Differentiation, implicit functions, and applications to generalized well-posedness

TL;DR

This paper extends an implicit function theorem to more general settings involving projective limits of Banach spaces, providing new tools for analyzing smooth dependence in nonlinear differential equations.

Contribution

It generalizes a previous implicit function theorem to maps between projective limits of Banach spaces, enabling broader applications in differential equations.

Findings

Proved existence and differentiability of solutions in generalized settings.

Applied theorems to demonstrate smooth dependence on initial/boundary data.

Provided examples in nonlinear PDEs illustrating the theory.

Abstract

This article is centered around generalizing a previous implicit function theorem of the author to be applicable for maps f:E sqcap F to F which can be lifted to Keller C^k_pi maps f_i:E sqcap F_i to F_i with F_i Banach and F=projlim F_i . We prove theorems about existence and differentiability of functions g satisfying f(x,g(x))=b constant. In addition to these abstract theorems, we give several examples of applications to proving smooth dependence of the solution on initial/boundary values and the nonlinearity in nonlinear (partial) differential equations.