Let $U\subset R^n$ be a bounded domain with smooth boundary. Prove that there does not exist any solution to the boundary-value problem
$$-\Delta u=0 \text{ in }U,\frac{\partial u}{\partial \nu}=1\text{ on }\partial U $$
My attempt:
I'm trying to get a contradiction by supposing there is such a $u$.
So,
$-u\Delta u=0$
$\int\limits_{U}-u\Delta u=0$
Then by applying Green's identity
$\int\limits_{U}|Du|^2dx-\int\limits_{\partial U}u\frac{\partial u}{\partial\nu}ds=0$
$\int\limits_{U}|Du|^2dx-\int\limits_{\partial U}uds=0$
But after this I cannot see how to proceed.
Appreciate your help
Best Answer
Intuition:
Recall that the Laplace equation is the stationary heat equation. So now think about the heat equation. If $\frac{\partial u}{\partial \nu}$ is positive all over in the heat equation, then heat must be coming into the domain in net. So there physically can't be a stationary solution to the heat equation with a Neumann boundary condition with a nonzero integral, because for there to be a stationary solution, it would need to maintain the same total heat all the time.
So what's the rate of change of the total heat in the domain in the heat equation? It is
$$\frac{d}{dt} \int_U u dx=\int_U \frac{\partial u}{\partial t} dx = \int_U \Delta u dx$$
which for a stationary solution is zero. So we should show that if $\frac{\partial u}{\partial \nu}=1$ everywhere, then $\int_U \Delta u dx \neq 0$.
Math:
If we had a solution to the Laplace equation with this boundary condition, then using the divergence theorem (not the Green's identity) we would have:
$$0=\int_U \Delta u dx = \int_{\partial U} \frac{\partial u}{\partial \nu} dS>0$$
which is a contradiction, so there is no such solution.