About properly posed problems.

  Another big difference between the Laplace equations are the boundary conditions that give rise to properly posed problems. For the wave equation we had two initial conditions both specified at the initial time. Instead the Laplace equation gave rise to a boundary value in which the temperature was specified at both the smallest and the largest value of y.

Of course, a boundary value problem could also be formulated for the wave equation: it would mean specifying the position of the string at a final time in addition to at the starting time. In that case the initial string velocity would not be given. However, there is no unique or physically relevant solution to such a problem. For example, for c=1 and $\ell=\pi$, if we specify that the string has no deflection at times t=0 and $t=\pi$, the obvious solution is that the string does not move and u is always zero. But u could also be any multiple of $\sin t \sin x$ or any higher harmonic of that solution. In other words, the solution is not unique; the boundary value problem for the wave equation is ill-posed.

Conversely, for the Laplace equation, we could not specify both the temperature T and vertical heat flux $\partial T/\partial y$at the bottom boundary, and nothing at the top boundary. To see that, assume that we take the temperature on the bottom boundary to be zero, and the temperature gradient to have a jump singularity. This is impossible, since no temperature distribution on the upper boundary can give rise to such conditions at the lower boundary. After all, even if the temperature on the top boundary would be singular, the singularity would immediately smooth out, so that there is no way to create a singular heat flux at the bottom boundary.