Foundations of the calculus of variations in generalized function algebras
Functional Analysis
2008-09-11 v1 Optimization and Control
Abstract
We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we introduce the first and second variation of a variational problem. We then derive necessary (Euler-Lagrange equations) and sufficient conditions for extremals. The concept of association is used to obtain connections to a distributional description of singular variational problems. We study variational symmetries and derive an appropriate version of N\"other's theorem. Finally, a number of applications to geometry, mechanics, elastostatics and elastodynamics are presented.
Cite
@article{arxiv.0707.1842,
title = {Foundations of the calculus of variations in generalized function algebras},
author = {Sanja Konjik and Michael Kunzinger and Michael Oberguggenberger},
journal= {arXiv preprint arXiv:0707.1842},
year = {2008}
}