Energy quantization for a singular super-Liouville boundary value problem

In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields a Noether current from which strong estimates can be derived. Here, however, the conical singularities destroy conformal invariance. Therefore, we develop another, more general, method that uses the vanishing of the Pohozaev constant for such solutions to deduce the removability of boundary singularities.

associated Noether current whose algebraic structure can be turned into estimates. In the case of conformal invariance this Noether current is a holomorphic quadratic differential. For harmonic map type problems, finiteness of the energy functional in question implies that that differential is in L 1 . This then can be used to obtain fundamental estimates. For other problems, however, like (super-) Liouville equations, finiteness of the energy functional is not sufficient to get the L 1 bound of that differential and hence this is an extra assumption leading to the removability of local singularities (Prop 2.6, [23]).
But for (super-) Liouville equations on surfaces with conical singularities, we do not even have conformal invariance, because the scaling behavior at the singularities is different from that at regular points, see [27]. It turns out, however, that for an important class of two-dimensional geometric variational problems, there is a condition that is weaker than conformal invariance, the vanishing of a so-called Pohozaev constant (i.e. the Pohozaev identity), that is not only sufficient but also necessary for the blowup analysis. This Pohozaev constant on one hand measures the extent to which the Pohozaev identity fails and on the other hand provides a characterization of the singular behavior of a solution at an isolated singularity. Such kind of quantity appears also in other two dimensional geometric variational problems and can be applied to study the qualitative asymptotic behaviour of solutions defined on degenerating surfaces [20,40,41]. This vanishing condition is already known to play a crucial role in geometric analysis (see e.g. [34]), but for super-Liouville equations, as mentioned, this identity by itself suffices for the blow-up analysis.
In this paper, we shall apply this strategy to the blow-up analysis of the (super-) Liouville boundary problem on surfaces with conical singularities. To this purpose, let M be a compact Riemann surface with nonempty boundary ∂ M and with a spin structure. We also denote this compact Riemann surface as (M, A, g), where g is its Riemannian metric with the conical singularities of divisor A = m j=1 α j q j (for definition of A, see Sect. 2). Associated to the metric g, one can define the gradient ∇ and the Laplace operator in the usual way.
We then have our main object of study, the super-Liouville functional that couples a real-valued function u and a spinor ψ on M E B (u, ψ) = (1) where K g is the Gaussian curvature in M, and h g is the geodesic curvature of ∂ M and c is a given positive constant. The Dirac operator / D is defined by / Dψ := 2 α=1 e α · ∇ e α ψ, where {e 1 , e 2 } is an orthonormal basis on T M, ∇ is the Levi-Civita connection on M with respect to g and · denotes Clifford multiplication in the spinor bundle M of M. Finally, ·, · g is the natural Hermitian metric on M induced by g. We also write | · | 2 g as ·, · g . For the geometric background, see [28] or [19].
The Euler-Lagrange system for E B (u, ψ) with Neumann / chirality boundary conditions is ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ − g u = 2e 2u − e u ψ, ψ g − K g , in M o \ {q 1 , q 2 , . . . , q m }, / D g ψ = −e u ψ, in M o \ {q 1 , q 2 , . . . , q m }, ∂u ∂n = ce u − h g , on ∂ M \ {q 1 , q 2 , . . . , q m }, Here B ± are the chirality operators (see Sect. 2 for the definition). When ψ = 0 and (M, g) is a closed smooth Riemann surface, we obtain the classical Liouville functional The Euler-Lagrange equation for E(u) is the Liouville equation Liouville [32] studied this equation in the plane, that is, for K g = 0. The Liouville equation comes up in many problems of complex analysis and differential geometry of Riemann surfaces, for instance the prescribing curvature problem. The interplay between the geometric and analytic aspects makes the Liouville equation mathematically very interesting. When ψ = 0 and (M, g) again is a closed smooth Riemann surface, we obtain the super-Liouville functional E (u, ψ) = M 1 2 |∇u| 2 + K g u + ( / D + e u )ψ, ψ g − e 2u dv.
The Euler-Lagrange system for E(u, ψ) is The supersymmetric version of the Liouville functional and equation have been studied extensively in the physics literature, see for instance [1,15,33]. As all supersymmetric functionals that arise in elementary particle physics, it needs anticommuting variables.
Motivated by the super-Liouville functional, a mathematical version of this functional that works with commuting variables only, but otherwise preserves the structure and the invariances of it, was introduced in [21]. That model couples the bosonic scalar field to a fermionic spinor field. In particular, the super-Liouville functional is conformally invariant, and it possesses a very interesting mathematical structure.
The analysis of classical Liouville type equations was developed in [2,4,29,30] etc, and the corresponding analysis for super-Liouville equations in [21,23,26]. In particular, the complete blow-up theory for sequences of solutions was established, including the energy identity for the spinor part, the blow-up value at blow-up points and the profile for a sequence of solutions at the blow-up points. For results by physicists about super-Liouville equations, we refer to [1,15,33] etc.
When (M, A, g) is a closed Riemann surface (without boundary) with conical singularities of divisor A and with a spin structure, we obtain that E (u, ψ) = M 1 2 |∇u| 2 + K g u + ( / D + e u )ψ, ψ g − e 2u dv g .
The Euler-Lagrange system for E(u, ψ) is This system is closely related to the classical Liouville equation, or the prescribing curvature equation on M with conical singularites (see [12,36]). In addition, [3,4,[6][7][8]35] studied the blow-up theory of the following Liouville type equations with singular data: where (M, g) is a smooth surface and the singular data appear in the equation, which is the asymptotic behavior associated to the m interior punctures. For system (3), [27] provides an analytic foundation and the blow-up theory. For Liouville boundary problems on (M, g) with or without conical singularites, there are also lots of results on the blow-up analysis, see [9,16,22,43,44]. For super-Liouville boundary problems on a smooth Riemann surface M, the corresponding results can be found in [24,25].
In this paper, we aim to provide an analytic foundation and to establish the blowup analysis for the system (2). Our main result is the following energy quantization property for solutions to (2): From the energy quantization property, one can deduce the concentration properties of conformal volume and the compactness of solutions. It turns out that understanding of this property is the key step to study existence from a variational point of view by a refined Moser-Trudinger inequality, see e.g. [10,11,18].
If we assume that the points q 1 , q 2 , . . . , q l are in M o for 1 ≤ l < m and the points q l+1 , q l+2 , . . . , q m are on ∂ M for the surface (M, A, g) with the divisor A = m j=1 α j q j , α j > 0, we have the following Gauss-Bonnet formula is the degree of A, see [36]. From (2) we obtain that M 2e 2u n − e u n |ψ n | 2 Then we can use Theorem 1.1 to get the following: In particular, then the blow-up set 1 contains at most one point.
then the blow-up set 1 = ∅.
To show Theorem 1.1, a key step is to compute the blow-up value at the blow-up point p ∈ 1 for a blow-up sequence (u n , ψ n ). Here B M R ( p) is a geodesic ball of (M, g) at p. For this purpose, we need to study the following local super-Liouville boundary value problem (see Sect. 3): Here α ≥ 0, V (x) is in C 1 loc (D + r ∪ L r ) and satisfies 0 < a ≤ V (x) ≤ b. L r and S + r here and in the sequel are portions of ∂ D + r , which are defined in section 3. Then we have the following Brezis-Merle type concentration compactness theorem: Theorem 1.3 Let (u n , n ) be a sequence of regular solutions to (4) satisfying D + r |x| 2α e 2u n + | n | 4 dx + L r |x| α e u n ds < C.
Then, we have 2 ⊂ 1 . Moreover, (u n , n ) admits a subsequence, still denoted by (u n , n ), that satisfies . b) For u n , one of the following alternatives holds: iii) 1 is finite, nonempty and either To show the quantization property of the blow-up value, we need to rule out (5) in the above theorem. When the spinor field is vanishing, namely, in the case of Liouvilletype problems, a technique based on Pohozaev identity was introduced in [7] to prove the vanishing of the mass in the neck region for blow-up sequences of solutions with interior conical singularities. In the case of super-Liouville-type problems, we need to overcome some new difficulties caused by the spinor part. For the case of interior conical singularities, this was achieved in [27]. In the present paper, we shall handle the boundary conical singularities case. To this end, the decay estimates of the spinor part n , the Pohozaev identity of the local coupled system (4) and the energy identity of n , which means there is no energy contribution on the neck domain, play the essential roles. The corresponding theorem is the following: Denote by 1 = {x 1 , x 2 , . . . , x l } the blow-up set of u n . Then there are finitely many for i = 1, 2, . . . , I , and k = 1, 2, . . . , K i , and α ≥ 0, or there are finitely many solutions (u j,l , j,l ) that satisfy for j = 1, 2, . . . , J , and l = 1, 2, · · · , L j , and α ≥ 0. Here S 2 c is a portion of the sphere cut out by a 2-plane with constant geodesic curvature c . After selection of a subsequence, n converges in C ∞ loc to on (B + r ∪ L r )\ 1 and we have the energy identity: A crucial step in proving the above theorem is to show the removability of isolated singularities at the boundary, which is equivalent to the vanishing of the Pohozaev constant (see Theorem 4.5). Once the energy identity for the spinor part (9) is established, we can then obtain Theorem 1.5 Let (u n , n ) be solutions as in Theorem 1.3. Assume that (u n , n ) blows up and the blow-up set 1 = ∅. Then u n → −∞ uniformly on compact subsets of (D + r ∪ L r ) \ 1 . Furthermore, In the end, with the help of the Pohozaev identity (see Proposition 4.1) and the Green function at some singular points, we have the following: Theorem 1.6 Let (u n , n ) be solutions as in Theorem 1.3. Assume that (u n , n ) blows up and the blow-up set 1 = ∅. Let p ∈ 1 and assume that p is the only blow-up point in D + δ 0 ( p) for some δ 0 > 0. If there exists a positive constant C such that

Preliminaries
In this section, we will first recall the definition of surfaces with conical singularities, following [36]. Then we shall recall the chirality boundary condition for the Dirac operator / D, see e.g. [17]. In particular, we will see that under the chirality boundary conditions B ± , the Dirac operator / D is self-adjoint. A conformal metric g on a Riemannian surface M (possibly with boundary) has a conical singularity of order α (a real number with α > −1) at a point p ∈ M ∪ ∂ M if in some neighborhood of p g = e 2u |z − z( p)| 2α |dz| 2 where z is a coordinate of M defined in this neighborhood and u is smooth away from p and continuous at p. The point p is then said to be a conical singularity of angle θ = 2π(α+1) if p / ∈ ∂ M and a corner of angle θ = π(α+1) if p ∈ ∂ M. For example, a (somewhat idealized) American football has two singularities of equal angle, while a teardrop has only one singularity. Both these examples correspond to the case −1 < α < 0; in case α > 0, the angle is larger than 2π , leading to a different geometric picture. Such singularities also appear in orbifolds and branched coverings. They can also describe the ends of complete Riemann surfaces with finite total curvature. If (M, g) has conical singularities of order α 1 , α 2 , . . . , α m at q 1 , q 2 , . . . , q m , then g is said to represent the divisor A = m j=1 α j q j . Importantly, the presence of such conical singularities destroys conformal invariance, because the conical points are different from the regular ones.
The chirality boundary condition for the Dirac operator / D is a natural boundary condition for spinor part ψ. Let M be a compact Riemann surface with ∂ M = ∅ and with a fixed spin structure, admitting a chirality operator G, which is an endomorphism of the spinor bundle M satisfying: and y∇ X (Gψ) = G∇ X ψ, for any X ∈ (T M), ψ, ϕ ∈ ( M). Here I denotes the identity endomorphism of M and (·) denotes the space of sections of a given bundle. We usually take G = γ (ω 2 ), which denotes the Clifford multiplication by the complex volume form ω 2 = ie 1 e 2 , where e 1 , e 2 is a local orthonormal frame on M.
Denote by the restricted spinor bundle with induced Hermitian product. Let − → n be the outward unit normal vector field on ∂ M. One can verify that − → n G : Hence, we can decompose S = V + V − , where V ± is the eigensubbundle corresponding to the eigenvalue ±1. One verifies that the orthogonal projection onto the eigensubbundle V ± : defines a local elliptic boundary condition for the Dirac operator / D , see e.g. [17]. We say that a spinor ψ ∈ L 2 ( ( M)) satisfies the chirality boundary conditions B ± if It is well known (see e.g. [17]) that if ψ, φ ∈ L 2 ( ( M)) satisfy the chirality boundary conditions B ± then − → n · ψ, ϕ = 0, on ∂ M.
In particular, It follows that the Dirac operator / D is self-adjoint under the chirality boundary conditions B ± .
It may be helpful if we recall that on a surface the (usual) Dirac operator / D can be seen as the (doubled) Cauchy-Riemann operator. Consider R 2 with the Euclidean metric ds 2 + dt 2 . Let e 1 = ∂ ∂s and e 2 = ∂ ∂t be the standard orthonormal frame. A spinor field is simply a map : R 2 → 2 = C 2 , and the actions of e 1 and e 2 on spinor fields can be identified by multiplication with matrices If := f g : R 2 → C 2 is a spinor field, then the Dirac operator is Therefore, the elliptic estimates developed for (anti-) holomorphic functions can be used to study the Dirac equation.
If M be the upper-half Euclidean space R 2 + , then the chirality operator is G = ie 1 e 2 = 1 0 0 −1 . Note that − → n = −e 2 , we get that By the standard chirality decomposition, we can write ψ = ψ + ψ − , and then the boundary condition becomes Without loss of generality, in the sequel, we shall only consider the chirality boundary condition B = B + .
We have the following geometric property: In other word, if we write that ϕ * (g) = λ 2 g, where λ > 0 is the conformal factor of the conformal map ϕ, and set

The local singular super-Liouville boundary problem
In this section, we shall first derive the local version of the super-Liouville boundary problem. Then we shall analyze the regularity of solutions under the small energy condition. First we take a point p ∈ M o , choose a small neighborhood U ( p) ⊂ M o , and define an isothermal coordinate system x = (x 1 , x 2 ) centered at p, such that p corresponds to x = 0 and g = e 2φ |x| 2α (dx 2 where φ is smooth away from p, continuous at p and φ( p) = 0. We can choose such a neighborhood small enough so that if p is a conical singular point of g, then U ( p) ∩ A = {p} and α > 0, while, if p is a smooth point of g, then U ( p) ∩ A = ∅ and α = 0. Consequently, with respect to the isothermal coordinates, we can obtain the local version of the singular super-Liouville-type equations, which has no boundary condition since p is a interior point of M. Here = |x| The detailed arguments can be found in the section 3 of [27]. We also assume that (u, ) satisfy the energy condition: We put D r := D r (0). We say that (u, ) is a weak solution of (12) and (13), if u ∈ W 1,2 (D r ) and ∈ W 1, 4 3 ( ( D r )) satisfy for any φ ∈ C ∞ 0 (D r ) and any spinor ξ ∈ C ∞ ∩ W 1, 4 3 0 ( ( D r )). A weak solution is a classical solution by the following: Proposition 3.1 Let (u, ) be a weak solution of (12) and (13).
For p ∈ ∂ M, we also can choose a small geodesic ball U ( p) ⊂ M and define an isothermal coordinate system x = (x 1 , x 2 ) centered at p, such that p corresponds to , where φ is smooth away from p and continuous at p. We can choose such a geodesic ball small enough so that if p is a conical singular point of g, then U ( p) ∩ A = {p} and α > 0, while, if p is a smooth point of g, then U ( p) ∩ A = ∅ and α = 0. Set L r = ∂ D + r ∩ ∂R 2 + , and S + r = ∂ D + r ∩ R 2 + . Also in the sequel, we will set L r (x 0 ) = ∂ D + r (x 0 )∩∂R 2 + , and S + r (x 0 ) = ∂ D + r (x 0 )∩R 2 + . Consequently, with respect to the isothermal coordinates, (u, ψ) satisfies Here = ∂ 2 x 2 x 2 is the usual Laplacian, and the Dirac operator / D can be seen as doubled Cauchy-Riemann operator, B is the chirality boundary operator of spinors.
Note that the relation between the two Gaussian curvatures and between the two geodesic curvatures are respectively − φ = e 2φ |x| 2α K g , ∂φ ∂n = e φ |x| α h g .
By standard elliptic regularity we conclude that φ ∈ W 2, p loc (D + r ∪ L r ) for some p > 1 if α ≥ 0 and if the curvature K g and h g of M is regular enough. Therefore, by Sobolev embedding, φ ∈ C 1 loc (D + r ∪ L r ). If we denote V (x) = e φ , W 1 (x) = e 2φ |x| 2α K g and for all p > 1 if the curvature K g and h g of M is regular enough. Therefore, the Eq. (14) can be rewritten as: It is easy to see that w(x) is in Here Thus we get the local system (4) of the boundary problem (2).
As the interior case, we can also define (u, ) be a weak solution of (4) if u ∈ For weak solutions of (4) we also have the following regularity result.

Proposition 3.2 Let (u, ) be a weak solution of (4) with the energy condition
Note that when α = 0 this proposition has been proved in [24]. When α > 0, to get the L 1 integral of u + , we need a trick which was introduced in [7] and also was used in [27]. That is, by using the fact that for some t > 0 we can choose s = t t+1 ∈ (0, 1) when α > 0 and s = 1 when α = 0 such that Once we get the L 1 integral of u + , we can get the conclusion of Proposition 3.2 by use the same argument in [24]. We omit the proof here.
). Next we consider the convergence of a sequence of regular solutions to (4) under a smallness condition for the energy. We assume that (u n , n ) satisfy that with the energy condition for some constant C > 0. First, we study the small energy regularity, i.e. when the energy D + r |x| 2α e 2u n dx and L r |x| α e u n dx are small enough, u n will be uniformly bounded from above. Our Lemma is: ) are uniformly bounded.
Proof As the same situation as in Proposition 3.2, we can no longer use the inequality 2 u + n < e 2u n to get the uniform bound of the L 1 -integral of u + n when α > 0. But notice that there exists a constant t > 0 such that Setting s = t t+1 ∈ (0, 1), then we obtain Then by a similar argument as in the proof of Lemma 3.5 in [24] we can prove this Lemma.
When the energy D + r 2V 2 (x)|x| 2α e 2u n + L r V (x)|x| α e u n ds is large, in general, blow-up phenomenon may occur, i.e., Theorem 1.3 holds.

Remark 3.4 Let
Then, by using similar arguments as in [7], the two blow-up sets of u n and v n are the same. To show this conclusion, it is sufficient to show the point x = 0 is a blow-up point for u n if and only if it is a blow-up point for v n . In fact, if 0 is the only blow-up point for v n in a small neighbourhood then, it is easy to see that 0 is also the only blow-up point for u n in a small neighbour- In converse, we assume that 0 is the only blow-up point for u n in a small neighbourhood D + δ 0 ∪ L δ 0 such that (19) is holds. We argue by contradiction and suppose that there exists a uniform constant C, such that v n (x) ≤ C for any x ∈ D + δ 0 . First, we can obtain that there exists a uniform constant C, such that | n | 2 (x) ≤ C for any x ∈ D Then n satisfies 2 )) and in particular n is uniformly bounded . Then by Harnack type inequality of Neumann boundary problem (see Lemma A.2 in [24]), it follows that inf D + δ 0 2 u n → +∞. Thus we get a contradiction since the blow-up set of u n is finite.

Removability of local sigularities
The Pohozaev identity is closely related to the removability of singularities. In this section, we shall first establish the Pohozaev identity for regular solutions to (4). Then for solutions defined on a domain with isolated singularity, we define a constant which is called the Pohozaev constant. The most important is that a necessary and sufficient condition for the removability of local singularities is the vanishing of Pohazaev constant.

Proposition 4.1 (Pohozaev identity) Let (u, ) be a regular solution of (4), that is
Then we have the following Pohozaev identity where ν is the outward normal vector to S + R , andx is the reflection point of x about ∂R 2 + .
Proof The case of α = 0 and V ≡ 1 has already been treated in [25]. The calculation of the Pohozaev identity is standard. Since in the sequel we will need to calculate the Pohozaev identity for different equations, for reader's convenience, we give the detailed proof for this general case. First, we multiply the first equation by x · ∇u and integrate over D + R to obtain It follows from direct computations that Therefore we have On the other hand, for x ∈ R 2 + , we denote x = x 1 e 1 + x 2 e 2 under the local orthonormal basis {e 1 , e 2 } on R 2 + . Using the Clifford multiplication relation for any spinors ψ, ϕ ∈ ( M). We know that ψ, e i · ψ + e i · ψ, ψ = 0 ( 2 2 ) for any i = 1, 2. Using the chirality boundary condition of , we extend (u, ) to the lower half disk D − R . Assumex is the reflection point of x about ∂R 2 + , and define Then it follows from the argument in Lemma 3.4 of [24] that we obtain Here Using the Schrödinger-Lichnerowicz formula / Then we multiply (25) by x · (where · denotes the Clifford multiplication) and integrate over D R to obtain On the other hand, by partial integration, and similarly Furthermore we also have Therefore we obtain Putting (21) and (26) together, we obtain our Pohozaev type identity (20).
Pohozaev type identity is shown to be closely related to the removability of local singularities of solutions. For a solution of (12) and (13), we defined in [27] the following Pohozaev constant: ) be a solution of (12) and (13). For 0 < R < r , we define the Pohozaev constant with respect to the Eqs. (12) with the constraint (13) as follows: (12) and (13).
We also proved in [27] that a local singularity is removable if the Pohozaev identity (27) holds, that is, iff the Pohozaev constant vanishes. Theorem 4.3 [27].
) be a solution of (12) and (13). Then there is a constant γ < 2π(1 + α) such that where h is bounded near 0. The Pohozaev constant C(u, ) and γ satisfy: For the singular boundary problem (4), we can define the Pohozaev constant in a similar way: ) be a solution of (4) and (15). For 0 < R < r , we define the Pohozaev constant with respect to the Eqs. (4) with the constraint (15) as follows: The removability theorem of a local singularity at the boundary is following: ) be a solution of (4) and (15), then there is a constant γ < π(1 + α) such that where h is bounded near 0. The Pohozaev constant C(u, ) and γ satisfy: To prove Theorem 4.5, we need to derive the decay of spinor part near the singular point. For the case of α = 0 and V (x) = 1, this is shown in [25]. By using similar arguments, we can also get the following lemma for the general case: Then for any x ∈ B for some positive constant C. Here ε is any sufficiently small positive number.
Proof of Theorem 4.5: By the conformal invariance, we assume without loss of generality that B + r |x| 2α e 2v dx ≤ ε 1 and |c| L r |x| α e v ds ≤ ε 2 where ε 1 and ε 2 are as in Lemma 4.6. By standard potential analysis, it follows that there is a constant γ such that lim By D + r |x| 2α e 2u dx < C, we obtain that γ ≤ π(1+α). Furthermore, by using Lemma 4.6 and by a similar argument as in the proof of Proposition 5.4 of [23], we can improve this to the strict inequality γ < π(1 + α). Next we set Since we can extend w to B r \{0} evenly to get a harmonic function w in B r \{0}, then we obtain that Duo to w is harmonic in B 1 \{0} we have with a smooth harmonic function w 0 in B r . Therefore we have u = − γ π log |x| + v + w 0 near 0.

By (30), we have
where o r (1) → 0 as r → 0. We also have Putting all together and letting r → 0, we get Since C(u, ) = 0 for (u, ), therefore we get γ = 0. This implies that the local singularity of (u, ) is removable.

Bubble energy
After a suitable rescaling at a boundary blow-up point, we will obtain a bubble, i.e. an entire solution on the upper half-plane R 2 + with finite energy. In this section, we will investigate such entire solutions. We will first show the asymptotic behavior of an entire solution and compute the bubble energy, and then show that an entire solution can be conformally extended to a spherical cap, i.e., the singularity at infinity is removable.
The considered equations are The energy condition is First, let us notice that if (u, ψ) is a weak solution of (31) and (32) with u ∈ H 1,2 loc (R 2 + ) and ψ ∈ W 1, 4 3 loc ( ( R 2 + )), by using similar arguments as in the proof of Proposition 3.2, we have u + ∈ L ∞ (R 2 + ). Consequently, it follows that u ∈ C 2 loc (R 2 . We call (u, ψ) a regular solution of (31) and (32) And, by change of variable, we can choose r 0 small enough such that (v, φ) satisfies (34) Applying Lemma 4.6 to (33) and (34), and by the Kelvin transformation, we obtain the asymptotic estimate of the spinor ψ(x) and Next, we will show that d = 2(1 + α)π and ξ 0 is a well-defined constant spinor. (u, ψ) be a regular solution of (31) and (32) and let c be a nonnegative constant. Then we have

Proposition 5.1 Let
ψ(x) = − 1 2π x where · is the Clifford multiplication, C is a positive universal constant, and I is the identity. In particular we have d = 2(1 + α)π and ξ 0 is well defined.
Proof We shall apply standard potential analysis to prove this proposition. Similar arguments can be found in [13,21,22] and the references therein. The essential facts used in this case are the Pohozaev identity and the decay estimate for the spinor. For readers' convenience, we sketch the proof here.
Step 1. lim |x|→∞ Consider We extend v(x) to R 2 by even reflection such that v(x) is harmonic in R 2 . From Lemma 5.1 we know v(x) ≤ C(1 + ln(|x| + 1)) for some positive constant C. Thus v(x) is a constant. This completes the proof of Step 1. Since R 2 + |x| 2α e 2u dx < ∞, we get that d ≥ π(1 + α). Furthermore, similarly as in the case of the usual Liouville or super-Liouville equation, we can show that d > π(1 + α).
Therefore the asymptotic estimates (35) and (36) of the spinor ψ(x) hold. By using the standard potential analysis we can obtain that for some constant C > 0. Thus we get the proof of (37). Furthermore, we can show that d = 2π(1 + α). For sufficiently large R > 0, the Pohozaev identity for the solution (u, ψ) gives By the asymptotic estimates (35), (36) and (37)  Let R → +∞ in (39), we get that It follows that d = 2π(1 + α).
This implies that the constant spinor ξ 0 is well defined. By using the chirality boundary condition of spinor, we extend (u, ψ) to the lower half plane R 2 + (see (23) and (24)) to get Here A(x) is defined by The constant spinor ξ 1 is also well defined. From the asymptotic estimates (35) and (40) and a similar argument in [27] we obtain for |x| near ∞. Since |y| α e u(y) ie 1 · ψ(y)dy Hence we obtain from (41) ψ(x) = − 1 2π x for |x| near ∞.
Thus we finish the proof of Step 3 and we complete the proof of the Proposition.
Proposition 5.1 indicates that the singularity at infinity of regular solutions for (31) and (32) can be removed as in many other conformally invariant problems. (u, ψ) be a regular solution of (31) and (32) . Then (u, ψ)

Theorem 5.2 Let
Since α = 2π(1 + α), it follows that v is bounded near the singularity 0. Recall that φ is also bounded near 0, we can apply elliptic theory to obtain that (v, φ) is regular on R 2 + .

Energy identity for spinors
The energy identity for spinor part of solutions to the super-Liouville equations on closed Riemann surfaces was derived in [23,27]. In this section, we shall prove an analogue for the singular super-Liouville boundary problem, i.e. Theorem 1.4. For harmonic maps in dimension two and J-holomorphic curves as well as for solutions of certain nonlinear Dirac type equations, similar results are derived in [14,38,39,42] and the references therein.
To prove Theorem 1.4, we shall derive the local estimate for the spinor part on an upper half annulus. Since we can extend (u, ) to the lower half disk D − r by the chirality boundary condition of , the proof of this local estimate can be established by using the result of Lemma 3.1 of [23]. Here we just state the Lemma and omit the proof. For 0 < r 1 < 2r 1 < r 2 2 < r 2 < r , consider the annulus A r 1 ,r 2 = {x ∈ R 2 |r 1 ≤ |x| ≤ r 2 } and the upper half annulus A + r 1 ,r 2 = A r 1 ,r 2 ∩ R 2 + . Then we have for a positive constant C 0 and some universal positive constant C.
Proof of Theorem 1. 4 We will follow closely the argument for the energy identity of harmonic maps, see [14], or for super-Liouville equations, see [7,23,25,27] . Since the blow-up set 1 is finite, we can find small disk , n converges strongly to in L 4 . So, we need to prove that there are (u i,k , ξ i,k ), which are solutions of (7), i = 1, 2, . . . , I ; k = 1, 2, . . . , K i , such that or, we need to prove that there are (u j,l , ξ j,l ), which are solutions of (8), j = 1, 2, . . . , J ; l = 1, 2, . . . , L j , such that When p ∈ (D + r ) o , from [27], we know that (44) holds. So, without loss of generality, we assume that p ∈ L r and there is only one bubble at each blow-up point p. Furthermore, we may assume that p = 0. The case of p = 0 can be handled in an analogous way and in fact this case is simpler, as |x| α is a smooth function near p. Then what we need to prove is that there exists a bubble (u, ξ) as (7), such that or there exists a bubble (u, ξ) as (8) such that such that Next we rescale functions (u n , n ) at the blow-up point p = 0 and then try to get the bubble of (u n , n ). To this purpose, we let x n ∈ D + δ such that u n (x n ) = max D + δ u n (x).
Write x n = (s n , t n ). It is clear that x n → p and u n (x n ) → +∞. Define λ n = e − un (xn ) α+1 . We know λ n , |x n | and t n converge to 0 as n → 0, but their rates of converging to 0 may be different. Next we will distinguish three cases. Case I. |x n | λ n = O(1) as n → +∞. In this case, we define the rescaling functions with the energy condition We know that Notice that the maximum point of u n (x), i.e. x n λ n , is bounded, namely | x n t n | ≤ C. So by taking a subsequence, we can assume that x n t n → x 0 ∈R 2 + with |x 0 | ≤ C. Therefore it follows from Theorem 1.3 that, by passing to a subsequence, ( u n , n ) converges in with the energy condition R 2 + (|x| 2α e 2 u + | | 4 )dx + ∂R 2 + |x| α e u dσ < ∞. By Proposition 5.1, there holds By the removability of a global singularity (Theorem 5.2), we get a bubbling solution on S 2 c .
Case II. |x n | λ n → +∞ as n → +∞. In this case, we must have u n (y n ) := u n (x n ) + (α + 1) ln |x n | = (α + 1) ln |x n | − (α + 1) ln λ n → +∞. (49) Therefore we can rescale twice to get the bubble. First, we define the rescaling functions Set that y n = x n |x n | . We assume that y 0 = lim n→∞ x n |x n | . By (49), we know y 0 is a blow-up point of (u n , n ). We can set δ n = e −u n (y n ) , and ρ n = e −un (xn ) |x n | α = λ n ( λ n |x n | ) α . It is clear that δ n → 0, ρ n → 0 and |x n | ρ n → +∞ as n → ∞. We define the rescaling functions u n (x) = u n (δ n x + y n ) + ln δ n = u n (x n + ρ n x) − u n (x n ) n (x) = = δ 1 2 n n (δ n x + y n ) = ρ 1 2 n n (x n + ρ n x) for any x such that y n + δ n x ∈ D + R (y n ) with any R > 1. By a direct computation, we have It is clear that Now we proceed by distinguishing two subcases.
Case II.1 t n ρ n → +∞ as n → ∞. Notice that | x n |x n | + ρ n |x n | x| → 1 as n → ∞ in C 0 loc (R 2 ). It follows from Theorem 1.3 that, by passing to a subsequence, ( u n , n ) converges in C 2 with the energy condition R 2 e 2 u + | | 4 dx < ∞. By Proposition 6.4 in [21], there holds By the removability of a global singularity (Theorem 6.5 in [21]), we get a bubbling solution on S 2 .
Case II.2 t n ρ n → as n → ∞. Similar in the Case II.1, we have from Theorem 1.3 that, by passing to a subsequence, ( u n , n ) converges in C 2 with the energy condition R 2 − e 2 u + | | 4 dx + ∂R 2 − e u dσ < ∞. By Proposition 6.4 in [21], there holds By the removability of a global singularity (Theorem 6.5 in [25]), we get a bubbling solution on S 2 c . It is well know, in order to prove (46) or (47), we need to prove that there is no any energy of n in the neck domain, i.e.
where A + δ,R,n is the neck domain which is defined latter. To this purpose, we shall proceed separately for Case I, Case II.1 and Case II.2.
For Case I, we define the neck domain is We have two claims. Claim 1 For any ε > 0, there is an N > 1 such that for any n ≥ N , we have |x| α e u n < ε; ∀r ∈ [eλ n R, δ].
To prove this claim, we note two facts. The first fact is: for any T > 0, there exists some N (T ) such that for any n ≥ N (T ), we have Actually, since (u n , n ) has no blow-up point in D + δ \{ p}, then | n | is uniformly bounded in D + δ \D + δe −T , and u n will either be uniformly bounded in D + δ \D + δe −T or uniformly tend to −∞ in D + δ \D + δe −T . So if u n uniformly tends to −∞ in D δ \D δe −T , it is clear that, for any given T > 0, we have an N (T ) big enough such that when n ≥ N (T ) Moreover, since n converges to in L 4 loc ((D + r ∩ L r ) \ 1 ) and hence For any small ε > 0, we may choose δ > 0 small enough such that D + δ | | 4 < ε 4 , then for any given T > 0, we have an N (T ) big enough such that when n ≥ N (T ) Consequently, we get (53). If (u n , n ) is uniformly bounded in D + δ \D + δe −T , then we know (u n , n ) converges to a weak solution (u, ) strongly on compact sets of D + δ \ {p}. Therefore, we can also choose δ > 0 small enough such that, for any given T > 0, there exists an N (T ) big enough, when n ≥ N (T ), (53) holds.
The second fact is: For any small ε > 0, and T > 0, we may choose an N (T ) such that when n ≥ N (T ) if R is big enough. Now we can prove the claim. We argue by contradiction by using the above two facts. If there exists ε 0 > 0 and a sequence r n , r n ∈ [eλ n R, δ], such that Then, by the above two facts, we know that δ r n → +∞ and λ n R r n → 0, in particular, r n → 0 as n → +∞. Rescaling again, we set v n (x) = u n (r n x) + (1 + α) ln r n , ϕ n (x) = r 1 2 n (r n x) It is clear that (|x| 2α e 2v n + |ϕ n | 4 ) + And (v n , ϕ n ) satisfies for any R > 0 According to Theorem 1.3, there are three possible cases: 1. There exists some q ∈ Q n = (D + |x| α e v n ≥ ε 0 > 0 for any small r > 0. In such a case, we still obtain the second "bubble" by the rescaling argument. Thus we get a contradiction.

For any
. Then, it is clear that ϕ n converges to a spinor ϕ in L 4 We translate ϕ to be a harmonic spinor on R 2 + \ {0} satisfying the corresponding chirality boundary condition and then extend it as in (24) to a harmonic spinor ϕ on R 2 \ {0} with bounded energy, i.e., ||ϕ|| L 4 (R 2 ) < ∞. As discussed in [23], ϕ conformally extends to a harmonic spinor on S 2 . By the well known fact that there is no nontrivial harmonic spinor on S 2 , we have that ϕ ≡ 0 and hence ϕ n converges to 0 in L 4 loc (R 2 + \ {0}). This will contradict (54) 3. For any R > 0, there is no blow-up point in ). In such a case (v n , ϕ n ) will converge to (v, ϕ) strongly with finite energy. It is clear that (v, ϕ) is regular.
Next we need to remove the singularities of (v, ϕ) and then obtain the second bubble of the system. Consequently we get a contradiction. To this purpose, let us use the Pohozaev identity of (u n , n ) in D + δ , it follows for any ρ with r n ρ < δ ((s, 0))|s| α se u n | s=r n ρ s=−r n ρ rn ρ x · ∇V (x)|x| α e u n | n | 2 dv Hence for rescaling functions (v n , ϕ n ) we have This implies that the associated Pohozaev constant of (v n , ϕ n ) satisfies ((r n s, 0)) ∂s |s| α se v n ds +cV ((r n s, 0) Since, for any ρ > 0, D + ρ |x| 2α e 2v n + |ϕ n | 4 dv + L ρ |x| α e v n ds < C, it is easy to check that ((r n s, 0)) ∂s |s| α se v n ds = 0.
This implies that Here β = lim r →0 lim n→∞ D + r 2V 2 (r n x)|x| 2α e 2v n − V (r n x)|x| α e v n |ϕ n | 2 dv + L r cV (r n x)|x| α e v n ds , On the other hand, we use the fact that as n → ∞. By using Green's representation formula for u n in D + ρ and then take n → ∞, we have It is clear that γ (x) is in C 1 (D + ρ ) and φ satisfies By similar arguments as the proof of Proposition 4.5, we can obtain that This implies that Noticing that D + ρ |x| 2α e 2v dx < ∞, we have β ≤ (1 + α)π. Therefore we obtain that β = 0, i.e. C(v, ϕ) = 0, and the singularity at 0 of (v, ϕ) is removed by Proposition 4.5. Furthermore, the singularity at ∞ of (v, ϕ) is also removed by Theorem 5.2. Thus we get another bubble on S 2 c , and we get a contradiction to the assumption that m = 1. Concequently we complete the proof of the claim 1. Claim 2 We can separate A + δ,R,n into finitely many parts where N k ≤ N 0 for N 0 is a uniform integer for all n large enough, A + k = D + r k−1 \ D + r k , r 0 = δ, r N k = λ n R, r k < r k−1 for k = 1, 2, . . . , N k , and C 0 is a constant as in Lemma 6.1.
The proof of this claim is very similar to those in [23,25,41] and the argument is now standard, so we omit it. Now we apply Claim 1 and Claim 2 to prove (52). Let ε > 0 be small, and let δ be small enough, and let R and n be big enough. We apply Lemma 6.1 to each part A + k to obtain

Therefore we have
Since ε is small, we may assume ε ≤ 1. Then we get With similar arguments, and using (55), we have Summing up (55) and (56) on A + l we get Thus we have shown (52) in the first case. For Case II, according the blow-up process, we define the neck domain is Notice that Duo to the assumption that (u n , n ) has only one bubble at the blow-up point p = 0, (u n , n ) also has only one bubble at its blow-up point y 0 . Therefore, we have To prove (58), by using the similar argument as the case 1, we have the following facts: Fact II.1 For any small ε > 0, and T > 0, there exists some N (T ) such that for any n ≥ N (T ) we have for sufficiently large S.
Fact II.2 For any small ε > 0, and T > 0, we may choose an N (T ) such that when n ≥ N (T ) if R is large enough. Buy using the above two facts, we need to prove the following claim: Claim II.1 For any ε > 0, there is an N > 1 such that for any n ≥ N , we have Proof of Claim II. 1 We assume by a contradiction that there exists ε 0 > 0 and a sequence r n , r n ∈ [eρ n R, |x n |S], such that Then, by Facts II.1 and II.2, we know that |x n |S r n → +∞ and ρ n R r n → 0, in particular, r n → 0 as n → +∞. We assume that = lim n→∞ t n r n . Here is either a nonnegative real number or +∞. Next we proceed by distinguishing two cases.
with finite energy. Next we will remove two singular points at 0 and at ∞, and consequently we get the second bubble of the considered system. Thus we get a contradiction. To this purpose, let us computate the Pohozaev constant of (v, ϕ). Let start with the Pohozaev identity of (u n , n ). We multiply all terms in (4) by (x − x n )·∇u n and integrate over D + r n ρ (x n ). It follows for any sufficient small ρ that Hence for rescaling functions (v n , ϕ n ) we have Since the associated Pohozaev constant of (v n , ϕ n ) is c ∂ V ((r n s + s n , 0)) r n |s n | s + s n |s n | α ∂s se v n ds +cV ((r n s + s n , 0)) r n |s n | s + s n we have from (63) that C(v n , ϕ n ) = C(v n , ϕ n , ρ) = 0.
Since | x n |s n | + r n |s n | x| 2α is a smooth function in D + ρ , by the energy condition, we can easily to check that This implies that Here and C(v, ϕ) = C(v, ϕ, ρ) is the Pohozaev constant of (v, ϕ), i.e.
On the other hand, we use the fact that (v n , ϕ n ) converges to (v, ϕ) as n → ∞. By using Green's representation formula for u n in D + ρ and then take n → ∞, we have with φ being a bounded term and γ (x) being a regular term. Consequently, we can obtain that This implies that Noticing that D + ρ e 2v dx < ∞, we have β ≤ π . Therefore we obtain that β = 0, i.e. C(v, ϕ) = 0, and the singularity at 0 of (v, ϕ) is removed by Propostion 4.5. Furthermore, the singularity at ∞ of (v, ϕ) is also removed by Theorem 5.2. Thus we get another bubble on S 2 c , and we get a contradiction to the assumption that m = 1. Consequently we complete the proof of Claim II.1.
Next , similarly to Case I. we can prove the following: Claim II. 2 We can separate A + S,R,n (x n ) into finitely many parts where N k ≤ N 0 for N 0 is a uniform integer for all n large enough, . . , N k , and C 0 is a constant as in Lemma 6.1.
Then, by using Claim II.1 and Claim II.2 we can complete the proof of the result.

Blow-up behavior
In this section, we will show that u n → −∞ uniformly on compact subset of (D + r ∪ L r ) \ 1 in means of the energy identity for spinors. Thus we rule out the possibility that u n is uniformly bounded in L ∞ loc ((D + r ∪ L r ) \ 1 ) in Theorem 1.3. The following is the proof of Theorem 1.5.

Proof of Theorem 1.5:
We prove the results by contradiction. Assume that the conclusion of the theorem is false. Then by Theorem 1.3, u n is uniformly bounded in L ∞ loc ((D + r ∪ L r ) \ 1 ). Thus we know that (u n , n ) converges in C 2 on any compact subset of (D + r ∪ L r ) \ 1 to (u, ), which satisfies that with bounded energy |x| α e u ds < +∞.
Since the blow-up set 1 is not empty, we can take a point p ∈ 1 . Choose a small δ 0 > 0 such that p is the only point of 1 in D 2δ 0 ( p) ∩ (D + r ∪ L r ) = {p}. If p is the interior point of D + r , then we can choose δ 0 sufficiently small such that D 2δ 0 ( p) ⊂ (D + r ∪ L r ). Hence by Theorem 1.3 in [27] we can get a contradiction. Next we assume that p is on L r . Without loss of generality, we assume that p = 0. The case of p = 0 can be dealed with in an analogous way.
We shall first show that the limit (u, ) is regular at the isolated singularity p = 0, i.e. u ∈ C 2 (D + r ) ∩ C 1 (D + r ∪ L r ) and ∈ C 2 ( ( D + r )) ∩ C 1 ( ( (D + r ∪ L r ))) for some small r > 0. To this end, we shall using Theorem 4.5 for removability of a local singularity to remove the singularity. We know that the Phohozaev constant, denote C B (u, ), of (u, ) at p = 0 is for any 0 < ρ < δ 0 . On the other hand, since (u n , n ) are the regular solution, the Pohozaev constant C B (u n , n ) = C B (u n , n , ρ) satisfies 0 = C(u n , n ) = C(u n , n , ρ) Let n → ∞ and ρ → 0, by using that (u n , n ) converges to (u, ) regularly on any compact subset of D + 2δ 0 \ {0} and that the energy condition (17), to get Moreover, we can also assume that in the sense of distributions in D + δ ∪ L δ for any small δ > 0. Then, applying similar arguments as in the proof of the local singularity removability in Claim I.1, Theorem 1.4, we can show that C B (u, ) = 0, β = 0 and hence (u, ) is a regular solution of (4) on D + 2δ 0 with bounded energy |x| α e u ds < +∞.
Hence, we can choose some small δ 1 ∈ (0, δ 0 ) such that for any δ ∈ (0, δ 1 ), (65) Next, as in the proof of Theorem 1.4, we rescale (u n , n ) near p = 0. We let . Write x n = (s n , t n ). Then x n → p = 0 and u n (x n ) → +∞. Define λ n = e − un (xn ) α+1 . It is clear that λ n , |x n | and t n converge to 0 as n → 0. we will proceed by distinguishing the following three cases: Case I. |x n | λ n = O(1) as n → +∞. In this case, the rescaling functions are u n (x) = u n (λ n x) + (1 + α) ln λ n n (x) = λ for any x ∈ D + δ 1 2λn . Moreover, by passing to a subsequence, ( u n , n ) converges in Then for δ ∈ (0, δ 1 ) small enough, R > 0 large enough and n large enough, we have cV (x)|x| α e u n ds Here in the last step, we use the fact from Theorem 1.4 that the neck energy of the spinor field n is converging to zero. We remark that in the above estimate, if there are multiple bubbles then we need to decompose D + δ \D + λ n R further into bubble domains and neck domains and then apply the no neck energy result in Theorem 1.4 to each of these neck domains.
On the other hand, we fix some δ ∈ (0, δ 1 ) small such that (66) holds and then let n → ∞ to conclude that Here in the last step, we have used (65). Thus we get a contradiction and finish the proof of the Theorem in this case.
Case II. |x n | λ n → +∞ as n → +∞. In this case, as in the arguments in Theorem 1.4, we can rescale twice to get the bubble. First, we define the rescaling functions ū n (x) = u n (|x n |x) + (α + 1) ln |x n | n (x) = |x n | 1 2 n (|x n |x) for any x ∈ D + δ 1 2|xn | . Set y n := x n |x n | . Due toū n (y n ) → +∞, we set that δ n = e −ū n (y n ) and define the rescaling function u n (x) =ū n (δ n x + y n ) + ln δ n n (x) = δ 1 2 n¯ n (δ n x + y n ) for any δ n x + y n ∈ D + δ 1 2|xn | . Denote that ρ n = e −un (xn ) |x n | α = λ n ( λ n |x n | ) α and x n = (s n , t n ). Case II.1 t n ρ n → +∞ as n → ∞. Then, by passing to a subsequence, ( u n , n ) converges in C 2 loc (R 2 )×C 2 loc ( ( R 2 )) to some ( u, ) satisfying Therefore, for δ ∈ (0, δ 1 ) small enough, S, R > 0 large enough and n large enough, the fact that the neck energy of the spinor field n is converging to zero, we have D + δ (2V 2 (x)|x| 2α e 2u n − V (x)|x| α e u n | n | 2 )dx + Then, applying similar arguments as in Case I, we get a contradiction, and finish the proof of the Theorem in this case.
Case II.2 t n ρ n → as n → ∞. Then, by passing to a subsequence, ( u n , n ) converges in C 2 with the bubble energy Then, applying similar arguments as in Case II.1, we can get a contradiction, and finish the proof of the Theorem.

Blow-up Value
By means of Theorem 1.5, we can further compute the blow-up value at the blow-up point p, which is defined as cV (x)|x| α e u n ds .
We know from Theorem 1.5 that m( p) > 0. Now we shall determine the precise value of m( p) under a boundary condition.
Proof of Theorem 1.6: When p / ∈ L δ 0 ( p), It is clear that we can choose δ 0 sufficiently small such that D + δ 0 ( p) = D δ 0 ( p). Then we have m( p) = 4π according to the arguments in [27]. Next we assume that p ∈ L δ 0 ( p). Without loss of generality, we assume p = 0. The case of p = 0 can be handled analogously.

Energy quantization for the global super-Liouville boundary problem
In this section, we will show the quantization of energy for a sequence of blowingup solutions to the global super-Liouville boundary problem on a singular Riemann surface. Let (M, A, g) be a compact Riemann surface with conical singularities represented by the divisor A = m j=1 α j q j , α j > 0 and with a spin structure. We assume that ∂ M is not empty and (M, g) has conical singular points q 1 , q 2 , . . . , q m such that q 1 , q 2 , . . . , q l are in M o for 1 ≤ l < m and q l+1 , q l+2 , . . . , q m are on ∂ M. Writing g = e 2φ g 0 , where g 0 is a smooth metric on M, we can deduce from the results for the local super-Liouville equations: by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.