## G. Moreno - Monika Ewa Stypa

# Natural boundary conditions in geometric calculus of variations

created by moreno on 15 Sep 2020

[

BibTeX]

*preprint*

**Inserted:** 15 sep 2020

**Last Updated:** 15 sep 2020

**Year:** 2013

**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 $\R^{n+1}$, and the Lagrangian is first-order
(in particular, the hypersurface area).