There is a axial symetric region $z>0$, $z<a$, $r<R(z)$. There is a some system of differential equation (In my problem it is a Maxwell's equations) $L_i f=0$.
I need to find a function $R(z)$ (which define above) for such the solution of this system of differential equation $f(r,z,\phi)$ with this boundary condition
$$z>0,\quad z<a, \quad\text{and}\quad r<R(z)$$
maximaze the following integral $$ \int_0^a f^2 dz $$
It looks like the variation problem but with boundary condition variation. But I can not find nothing similiar in books. I gues this type problem a very common and should be solved if it posible. I will be heppy for to the useful idea or links.