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.