In this paper, I study the Seiberg-Witten equations on the product $\R \times Y$, where $Y$ is a compact 3-manifold with boundary.
Since I describe the geometric setup and motivation for these equations in the introduction of the paper for quite some length, I will focus mainly on the analytic aspects of these equations here. In fact, it is the analysis that is the heart of the issue in this paper, and it turns out to be excruciatingly technical because we have a boundary.
Let me recall what it is that we want for our equations as a PDE. First, we want the equations to be elliptic modulo guage, so that a solution to the Seiberg-Witten equations that a priori only belongs to some Sobolev space is smooth modulo gauge. Second, we want to the moduli space of solutions to be compact. Third, we want the linearized operator associated to the equations modulo gauge to be Fredholm. When $Y$ is closed, these conditions ensure that the Seiberg-Witten equations yield a well-defined monopole Floer theory for $Y$. (Indeed, for there to be a Floer theory, we need a well-defined differential that "counts" the number of solutions connecting two monopoles on $Y$. This in turn requires suitable compactness properties of the moduli space of Seiberg-Witten equations on $\R \times Y$.) We thus wish the same properties to hold when $Y$ has a boundary.
In this case, however, we must impose boundary conditions for the equations. The natural boundary conditions to impose are Lagrangian boundary conditions, as explained in the paper. Here, the Lagrangians considered are those arising from Part I. Abstracting the situation, our equations have the following shape. We have a path $u: \R \to \mathcal{X}_Y$ into a Banach space of configurations on $Y$ (namely connections and spinors on $Y$ in some function space). If $\mathcal{X}_\Sigma$ is the space of boundary values of $\mathcal{X}_Y$, we specify a Lagrangian submanifold $\frak{L} \subset \mathcal{X}_\Sigma$ and ask that $u(t)|_\Sigma$ lie inside $\frak{L}$ for all $t \in \R$. This, together with the Seiberg-Witten equations in a suitable gauge, yields the following system of equations:
\begin{align}
\frac{d}{dt}u &= -Du + N(u) \tag{1}\\
u(t)|_{\Sigma} & \in \frak{L}, \qquad \forall t \in \R. \tag{2}
\end{align}
Here $D$ is a Dirac operator on $Y$ and $N(u)$ is a quadratic nonlinearity.
The resulting nonlinear boundary value problem is highly nontrivial, and the tools I use to establish the aforementioned desirable analytic properties of this PDE include pseudodifferential operators and nonlinear functional analysis. To give a hint as to why the analysis is so involved (as compared to the other gauge-theoretic PDE in the literature), if we formally think of the Seiberg-Witten equations as an flow (with the time variable being the $\R$ factor of $\R \times Y$) then we are flowing a connection and spinor in a nonlinear space, the space of configurations on $Y$ subject to a nonlinear boundary condition. This is where all the difficulty lies; the quadratic nonlinearity $N(u)$ is completely tame.
As a result, let me list some of the following issues I had to consider:
(1) First, the Lagrangian submanifold $\frak{L}$ is an infinite dimensional Banach manifold (even modulo gauge). Thus, it is difficult to understand this space explicitly, and hence, the nature of the nonlinear boundary condition. This is in contrast, say, the flows that occur in Riemannian geometry, which happen on a finite dimensional Riemannian manifold. Thus, the typical nonlinear PDEs one encounters are such that if the domain is nonlinear, it is finite dimensional (it is a Riemannian manifold), and if the domain is infinite dimensional, it is linear (it is a Banach space). However, in the our situation, the domain is an infinite dimensional Banach manifold.
(2) The tangent space to the Lagrangian $\frak{L}$ is (approximately) pseudodifferential, i.e., it is the range of a pseudodifferential projection on the Banach space $\mathcal{X}_\Sigma$. But because of the ``slicewise" nature of (2), the linearization of the boundary condition (2) is only ``slicewise" pseudodifferential, and hence is
not pseudodifferential as an operator on $\R \times \Sigma$. Thus, the (linearized) boundary condition is neither a local or pseudodifferential boundary condition, and hence is nonstandard from this point of view.
In some sense, one can think of (2), for each $t \in \R$, as a ``nonlinear APS" boundary condition, since a tangent space to $\frak{L}$ is an Atiyah-Patodi-Singer (APS) like boundary condition. (That is, the tangent space is closely related to certain spectral subspaces of $\mathcal{X}_\Sigma$).
(3) It turns out that less standard function spaces (those which are not Sobolev spaces) arise in studying our PDE, and I was forced to do nonlinear analysis on Besov spaces and other types of function spaces, including Banach space valued function spaces (indeed, one can already see that $u: \R \to \mathcal{X}_Y$ lives in a function space on $\R$ with values in the Banach space $\mathcal{X}_Y$). Working with these latter spaces requires tools from nonlinear interpolation theory and estimates that work in the Banach space valued function space setting (including e.g. the Gagliardo-Nirenberg inequality). As for Besov spaces, these spaces naturally arise because they are boundary values of Sobolev spaces, and thus they inevitably occur for our boundary value problem.
(Note: Other nonstandard function spaces I had to consider were anisotropic Besov spaces. This led to the following
paper which studies these spaces in a more abstract setting.)
It was only by handling these issues and others that I was able to prove the basic theorems concerning the PDE (1)-(2).