Anti-self-Dual instantons with Lagrangian boundary conditions. I: Elliptic theory (Q2571127)

On an oriented 4-manifold, the anti-self-dual instantons, i.e., connections satisfying \(F_{A}+\ast F_{A}=0\), are special solutions of the Yang-Mills equation. The present paper, part of a program proposed by Salamon for the proof of the Atiyah-Floer conjecture for homology 3-spheres, is devoted to nonlocal Lagrangian boundary conditions for anti-self-dual instantons on 4-manifolds with a space-time splitting of the boundary. Assuming \(L^p\)-bounds on the curvature for \(p>2\) the basic regularity and compactness properties are established as well as the Fredholm theory in a compact model case.