A has the same property for all admissible $v$ with $\|v-\bar u\|_\infty<\rho$; its derivative need not be close. Since the supremum <norm> is bounded by the Sobolev <norm>, this neighborhood contains a sufficiently small weak neighborhood. Thus \b[every strong local minimizer is a <weak local minimizer of a variational integral>].

Solution

Past exam of the partial differential equations for pure and applied mathematicians and calculus of variations course of the University of Oxford / 2007 / B / 7 / a / ii / Solution

 Solution (source code)