Answer Happy

(a)/Let a be a given positive real number and let h: [0, π/2] → R be a given function. Suppose that u(r, 0) solves Laplace's equation, Au(r, 0) = 0 in the two-dimensional open wedge W, which is defined in polar coordinates (r, 0) as follows: W := {(r,0) 10<0<T/2, 0≤r <a}. Suppose also that u(r, 0) satisfies the following three boundary conditions: u(r, 0) = 0, 0≤r≤a, u(r, π/2) = 0, 0≤r≤a, u(a,0)=h(0), 0≤0 ≤ π/2. Use the method of separation of variables to derive a formula for the function u(r, 0). (b) State the maximum / minimum principle for Laplace's equation. (c) Show how the maximum / minimum principle can be used to prove that there cannot be more than one solution to part (a) above.