18.3.3.5 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. To focus your thoughts, you can take an example domain $\Omega$ to be the inside of a sphere, and $\delta\Omega$ as its surface.

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:

Write the integral of the partial differential equation over the domain $\Omega$ (like the volume of the sphere). Then use the divergence theorem to convert it to an integral over $\delta\Omega$ (like the surface of the sphere). Identify the integrand in each integral, and hence the integrals themselves.

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.