How to calculate the potential and electric field everywhere if we know the electrostatic potential on the surface of a cylindrical? [closed]

I know that electric field $$E=-∇V.$$ But since the only given parameter is the potential on the surface. Is it possible to calculate the potential everywhere (positions besides the surface)?

• You should put the shape in the question. Don't ask the full question in the title. Is the cylindrical shape finite or infinitely long? The title mentions the electric field, but the question text asks about the potential. Nov 4 '18 at 13:42
• I think more information about the problem needs to be given. Where are you wanting to determine the field? Is this a physical cylinder with certain properties? Nov 4 '18 at 16:07

There is not a unique solution without more given information. To make a simpler contrived analogy, it is like asking to find the function $$f(x)$$ and its derivative $$f'(x)$$ such that $$f(3)=5$$. Multiple functions have this property, so there is not a unique solution.
For example, if you are given the potential on the cylinder and are only interested in the field outside of the cylinder where the charge density $$\rho=0$$, then you are essentially solving the Laplace equation $$\nabla^2V=0$$, and that solution would be unique.