On an Iterative Scheme for the Spinorial Yamabe Equation on Manifolds with Boundary

We study the existence of solutions to the spinorial Yamabe equation -- that is, the Euler--Lagrange equation associated with the conformal invariant introduced by S. Raulot -- for compact manifolds with boundary. For the inhomogeneous equation, we employ an iterative scheme to establish existence under smallness assumptions on the relevant parameters. Using bootstrapping methods, we extend the regularity of the solution to $C^\infty$ away from its zero set in the interior, and up to the boundary in the case of Shapiro--Lopatinski boundary conditions.