On the Poisson Equation on a Surface with a boundary condition in co-normal direction

This paper considers the existence of weak and strong solutions to the Poisson equation on a surface with a boundary condition in co-normal direction. We apply the Lax-Milgram theorem and some properties of $H^1$-functions to show the existence of a unique weak solution to the surface Poisson equation when the exterior force belongs to $L_0^p$-space, where $H^1$- and $L_0^p$- functions are the ones whose value of the integral over the surface equal to zero. Moreover, we prove that the weak solution is a strong $L^p$-solution to the system. As an application, we study the solvability of ${\rm{div}_\Gamma } V = F$. The key idea of constructing a strong $L^p$-solution to the surface Poisson equation with a boundary condition in co-normal direction is to make use of solutions to the surface Poisson equation with a Dirichlet boundary condition.