Natural boundary conditions in geometric calculus of variations

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).