Elliptic bootstrapping and the nonlinear Cauchy–Riemann equation

Abstract

The goal of this paper is to deduce a nonlinear elliptic regularity result from a linear one. In particular, elliptic bootstrapping is a powerful method to determine the regularity of a solution to a partial differential equation. We apply elliptic bootstrapping and linear elliptic regularity to the nonlinear Cauchy–Riemann equation. In doing so, we generalize the fundamental analytic result that holomorphic functions are automatically smooth. In particular, we show that, under certain conditions, the same is true for so-called J-holomorphic functions. We conclude by discussing how this nonlinear regularity result relates to ideas in symplectic geometry.