Smooth counterexamples to strong unique continuation for a Beltrami system in C2\mathbb{C}^2

We construct an example of a smooth map $\mathbb{C}\to\mathbb{C}^2$ which vanishes to infinite order at the origin, and such that the ratio of the norm of the $\bar z$ derivative to the norm of the $z$ derivative also vanishes to infinite order. This gives a counterexample to strong unique continuation for a vector valued analogue of the Beltrami equation.Comment: 21 page

On the existence of solutions to nonlinear systems of higher order Poisson type

In this paper, we study the existence of higher order Poisson type systems. In detail, we prove a Residue type phenomenon for the fundamental solution of Laplacian in \RR^n, n\ge 3. This is analogous to the Residue theorem for the Cauchy kernel in \CC. With the aid of the Residue type formula for the fundamental solution, we derive the higher order derivative formula for the Newtonian potential and obtain its appropriate \s C^{k, \alpha} estimates. The existence of solutions to higher order Poisson type nonlinear systems is concluded as an application of the fixed point theorem.Comment: 33 page