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.
...
1
vote
0answers
30 views
Minimum is attained in a subset of a Sobolev space
Let $\Omega \subset \mathbb R^n$. I have a functional of the form,
$$\int_{\Omega}f(x,u,\nabla u)dx$$
where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian ...
2
votes
0answers
42 views
Differential of a Sobolev map between manifolds
Let $\Sigma, M$ be compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by
$$W^{k,p}(\Sigma,M) = \{ u \in ...
3
votes
2answers
185 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 ...
