# Tagged Questions

0answers
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### 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. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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