Absolutely continuous solutions

Abstract

The theory of the calculus of variations at the turn of the twentieth century lacked a critical component: it had no existence theorems. These constitute an essential ingredient of the deductive method, the approach whereby one combines existence, rigorous necessary conditions, and examination of candidates to arrive at a solution. When it applies, it often leads to the conclusion that a global minimum exists, whereas the classical methods assert only the existence of a local minimum. In mechanics, a local minimum is a meaningful goal, since it generally corresponds to a stable configuration of the system. In many modern applications however (such as in engineering or economics), only global minima are of real interest. Along with the quest for the multiplier rule (which we discuss in the next chapter), it was the longstanding question of existence that dominated the scene in the calculus of variations in the first half of the twentieth century. The key step in developing existence theory is to extend the context of the basic problem to admit absolutely continuous functions.