Partial integrability of almost complex structures and the existence of solutions for quasilinear Cauchy–Riemann equations

Chong-Kyu Han and Jong-Do Park

Vol. 265 (2013), No. 1, 59–84

DOI: 10.2140/pjm.2013.265.59

Abstract

We study the local solvability of the system of quasilinear Cauchy–Riemann equations for d unknown functions in n complex variables, which is a system of elliptic type and overdetermined if n ≥ 2. We consider an associated almost complex structure on ℂ^n+d and its partial integrability and prove by using the Newlander–Nirenberg theorem and its algebraic generalizations that the existence of a pseudoholomorphic function on the zero set is equivalent to the local solvability of the original quasilinear system. We discuss an algorithm for finding pseudoholomorphic functions on the zero set and then present examples.

Keywords

overdetermined system, elliptic PDE system, almost complex structure, nonlinear Cauchy–Riemann equations

Mathematical Subject Classification 2010

Primary: 32W05, 35N10

Secondary: 32Q60, 35J60

Milestones

Received: 28 June 2012

Revised: 26 September 2012

Accepted: 3 October 2012

Published: 28 August 2013