Consider $$ \begin{cases} -\Delta u &= f\quad \text{in} \ \ \Omega\\ \frac{\partial u}{\partial n} &= g \ \ \ \text{on} \ \ \ \ \partial \Omega, \end{cases} $$ where $\Omega \subset \mathbb R^n$ is a bounded domain with boundary $\partial \Omega$ , $\Delta$ is the laplace operator, $f$ and $g$ are given smooth functions and $\frac{\partial u}{\partial n}$ denoes the outer normal derivative of $u$. What is the necessary and sufficient condition for the following problem to admit a solution?
I am using Gauss divergence theorem in $k$-dimensional space $\mathbb R^k$ which states the following:
Let $F(X)$ be a continuously differentiable vector field in a domain $D \subset \mathbb R^k$. Let $R \subset D $ be a closed, bounded region whose boundary is a smooth surface, $\Sigma \subset D$ . For each point $x \in \Sigma$ , let $\frac{\partial u}{\partial n}$ be the unit normal vector on $\Sigma$ with respect to the region $R$. Then, with $dX ≡ dx_1dx_2\dots dx_k$ and with $d\sigma$ indicating integration with respect to surface area on $\Sigma$ $$\int_R \nabla \cdot F(X)dX=\int_{\sum}F(X)\cdot n\, d\sigma.$$
Here, $\Sigma = \partial \Omega$ and $R = \Omega$ and $$\frac{\partial u}{\partial n} = \frac{\partial F}{\partial x_1}e_1 + \frac{\partial F}{\partial x_2}e_2 \dots + \frac{\partial F}{\partial x_n}e_n = F(X)\cdot n.$$ We get $$\int_{\Omega} \nabla \cdot \frac{\partial u}{\partial n} dX = \int_{ \partial\Omega}\frac{\partial u}{\partial n} dS = \int_{\Omega} \nabla\cdot \nabla u \ \ dX$$ iff $$\int_{\Omega} \Delta u \ \ dX = \int_{\partial \Omega} \frac{\partial u}{\partial n} \ \ dS$$ Since $\nabla \cdot\nabla u = \Delta$, $$\int_{\Omega} -f \ \ dX = \int_{\partial \Omega} g \ \ dS.$$
Please check my Solution, if you find any mistake, then correct me.
Thank you.
What do you want to prove here? In this context you look for a solution $u$ for the boundary value problem $- \Delta u = f$ in $\Omega$ and $\frac{\partial u}{\partial n} = g$ on $\partial \Omega$ i.e. you consider the Poisson equation with von Neumann boundary conditions. The existence and uniqueness of a solution can be proved under certain assumptions concerning the regularity of the data $f$ and $g$ and the domain $\Omega$ using Gauß or Green's formulas.