Existence theorems for parametric problems in the calculus of variations and approximation

Abstract

In this paper, we investigate the parametric growth condition which arises in connection with existence theorems for parametric problems of the calculus of variations. In particular, we study conditions under which the length of a curve is dominated in a suitable sense by its “cost". We show that we may restrict our attention to local growth conditions on a particular set. Then we link the growth conditions to a certain approximation problem on this set. Finally, we prove that under suitable topological restrictions related to dimension theory, the local and global problems can be solved.