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Consider $$-\triangle u = f \ \ \ \ \ \text{in} \ \ \ \Omega$$

.$$\frac{\partial u}{\partial n} = g \ \ \ \text{on} \ \ \ \ \partial \Omega $$

Where $\Omega \subset \mathbb R^n$ is a bounded domain with boundary $\partial \Omega$ , $\triangle$ is the laplace operator, $f$ and $g$ are givem smooth function 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 $k$ - dimmensional space $\mathbb R^k$ which states that

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, $\sum \subset D$ . For each point $x \in \sum$ , let $\frac{\partial u}{\partial n}$ be the unit normal on $\sum$ with respect to the region $R$. Then Then, with $dX ≡ dx_1dx_2\dots dx_k$ and with $d\sigma$ indicating integration with respect to surface area on $\sum$

$$ \int_R \nabla . F(X) dX = \int_{\sum} F(X) . n \ \ d\sigma $$

here $\sum = \partial \Omega$ and $R = \Omega$ and $\frac{\partial u}{\partial n} = \frac{\partial F}{\partial x_1}e_1 + \frac{\partial F}{\partial x_2}e_2 \dots + \frac{\partial F}{\partial x_n}e_n = F(X). n$ , we get

$$\int_{\Omega} \nabla . \frac{\partial u}{\partial n} dX = \int_{ \partial\Omega}\frac{\partial u}{\partial n} dS = \int_{\Omega} \nabla. \nabla u \ \ dX$$

iff

$$\int_{\Omega} \triangle u \ \ dX = \int_{\partial \Omega} \frac{\partial u}{\partial n} \ \ dS$$

Since $\nabla . \nabla u = \triangle $

$$\int_{\Omega} -f \ \ dX = \int_{\partial \Omega} g \ \ dS$$

Please check my Solution, if you feel any mistake, then correct me.

Thank you

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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.

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  • $\begingroup$ By the way: useful literature in this context are the books from Lawrence Evans and of course Folland's book... $\endgroup$
    – mkk030572
    Jan 1 '16 at 13:54
  • $\begingroup$ I want to find a condition which is necessary and sufficient for the exisitence of a solution for $f$ and $g$. $\endgroup$
    – Struggler
    Jan 1 '16 at 14:27
  • $\begingroup$ That is not that simple, but refer to the books I mentioned and you will find conditions which imply existence and uniqueness of $u$. $\endgroup$
    – mkk030572
    Jan 1 '16 at 14:37
  • $\begingroup$ ok, I will see it, $\endgroup$
    – Struggler
    Jan 1 '16 at 14:59
  • $\begingroup$ tell me about my solution , it is correct or not $\endgroup$
    – Struggler
    Jan 2 '16 at 14:45

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