Elliptic bootstrapping and the nonlinear Cauchy–Riemann equation
PDF

How to Cite

Zhang, J. (2024). Elliptic bootstrapping and the nonlinear Cauchy–Riemann equation. Columbia Journal of Undergraduate Mathematics, 1(1). https://doi.org/10.52214/cjum.v1i1.12911

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.

https://doi.org/10.52214/cjum.v1i1.12911
PDF
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Copyright (c) 2024 Jessica J. Zhang