p.897

Cake Eating, Chattering, and Jumps: Existence Results for Variational Problems

Paul Romer

Abstract

This paper establishes a simple existence result for solutions to variational problems of the form @?^@?"0 G(x, x, t) dt or @?^@?"0 G (x, x, x, t)dt. The key assumptions are that G have an integrable upper bound, that it satisfy a growth condition, and that it be concave as a function of the highest order derivative in the problem, other arguments held constant. The discussion illustrates why three well known types of problems fail to have solutions. For two of these--chattering and cake eating--extended solution concepts are contrasted with simple modifications that restore the existence of a conventional solution. In a third case--state variables with jumps--the source of the difficulty is fundamental. For these problems a natural extended solution, analogous to the extension from probability density functions to general distribution functions, is suggested.