10 03/26 F

1. See whether any terms must be changed in expression (5) in the notes on elliptic equations in the three dimensional case. Then determine how (6) differs from the two-dimensional case.

2. 3.30. In (a), use the property mentioned in class that the minimum of a harmonic function must occur on the boundary. In (b), try $1-y$ in the domain $\Omega$ given by $y\ge0$. In (c), consider the functions $s=v-u$ and $t=w-v$.

3. 4.19. Plane wave solutions are solutions that take the form (2) in solved problem 4.12, with $\vec\alpha$ a constant vector and $\mu$ a constant. This sort of solutions are a multi-dimensional generalization of the $f(x-at)$ moving “wave” solution of the one-dimensional wave equation. In fact, if you take $\vec\alpha$ to be a unit vector, it gives the oblique direction of propagation of the wave and $\mu$ gives the wave propagation speed. However, in this case you will see that the function $F$ cannot be an arbitrary function unless $b=0$. You may want to do the case $b=0$ separately. And also split up the cases for $\mu$.