Piecewise C¹ Extremal Functions

Abstract

In many problems examined thus far we have required continuous differentiability of the function y (or Y) defining the classes for optimization. Already with the example of the minimal surface of revolution from §1.4(a) we have argued that for some configurations, the minimizing curve (if it exists) should exhibit“ corners,” and it is natural to wonder whether cornered curves ŷ and Ŷ such as those shown in Figure 7.1 might not give improved results for other problems. Such curves are represented readily by functions which are piecewise continuously differentiable, or piecewise C¹.