APM462: Homework 5 sol Let f : Rn R be a C1 function and consider the functional: F [u()] = f Z b a 1(x)u(x)dx, . . . , Z b a n(x)u(x)dx ! where for each i = 1, ..., n, i(x) is 1 for x [a (i1) ba n , a i ba n ] and 0 elsewhere. As usual the space A = {u : [a, b] R | u C1, u(a) = A, u(b) = B}. Find the first order condition for a minimizer u() of F in A. Hint: start by computing the "directional derivative" 0 = d ds |s=0 F [u() sv()] = , where v() is a test function