_1,n=2e^2uⁿ−e^uⁿ|ψ_n|², inB_r⁺,

∂u_1,n

∂n =0, onΓ₁,

u_1,n=0, onΓ₂,

and ⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

−u_2,n=0, inB_r⁺,

∂u_2,n

∂n =ce^uⁿ, onΓ1, u2,n=0, onΓ2. Extendingu1,nevenly we have

−u1,n=2e^2uⁿ, inBr, u_1,n=0, on∂B_r.

Sinceε₁< π, we can chooseδ₁>0 such that 4π−δ₁> (4ε1+2√

Cε₁)(2+δ₁). ByLemma 3.2we get

B_r

e⁽²⁺^δ¹⁾^|^u^1,n^|C

for some constantC. In particular we have

Br⁺

e⁽²⁺^δ¹⁾^|^u^1,n^|C.

Foru2,n, sinceε2< π, byLemma 3.3we also can chooseδ2>0, δ3>0 such that

Br⁺

e⁽²⁺^δ²⁾^|^u^2,n^|C,

Γ₁

e⁽¹⁺^δ³⁾^|^u^2,n^|C.

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Now settingw_n=u_n−u_1,n−u_2,n, it follows

⎧⎨

⎩

wn=e^uⁿ|ψn|²0, inB_r⁺,

∂wn

∂n =0, onΓ₁.

Extendingw_n evenly, we havew_n are subharmonic functions inB_r. Then the mean value theorem for subharmonic functions implies that

w_n⁺

L^∞(B⁺_r

2

)Cw⁺_n

L¹(B⁺_r). Notice that

B⁺r

w⁺_n dx

Br⁺

u⁺_n + |u_1,n| + |u_2,n|dx

C

B_r⁺

e^2uⁿ+e⁽²⁺^δ¹⁾^|^u^1,n^|+e⁽²⁺^δ²⁾^|^u^2,n^|dx

C.

Therefore we have w_n⁺

L^∞(B⁺_r

2

)C.

Finally, we write

⎧⎨

⎩

−u_n=2e^2uⁿ−e^uⁿ|ψ_n|²=f_n, inB_r⁺,

∂u_n

∂n =ce^uⁿ=g_n, onΓ₁. The standard elliptic estimates imply that

u⁺_n

L∞(B⁺_r

4

)C since f_nL^q(B⁺_r

2

) C and g_nL^q(∂B⁺_r

2

)∩{t=0} C for some q > 1. Consequently, it follows that ψn_L∞(B⁺r

8

)C. 2 4. Blow-up behavior

When the energy

Me^2uⁿdx and

∂Me^uⁿdxare large, the blow-up phenomenon may occur as in the case of the Liouville equation. In this section we will analyze the asymptotic behavior of a sequence of regular solutions

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−un=2e^2uⁿ−e^uⁿψn, ψn −Kg, inM^o,

/

Dψ_n= −e^uⁿψ_n, inM^o,

∂un

∂n =ce^uⁿ−h_g, on∂M,

Bψ_n=0, on∂M,

(8)

with

M

e^2uⁿ+ |ψn|⁴dv+

∂M

e^uⁿdσC. (9)

The blow-up analysis was first introduced in [4] for the Liouville-type equation on an open bounded domain.

Later, similar results for the Toda system and the super-Liouville equation, the natural generalization of the Liouville

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equation, were obtained in[11] and in[12]respectively. Here we will provide the blow-up analysis for the Neu- mann boundary problem(8) under condition(9). The key point is a Harnack inequality for the non-homogeneous Neumann-type boundary problem for second-order elliptic equations. SeeLemma A.2inAppendix A.

Theorem 4.1.Let(un, ψn)be a sequence of regular solutions to(8)satisfying(9). Define Σ1=

x∈Mthere is a sequencey_n→xsuch thatu_n(y_n)→ +∞

, Σ2=

x∈Mthere is a sequenceyn→xsuch thatψn(yn)→ +∞

.

Then, we haveΣ₂⊂Σ₁. Moreover,(u_n, ψ_n)admits a subsequence, denoted still by(u_n, ψ_n), satisfying that:

a) |ψn|is bounded inL^∞_loc(M\Σ2).

b) Forun, one of the following alternatives holds:

i) unis bounded inL^∞(M).

ii) u_n→ −∞uniformly onM.

iii) Σ₁is finite, nonempty and either

unis bounded inL^∞_loc(M\Σ1) (10)

or

u_n→ −∞uniformly on compact subsets ofM\Σ₁. (11)

Proof. First of all, if x∈M\Σ1, then it follows from the equationDψ

/

_n= −e^uⁿψ_n that x∈M\Σ2. Therefore we haveΣ2⊂Σ1. It is clear that|ψn|are bounded inL^∞_loc(M\Σ2).

Since e^2uⁿ is bounded inL¹(M)ande^uⁿ is bounded inL¹(∂M), we may extract a subsequence fromu_n (still denotedu_n) such that

M

e^2uⁿϕ dv→

M

ϕ dμ,

∂M

e^uⁿφ dσ→

∂M

φ dϑ

for everyϕ∈C(M)andφ∈C(∂M). Hereμandϑare two nonnegative bounded measures. A pointx∈Mis called anε-regular point with respect toμandϑif there is a functionϕ∈C(M), suppϕ⊂B_r^M(x)⊂Mwith 0ϕ1 and ϕ=1 in a neighborhood ofxsuch that

M

ϕ dμ < ε, ifx∈M^o,

or

M

ϕ dμ < ε, and

∂M

ϕ dϑ < ε, ifx∈∂M.

HereB_r^M(x)is a geodesic ball at centerx.

We define

Ω(ε)= {x∈M|x is not anε-regular point with respect toμandϑ}. By

Me^2uⁿ< Cand

∂Me^uⁿ< C, we have thatΩ(ε)is finite. We divide the proof into three steps.

Step1.Σ1=Ω(ε0), whereε0=min{ε1, ε2}andε1, ε2as inLemma 3.5.

First we show thatΩ(ε0)⊂Σ1. Suppose thatx0∈Ω(ε0). Ifx0∈M^o, it is easy to show thatx0∈Σ1, see[12].

Next we assume thatx0∈∂M. We claim that for anyR >0, andB_R^M(x0)⊂M, limn→+∞u⁺_n_L∞(B_R^M(x0))= +∞.

We prove the claim by a contradiction. So we assume that there is someR₀>0 andB_R^M

0(x₀)⊂Mand a subsequence