I am a PhD student in Applied maths and I am looking forward to solve the elliptic problem: find u(x,y) such that:

-grad(K(grad(u))=dirac(0,0) on Omega=[0,1]*[0,1]; du/dn=0 on Omega\(0,0) : Neumann Boundary Conditions; u(0,0)=1.0 : Dirichlet BCS;

K is a 2*2 tensor.

The question is : How to prove the existence and uniqueness of a solution (L2?)? and what is the analytical solution to the problem?

Usually when there are both neumann and dirichlet conditions together, numerical methods are used. I couldn t find an analytical framwork which deals with this kind of problems. Most of the literature assuming in most cases that the right side is at least continuous and the boundaries are assumed to have a non zero measure.

I would be grateful for your comments and for any indications.

Thank you. Sadek

Could you use a Green's function?