9
$\begingroup$

In the jet bundle approach to differential equations

https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations

one identifies the equation with the set of a solution of the differential equations.

Now if I would formally want to capture a differential equation, I'd say it comes with a boundary condition as well. Like for example a Dirichlet/Neumann boundary conditions to a diffusion equation.

How are boundary conditions formally captured by the jet bundle approach to differential equations?

$\endgroup$
  • 1
    $\begingroup$ Boundary conditions are also conditions in jets, this is no different from the diff equation itself. The only difference is that you have jet bundle over a smooth manifold with boundary. $\endgroup$ – Moishe Kohan Sep 17 '13 at 2:02
  • $\begingroup$ @studiosus: So you say the boundary condition are not formalized further, but are just formulas to protect a certain solution out of the set $J^r(\pi)$? Then I think I don't fully grasp the need for all the overhead over the standard treatment of differential equations on general manifolds. $\endgroup$ – Nikolaj-K Sep 17 '13 at 8:40
  • $\begingroup$ That's right. My (admittedly personal and biased) feeling about this formalism is that the language of jets is just a bookkeeping device, designed to make the treatment of PDEs on manifolds independent of choices of local coordinates. This is quite useful for global (topological) considerations (like h-principle) but, from the analytic viewpoint, in the end of the day, you still have to write the traditional PDEs in local coordinates. $\endgroup$ – Moishe Kohan Sep 17 '13 at 9:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.