Boundary Value Problems for the $2^{nd}$-order Seiberg-Witten Equations

It is shown that the non-homogeneous Dirichlet and Neuman problems for the $2^{nd}$-order Seiberg-Witten equation admit a regular solution once the $\mathcal{H}$-condition (described in the article) is satisfied. The approach consist in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation.


Introduction
Let X be a compact smooth 4-manifold with non-empty boundary. In our context, the Seiberg-Witten equations are the 2 nd -order Euler-Lagrange equation of the functional defined in 2.2.1. When the boundary is empty, their variational aspects were first studied in [9] and the topological ones in [2]. Thus, the main aim is to obtain the existence of a solution to the non-homogeneous equations whenever ∂X = ∅. The non-emptyness of the boundary inflicts boundary conditions on the problem. Classicaly, this sort of problem is classified according with its boundary conditions in Dirichlet Problem (D) or Neumann Problem (N ).
For each α ∈ Spin c (X), there is a representation ρ α : SO 4 → Cl 4 , induced by a Spin c representation, and a pair of vector bundles (S α + , L α ) over X (see [11]). Let P SO4 be the frame bundle of X, so • S α = P SO4 × ρα V = S α + ⊕ S α − . The bundle S α + is the positive complex spinors bundle (fibers are Spin c 4 − modules isomorphic to C 2 ) It is called the determinant line bundle associated to the Spin c -struture α.
From now on, we considered on X a Riemannian metric g and on S α an hermitian structure h.
Let P α be the U 1 -principal bundle over X obtained as the frame bundle of L α (c 1 (P α ) = α). Also, we consider the adjoint bundles Ad(U 1 ) = P U1 × Ad U 1 ad(u 1 ) = P U1 × ad u 1 , where Ad(U 1 ) is a fiber bundle with fiber U 1 , and ad(u 1 ) is a vector bundle with fiber isomorphic to the Lie Algebra u 1 .
1.2. The Main Theorem. Let A α be (formally) the space of connections (covariant derivative) on L α , Γ (S + α ) is the space of sections of S + α and G α = Γ(Ad(U 1 )) is the gauge group acting on A α × Γ (S + α ) as follows: A α is an afim space which vector space structure, after fixing an origin, is isomorphic to the space Although each boundary problem requires its own configuration space, the superscripts D and N will be used whenever the distintion is necessary, since most arguments works for both sort of problems. The Gauge Group G α action on each of the configuration space is given by 1.1.

Basic Set Up
2.1. Sobolev Spaces. As a vector bundle E over (X,g) is endowed with a metric and a covariant derivative ▽, we define the Sobolev norm of a section φ ∈ Ω 0 (E) as In this way, the L k,p -Sobolev Spaces of sections of E is defined as In our context, in which we fixed a connection ▽ 0 on L α , a metric g on X and an hermitian structure on S α , the Sobolev Spaces on which the basic setting is made are the following; (G α is an ∞-dimensional Lie Group which Lie algebra is g = L 1,2 (X, u 1 )).
The Sobolev spaces above induce a Sobolev structure on C D α and on C N α . From now on, the configuration spaces will be denoted by C α by ignoring the superscripts, unless if it needed be.
The most basic analytical results needed to achieve the main result is the Gauge Fixing Lemma ) and the estimate 2.1, both extended by Marini,A. [12] to manifolds with boundary; notation: * f is the Hodge operator in the flat metric and the index τ denotes tangencial components.

Variational Formulation.
A global formulation for problems D and N is made using the Seiberg-Witten functional; where k g = scalar curvature of (X,g).
The G α -action on C α has the following properties; The tangent bundle T C α decomposes as In this way, the 1-form By performing the computations, we get (1) for every Λ ∈ A α , Therefore, by taking supp(Λ) ⊂ int(X) and supp(V ) ⊂ int(X), we restrict to the interior of X, and so, the gradient of the SW α -functional at (A, φ) ∈ C α is It follows from the G α -action on T C α that .
An important analytical aspect of the SW α -functional is the Coercivity Lemma proved in [9]; Proof. lemma 2.3 in [9]. The gauge transform is the Coulomb one given in the Gauge Fixing Lemma 2.1.1.
Considering the gauge invariance of the SW α -theory, and the fact that the gauge group G α is a infinite dimensional Lie Group, we can't hope to handle the problem in the general. So forth, we need to restrict the problem to the space The superscript D and N are being ignored for simplicity, although each one should be taken in account according with the problem. These choice of spaces is a a property of the G α action on C α , it is suggested by the Gauge Fixing Lemma and the Coercivity Lemma; this sort of propertie is not shared by most actions.
(2) for every V ∈ Γ (S + α ), Analougously, it follows that (A, φ) is a weak solution of the equation In order to pursue the strong L 1,2 -convergence for the sequence {(A n , φ n )} n∈Z , next we obtain an upper bound for || φ || L ∞ , whenever (A, φ) is a weak solution . (1) If σ = 0, then there exists a constant k X,g , depending on the Riemannian metric on X, such that (2) If σ = 0, then there exist constantc c 1 = c 1 (X, g) and c 2 = c 2 (X, g) such that Proof. Fix r ∈ R and suppose that there is a ball B r −1 (x 0 ), around the point Since r <| φ |, Hence, by 3.8 and 3.9, η ∈ L 1,2 . The directional derivative of SW α at direction η is given by By 2.4), However, Hence, Since r <| φ(x) |, whenever x ∈ B r −1 (x 0 ), it follows that There are two cases to be analysed independently; (1) σ = 0. In this case, we get The scalar curvature plays a central role here: if k g ≥ 0 then φ = 0; otherwise, Since X is compact, we let k X,g = max x∈X {0, [−k g (x)] 1/2 ), and so, The inequality 3.10 implies that Consider the polynomial A estimate for | φ | is obtained by estimating the largest real number w satisfying Q σ(x) (w) < 0. Q σ(x) being monic implies that lim w→∞ Q σ(x) (w) = +∞. So, either Q σ(x) > 0, whenever w > 0, or there exist a root ρ ∈ (0, ∞).
The first case would imply that contradicting 3.10. By the same argument, there exists a root ρ ∈ (0, ∞) such that Q σ(x) (w) chances its sign in a neighboorhood of ρ. Let ρ be the largest root in (0, ∞) with this propertie. By the Corollary A.0.11, there exist constants c 1 = c 1 (X, g) and c 2 such that Consequently, where C 1 , C 2 are constants depending on vol(B r −1 (x 0 )). The inequality 3.13 can be extended over X by using a C ∞ partition of unity. Moreover, if σ ∈ L ∞ , then where C 1 , C 2 are constants depending on vol(X).
The term X | ▽ A φ | 2 is bounded by 4.0.6 and A ∈ C 0 by 4.0.9.
In each of the last two lines above, the first terms are bounded by || A n − A || L 4 , while the term {(i) − (ii)} can be written as
Since the Laplacian defined on u 1 -forms is a elliptic operator, the fundamental inequality for elliptic operators claims that there exists a constant C k such that Consequently, F A being harmonic implies, for all k ∈ N, that In this case, since Θ ∈ L 1,2 , φ ∈ L ∞ and it follows that F A ∈ L 2,2 . Therefore, by the Sobolev embedding theorem F A ∈ L q , for all q < ∞. Proof. In 4.4, we must take care of the last terms; (1) F (▽ A φ) ∈ L p , for all 1 < p < 2. By Young's inequality, (2) There is no contribution from the divergent terms, since In the same way, The estimates above applied to 4.4 implies that
(2) The Sobolev class of φ is obtained by the bootstrap argument.