15. Nonsmooth extremals

Is it always satisfactory to consider only smooth solutions of the basic problem, as we have done in the previous chapter? By the middle of the 19th century, the need to go beyond continuously differentiable functions was becoming increasingly apparent. In fact, the calculus of variations may have been the first subject to acknowledge a need to consider nonsmooth functions. We begin by exhibiting a simple problem that has a natural solution in the class of piecewise-smooth functions, but has no solution in that of smooth functions. This shows that the very existence of solutions is an impetus for admitting nonsmooth arcs. The need also became apparent in physical applications (soap bubbles, for example, generally have corners and creases). Spurred by these considerations, the theory of the basic problem was extended to the context of piecewise-smooth functions. In this chapter, we develop this theory, but within the more general class of Lipschitz functions x. We extend the setting of the basic problem in one more way, by allowing x to be vector-valued.