the work on my paper would follow the next steps so everything will be clear for you.
first of all, the paper that I want to write is an extension from the linear case to the non-linear case as we discussed earlier.
i- you already have done the circular nonlinear case from equation (1) until equation (5) in p 1 file as in the attachment following the same approach in file p 0. The whole paper is based on the analytic solution which we should obtain too. Then, we must have the exact solution and stability.
ii- we should extend the Riemann mapping over close the unit circle for the whole space to the nonlinear case.
iii- we should prove the existence and we can do that in several ways:
a- following the same work of P 0 file we must convert the nonlinear Riemann-Hilbert problem into the nonlinear Robin's problem to change the boundary of the unit disk to nonlinear one. you can find that in lecture note three file under the name Remark 3.3, handling that by trying the fix point theory or argument ( we could use the theory of Banach space and if we use that then the nonlinear part has to be very small/ or we could use the Schouder's fix point theorem).
Notation: The elliptic equation is more relevant to the approach that we need to find in the lecture note three file.
b- In the linear case the paper P 0 solved the differential equation to find the existence for the unit disk, but in the nonlinear case we should use the Schwartz's function to do the extension .
The key of completing the work is to solve the Riemann-Hilbert problem by extending the solution and going over the unit disk for the nonlinear case.
I have uploaded some files where they can help to understand more about the work.
- Feel free to ask anything or more details.