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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 65, Issue 6

Issues

Natural Boundary Conditions in Geometric Calculus of Variations

Giovanni Moreno / Monika Ewa Stypa
Published Online: 2016-02-09 | DOI: https://doi.org/10.1515/ms-2015-0105

Abstract

In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of Y - the manifold of dependent and independent variables underlying a given problem - as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over Y . Explicit examples of natural boundary conditions are obtained when Y is an (n + 1)-dimensional domain in ℝn+1, and the Lagrangian is first-order (in particular, the hypersurface area).

Keywords: global analysis; calculus of variations; free boundary problems; jet spaces; flags

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About the article

Received: 2012-10-12

Accepted: 2013-02-18

Published Online: 2016-02-09

Published in Print: 2015-12-01


Citation Information: Mathematica Slovaca, Volume 65, Issue 6, Pages 1531–1556, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2015-0105.

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