Function-theoretic properties of solutions of semi-linear \({\bar \partial}\)-equations -- similarity principle (Q1108465)

The present paper is concerned with the function-theoretic properties and the local existence of solutions for a system of semi-linear first order partial differential equations in a complex-valued unknown function defined on a domain G (connected open set) in \({\mathbb{C}}^ n \)(n\(\geq 2)\). The system of semilinear equations is as follows: \[ {\bar \partial}_ jw=f_ j(z,w),\quad j=1,2,...,n, \] where \({\bar \partial}_ j=(1/2)(\partial /\partial x_ j+i\partial /\partial y_ j)\), \(z_ j=x_ j+iy_ j\) and \(z=(z_ 1,z_ 2,...,z_ n)\in {\mathbb{C}}^ n.\) When we rewrite this system of equations by the real form, then we obtain \[ \partial x_ ju-\partial y_ jv=2p_ j,\quad \partial y_ ju+\partial x_ jv=2q_ j, \] where \(w=u+iv\) and \(f_ j=p_ j+iq_ j\).