Representability of Elliptic Moduli Problems in Derived $C^{\infty}$-Geometry

We study moduli spaces of solutions of nonlinear Partial Differential Equations on manifolds in the framework of derived $C^{\infty}$-geometry. For an arbitrary smooth stack $S$, we define $S$-families of nonlinear PDEs acting between $S$-families of submersions over an $S$-family of manifolds and show that in case the family of PDEs is elliptic and the base family of manifolds is proper over $S$, then the moduli stack of solutions is relatively representable by quasi-smooth derived $C^{\infty}$-schemes over $S$. Along the way, we develop tools to analyse the local structure of families of mapping stacks between manifolds and explain how to compare mapping stacks in smooth and in derived geometry. To access the notion of a family of PDEs over an arbitrary smooth base stack, we introduce a formalism of stacks of relative jets. Finally, we show how natural ideas from (higher) topos theory can be leveraged to facilitate the application of nonlinear Fredholm analysis to derived stacks of solutions of elliptic PDEs.