Consider the following probleblem:

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


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

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

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


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

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?


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