Tagged Questions
1
vote
0answers
54 views
Sobolev trace theorem for manifolds with boundary
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.
...
4
votes
1answer
116 views
Green's function for the Yamabe problem
I'm currently reading the paper on the Yamabe problem by Lee and Parker, and am looking for a reference for Theorem 2.8.
Theorem 2.8 (Existence of the Green Function).
Suppose $M$ is a ...
3
votes
1answer
95 views
Submersion Theorem for Banach Spaces
I'm having difficulty proving a well-known result from functional analysis. Any hints would be greatly appreciated.
Fix a Fréchet differentiable map of Banach spaces $g: X \to B$. Assume that, at a ...
25
votes
2answers
587 views
PDEs on manifold: what changes from Euclidean case?
I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds.
For example, do things like Poincare's inequality ...
0
votes
1answer
36 views
meaning of “doubly inward-pointing”
I am currently trying to understand the $b$-calculus developed by R. Melrose. an important part of the theory is the stretched product of a manifold $X$ with boundary $\partial X$.
looking at the ...
3
votes
2answers
173 views
Green's Function for Operator
I'm trying to show that the Green's function for the Laplace operator $-\nabla^2$ is badly behaved at infinity. I.e.
$$\int d^dx|G(x,y)|^2=\infty$$ for $d=1,2,3$. What happens when $d>4$?
I know ...
3
votes
2answers
184 views
Does a diffeomorphism between manifolds induce an isomorphism of Sobolev spaces?
Let $M$ be a Riemannian manifold, and define the Sobolev spaces $H^k(M)$ to be the set of distributions $f$ on $M$ such that $Pf \in L^2(M)$ for all differential operators $P$ on $M$ of order less ...
