8 03/06 F

1. 3.44. This is mostly the uniqueness proof given in class, which can also be found in solved problems 3.14-3.16. However, here you will want to write out the two parts of the surface integral separately since the boundary conditions are a mixture of the two cases 3.14 and 3.15 (with $c=0$).

2. Show that the following Laplace equation problem has a unique solution, $u=0$:
   
   $$\mbox{PDE: } \nabla^2 u = 0\qquad \mbox{BC: } u(0,y)=u_y(x,0)=u_y(x,1)=u(1,y)+u_x(1,y)=0$$
   
   
   This is essentially the uniqueness proof given in class, which can also be found in solved problems 3.14-3.16. However, you will want to write the four parts of the surface integral out separately since the boundary conditions are a mixture of the three cases 3.14-3.16.

3. Show that the following Laplace equation problem has infinitely many solutions beyond $u=0$:
   
   $$\mbox{PDE: } \nabla^2 u = 0\qquad \mbox{BC: } u(0,y)=u_y(x,0)=u_y(x,1)=u(1,y)-u_x(1,y)=0$$
   
   
   Hint: Guess a very simple nonzero solution and check that it satisfies all boundary conditions and that its second order derivatives are zero. Since the equations are linear, any arbitrary multiple of this solution is also a solution. Verify whether or not the uniqueness proof of the previous section conflicts with the nonunique solution of this problem. Why would a slight difference in one boundary condition make a difference?

4. Find the Green’s function in three-dimensional unbounded space $R^3$. Use either the method of section 2.1 or 2.2 of the web page example as you prefer.