# Weak formulation of non-local Neumann problem

Consider the following probleblem:

$$-\Delta u +a(x)\int_{\Omega}b(z)u(z)dz = f \qquad \text{in \Omega}$$ $$\partial_{\nu}u=0 \qquad \text{in \partial\Omega }$$

where

$$\Omega\quad \text{is a bounded and Lipschitz domain,} \subset R^n \quad a,b\in L^\infty(\Omega) \quad f \in L^2(\Omega)$$

So I've reduced myself to the A.V.P.

$$a(u,v)=(f,v)_{L^2} \quad \forall v \in H^1(\Omega)$$

Where

$$a(u,v) = \int_{\Omega}\nabla u \nabla v dx \quad + \int_{\Omega}a(x)\left( \int_{\Omega}b(z)u(z)dz \right) v(x)dx$$

Proving that $a$ is continuos is trivial, and also the weak $H^1-L^2$ coercivity

so that, in order to have a solvability condition, I need to examine $Ker(a^*)$, that wuold be:

Find $u\in H^1(\Omega):$ $$a^*(u,v)= \int_{\Omega}\nabla u \nabla v dx \quad + \int_{\Omega}a(x)\left( \int_{\Omega}b(z)v(z)dz \right) u(x)dx=0 \quad \forall v \in H^1$$

Any idea on how to do this?