Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 9th 2017
   • (edited Nov 9th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis question will just show my ignorance, please bear with me: For $P \colon C^\infty(X) \to C^\infty(X)$ a differential operator (self-adjoint say), there is an evident linear map $$ DistribSolutions(P) \longrightarrow \left(C^\infty(X)/im(P)\right)^\ast $$ from distributions $u$ for which $P u = 0$ to linear duals on the cokernel of $P$. When is this surjective? (I'd be happy to add various qualifiers if necessary, say compactly supported distributions, or whatever it takes.)

   This question will just show my ignorance, please bear with me:

   For $P \colon C^\infty(X) \to C^\infty(X)$ a differential operator (self-adjoint say), there is an evident linear map

   $$DistribSolutions(P) \longrightarrow \left(C^\infty(X)/im(P)\right)^\ast$$

   from distributions $u$ for which $P u = 0$ to linear duals on the cokernel of $P$.

   When is this surjective?

   (I’d be happy to add various qualifiers if necessary, say compactly supported distributions, or whatever it takes.)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 11th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexWith help from Igor Khavkine, here is the sketch of an argument that this is indeed true at least for the case of "[[Green hyperbolic differential equations]]": [here](https://ncatlab.org/nlab/show/Green+hyperbolic+partial+differential+equation#DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions)

   With help from Igor Khavkine, here is the sketch of an argument that this is indeed true at least for the case of “Green hyperbolic differential equations”: here

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 11th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexI should say that I created some stubby minimum at _[[generalized solution]]_

   I should say that I created some stubby minimum at generalized solution

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 12th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have forwarded the question to MO [here](https://mathoverflow.net/q/285880/381). I know it's a straightforward check, but maybe somebody lends a hand.

   I have forwarded the question to MO here. I know it’s a straightforward check, but maybe somebody lends a hand.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 13th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexIgor Khavkine has kindly provided a proof, now [here](https://ncatlab.org/nlab/show/Green+hyperbolic+partial+differential+equation#ContinousLinearFunctionalsOnTheSolutionSpace).

   Igor Khavkine has kindly provided a proof, now here.