3

When is the Frobenius theorem used to prove existence for PDE on manifolds, as opposed to more analytical techniques? I apologize that my question is pretty vague, but it stems from confusion about what techniques are generally used in geometric PDE. Sometimes I see more analytic machinery (from, for instance, Evans' textbook on PDE) applied in coordinates. Others (Lee's books on manifolds/Riemannian manifolds) prove existence to PDE using the Frobenius theorem. Could someone shed some light on when one technique is used versus another, or any similarities/differences between them? (Perhaps I am making a distinction where there really isn't one?)

Chris
  • 5,231
  • 2
    It suffices to consider the situation for a PDE on an open set of $\mathbb{R}^n$. The assumptios of the Frobenius theorem is usually written in terms of vector fields or diferential forms. I suggest that you rewrite them explicitly as a system of PDEs. That will tell you how to recognize when a system of PDEs could be solved using Frobenius, In short, the system has to be maximally overdetermined. – Deane Jan 27 '21 at 18:17

0 Answers0