All Questions
16 questions with no upvoted or accepted answers
7
votes
0
answers
116
views
Bundles over Function Spaces
Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable Hilbert-...
- 71
6
votes
0
answers
493
views
Reference for the Banach Manifold structure of $C^k(M,N)$
I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:
Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^...
- 71
4
votes
0
answers
343
views
Diffeomorphism group action on the space of embeddings
Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is ...
- 2,789
4
votes
0
answers
172
views
Donnelly-Fefferman growth of eigenfunctions
Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...
- 41
4
votes
0
answers
241
views
"Partition" of a smooth function in $\mathbb R^2$
This is a question asking for reference.
I have a proof of the following.
Let $f=f(x,y)$ be a smooth function in $\mathbb R^2$ which vanishes at the origin. Then there exist smooth functions $f_1=...
- 231
3
votes
0
answers
153
views
Quasimode construction on a compact Riemannian manifold
Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
- 131
3
votes
0
answers
479
views
de Rham currents/distributions on manifolds with boundaries
My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De ...
- 2,792
3
votes
0
answers
406
views
Smooth perturbation of a positive self-adjoint operator with compact resolvent
Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...
- 31
2
votes
0
answers
126
views
Differential equations: trying to connect a nonlinear equation to a linear one
The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
- 383
2
votes
0
answers
255
views
Sobolev Multiplication on non-compact manifold
We know that for a compact Riemannian $n$-dim manifold $(M,g)$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that $L^p_k\times L^q_l⟶L^r_m$, where $1/r−m/n>1/p−k/m+1/...
- 1,915
2
votes
0
answers
127
views
Functional inequality under mean curvature flow
Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...
- 143
2
votes
0
answers
234
views
Concentration compactness on a compact setting
Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\{\varphi_k\}_k \in C^\infty(M)$ such that $\{\varphi_k\}_k$ satisfy the basic concentration ...
- 1,407
2
votes
0
answers
108
views
Quantitative estimate of heat dispersion - off diagonal estimates
Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = u(...
- 21
1
vote
0
answers
56
views
Smooth approximation in Sobolev spaces for surfaces with boundary
Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ with closure $\overline{\mathbb{D}}$, and let $\varphi:\partial \mathbb{D}\to \partial \mathbb{D}$ be any continuous homeomorphism. Let $\mu$ be a ...
- 163
1
vote
0
answers
75
views
Derivation of the vortex filament equation from Euler equation
How can the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
be derived from the Euler equation
$$\partial_t \...
- 277
0
votes
0
answers
70
views
Normal vector to a level set and fractional Laplacian
Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
- 99