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The Von Neumann Problem is as such:

$\Delta u = f(x,y,z)$ in $\ D$

$\frac {\partial u} {\partial n} = 0$ on bdy $\ D$.

Using this you can prove that for there to be a solution to this Von Neumann problem, $\int\int\int f(x,y,z) dx dy dz=0$ .

So I found a partial answer to this question. Help with the Neumann Problem

However, I am having trouble applying a physical interpretation to a situation where an object is being heated by a heat source $f(x,y,z)$.

On other sources on the web I found that heat is not lost, and so the triple integral of $f$ is $0$. But the heat flux is $0$ on bdy $D$, so how can there be no heat flux through the boundary yet the triple integral of $f$ be equal to $0$?

I would imagine that the triple integral of the heat source function be not zero on the domain if it is constantly dumping heat into $D$.

Nirbhav Chopra
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1 Answers1

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From my understanding, the f(x,y,z) term is designated the "source term," and represents the rate at which heat is emitted or absorbed. Because the Neumann condition dictates that the total heat flux is zero, as you mentioned, and the source term is zero as well, this means that the heat is being absorbed at the same rate it is being generated, so overall the volume integral balances the two.

Reference question 10 from this site:

https://www.math.upenn.edu/~deturck/m260/hw11sols.pdf

Nicholas Stern
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