HW 3

1. At a gas planet with variable viscosity, the travel speed is proportional to the radius r from the center, v(r) = α r . What is the fastest path from a point A = r_{1}, θ_{1}) to a point B = (r_{2}, θ_{2}), where θ is a polar angle in the plane that passes through the center and points A and B.

2. Show that the problem

min_{u(x)} ∫_{−π}^{π} [(u′)^{2}(1 − cos x)] dx, u(−π) = −1, u(π) = 1

does not have a regular (continuous) solution. Regularize the problem, find the solution, and plot the graphs of the extremals with several values of the regularization parameter.