1.6.3.5 Solution ppe-e

Question:

Show that the Laplace equation

$$\nabla^2 u = 0 \qquad\mbox{inside}\qquad\Omega$$

with the Neumann boundary condition

$$\frac{\partial u}{\partial n} = 1 \qquad\mbox{on}\qquad\delta\Omega$$

has no solution. That makes it an improperly posed problem.

Explain the lack of solution in physical terms. To do so, consider this a steady heat conduction problem, with $u$ the temperature, and the gradient of $u$ the scaled heat flux.

Generalize the derivation to determine the requirement that

$$\nabla^2 u = f \qquad\mbox{inside}\qquad\Omega$$

with the Neumann boundary condition

$$\frac{\partial u}{\partial n} = g \qquad\mbox{on}\qquad\delta\Omega$$

has a solution.

Answer:

Integrate the partial differential equation over the domain and then use the divergence theorem.

In the explanation, consider the net heat entering or leaving the domain. Note that the Laplace equation describes steady heat conduction, in which the temperature does not vary with time.