Natural Boundary Conditions in Geometric Calculus of Variations

Giovanni Moreno 1  and Monika Ewa Stypa 2
  • 1 Mathematical Institute in Opava Silesian University in Opava Na Rybníčku 626/1 746 01 Opava CZECH REPUBLIC
  • 2 Department of Mathematics Salerno University Via Ponte Don Melillo 84084 Salerno ITALY

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).

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Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process.

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