HW 4 for Calculus of Variations 2012

Andrej Cherkaev

March 2, 2012

1. Compute convex envelopes of the functions

f_{1}(u) = min {(u − 1)^{2}, (u + 1)^{2}}

f_{2}(u) = −\frac{1}{1 + u^{2}}

f_{3}(u_{1}, u_{2}) = \{ \begin{matrix}0 & if u_{1}^{2} + u_{2}^{2} = 0 \\ 1 + u_{1}^{2} + u_{2}^{2} & otherwise \end{matrix}

2. Consider variational problems

inf ∫_{0}^{1}(u^{2} + f_{i}(u′))dx, u(0) = a, i = 1, 2

Find a relaxed formulation, and write the equations for the minimizers. 3. Solve the equation for eikonal

(∇S)^{2} = 1, S = S(x_{1}, x_{2}), |x_{1} + |x_{2}| > 1, S = 0 if |x_{1} + |x_{2}| = 1

(Guess the solution, assumig that |x_{1} + |x_{2}| = 1 is the radiating curve, than check that the solution is correct. )