February 8/8: I proved existence ¡­

February 8/8: I proved existence and uniqueness of minimizers. The notion of a weak solution of the Euler-Lagrange equation was introduced. The result is that a minimizer of the integral function is such a weak solution, thereby revealing the (rigorous) connection between the minimization problem and the Euler-Lagrange PDE. The theory developed this week can be extended to convex systems of equations; I mentioned this and it is expected that you carry out the details yourself (which means essentially "chasing notations"). Next week's topics are polyconvex systems and minimization problems with constraints.