3
$\begingroup$

Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality $$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$ will hold.

What does the manifold need to satisfy for it to hold? Thanks

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ If the manifold $M$ is smooth and the boundary is compact then it certainly holds. It also holds in certain noncompact situations if $M$ and $\partial M$ have bounded geometry at $\infty$. $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 22, 2013 at 12:15
  • 1
    $\begingroup$ For $M$ compact with smooth boundary, you can find this in Proposition 4.5 in volume I of the 3-volume monograph of M.E. Taylor on Partial Differential Equations. $\endgroup$
    – user80744
    Commented Apr 22, 2013 at 12:23
  • $\begingroup$ Does $M$ really need to $C^\infty$ smooth like Taylor demands? $\endgroup$
    – user28178
    Commented Apr 24, 2013 at 9:27
  • 1
    $\begingroup$ No soup89, the manifold need not be $C^\infty$. $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 24, 2013 at 9:35

1 Answer 1

Reset to default
3
$\begingroup$

The book by Robert A. Adams has a 'ultimate' theorem about: trace theorem (resctricting to lower dimensions), extension theorem (extending to higher dimensions) and sobolev embedding.

It is Theorem 7.58. It is definitely more than you need here. It involves even fractional order Sobolev spaces.

The theorem is about Euclidean spaces. If you want a manifold version, we can always do it by hand. (Or just by waving your hand, especially when you are not really interested in controlling the constant.) Chop the manifold into pieces near the boundary and consider a coordinate system mapping a neighborhood to (a domain of ) half space, then use the partition of unity. The constant in your final inequality depends on a uniform bound of the coordinate system. Sorry, I can not be very precise here. However, the method works.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Which edition of Adams are you referring to ? $\endgroup$
    – user14319
    Commented Sep 11, 2016 at 13:03

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .