1. Let u(x, y) and v(x, y) be a solutions of the Laplace equation △u = △v = 0 in a bounded region Ω in the plane. If u > v on the boundary of Ω, what, if anything, can you conclude about the relationship between u and v inside Ω? Justify your assertion.

2. Let u(x, y) satify △u = u in a bounded region Ω ⊂ R2  with u(x, y) = 0 on the boundary of Ω .  Use Green’s identity to show that u(x, y) = 0 throughtout Ω .

3. If u(x, y) is a solution of the Laplace equation in the unit disk x2 + y2  < 1 with boundary conditions

u(x, y) = ,0(1)

Compute u(0, 0).

for x2 + y2  = 1,y > 0

for x2 + y2  = 1,y ≤ 0

4. Solve u北北 + uyy  = 1 in the annulus a < T < b with u(x, y) vanishing on both parts of the boundary T = a and T = b.

5. Prove the uniqueness of the Dirichlet problem △u = f in D the unit disk, u = g on ?D by the energy method. i.e., with w = u − v, multiply the Laplace equation for w by w itself and use the divergence theorem.

6. In a bounded region D in the plane, say △u = △v = 0 with u(x, y) = f (x, y) and v(x, y) = g(x, y) at all points on the boundary ?D . What information does the maximum principle give?

7. A function u is subharmonic in a bounded region D if △u ≥ 0 in D . Prove that its maximum value is attained on the boundary of D . Note that this is not true for the minimum value.

8. Solve △u = 0 in the disk {r < a} with the boundary condition u = 1 + 3 sin θ on r = a.

9.  Solve △u = 0 in the exterior {r > a} with the boundary condition u = 1 + 3 sin θ on r = a and the condition at infinity that u be bounded as r → ∞ .

10. Solve △u = 0 in the annulus 1 ≤ x2 + y2  ≤ 2 with u(x, y) = 1 on the circle x2 + y2  = 1 and u(x, y) = 7 on x2 + y2  = 2.