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

Author:
Robert M. Goor

Journal:
Trans. Amer. Math. Soc. **223** (1976), 347-365

MSC:
Primary 49A50

DOI:
https://doi.org/10.1090/S0002-9947-1976-0425716-3

MathSciNet review:
0425716

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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0425716-3

Keywords:
Parametric problem,
calculus of variations,
orientor field,
growth condition,
Fréchet curve,
parametric problems of optimal control

Article copyright:
© Copyright 1976
American Mathematical Society