8 03/07 F

1. Check whether the derivation for the expression (5) in the notes on elliptic equations also applies in the three dimensional case. If it does, determine how (6) differs from the two-dimensional case.

2. 3.29a,b. Use the expressions for the Laplacian found in table books and the chain rule of differentiation. You may assume that $a=1$ if you want.

3. Derive the Poisson integral formula for the Dirichlet problem in a sphere, as listed in the notes on elliptic equations Remember that you want to get rid of the unknown derivative $u_r$ on the circle and note that the source distribution does not disappear in this case.

4. 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 $t=w-v$.

5. 3.38. This does not require solution of the problem using the Poisson integral formula. You can just examine what symmetry properties the solution $u(x,y)$ should have to figure out the value at the origin.

6. 3.39. Again this does not require solution of the problem using the Poisson integral formula. You should be able to find the complete solution $u(x,y)$ by mere inspection.

7. 3.40. This is about the solution to the Dirichlet problem in a circle, which, as derived in class, is given by the Poisson integral formula (also listed in 3.37.) The question is really, suppose that the temperature $f$ on the boundary is a narrow spike centered at a position $\frac12(\phi_1+\phi_2)$, will it cause a nonzero temperature at every point in the inside, or just in a limited range? (Note that the Poisson integral uses $\phi$ as the angular coordinate on the boundary and $\theta$ for the angle at which $u$ is evaluated.)

8. 3.42. In both cases (a) and (b), you can assume that the solution $u$ is given by the stated sum. You may also assume that, (using standard Fourier series results from a table book,) in both cases,
   
   $$f=\sum_{m,n=1}^\infty b_{mn}\sin(mx)\sin(ny)$$
   
   
   but in case (a), in which $f=1$,
   
   $$b_{mn} = \left\{ \begin{array}{cl} \displaystyle\frac{16... ...are both odd} \\ 0 & \mbox{ otherwise} \end{array} \right.$$
   
   
   while in case (b), where $f=\cos(x)\cos(y)$,
   
   $$b_{mn} = \left\{ \begin{array}{cl} \displaystyle\frac{16... ...re both even} \\ 0 & \mbox{ otherwise} \end{array} \right.$$
   
   
   Plugging the sums into the PDE, in (a) there is a problem with equating left and right hand sides for some particular values of $m$ and $n$, while there is no problem in (b); in (b) suitable $a_{mn}$ values can be found, in (a) they cannot.