I have a 2D axisymmetric mesh on which I am solving the optimality system. Now to update the control variable on a boundary I need to solve a PDE with some conditions. I dont know how to do that while I am on 2D mesh. on the weak form formula below I know S but I want g.

g, S, and nu are vectors and the gradient is in the direction of S?
I guess personally I would just do it all in cartesian coords like
any other problem. You have just two unknowns g.x and g.y right?
Probably just search the documentation for the â€śvarfâ€ť examples
although â€śsolveâ€ť and â€śproblemâ€ť examples may be easier the varf may be more useful for now.
You can make a varf with the 2 components of g and nu as parameters and
it just uses S in the varf body. To get the value zero on boundary something like
â€śonâ€ť which should be in most of the varf examples. Using varf you can
extract the matrix and rhs and then dump them to see if they make sense.

Thanks for your answer. S and g are vectors but nu and alpha and beta are constants.
I know I should solve it but I have to solve it on line not 2D mesh. I dont know how to do that. I have my system of equation on 2d surface below but this integral is on the inlet boundary.