All Questions
8 questions
11
votes
2
answers
1k
views
Concentration compactness. Can this concept be stated in a theorem?
I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk.
When I approached the speaker ...
- 501
5
votes
2
answers
625
views
Reconstruction of second-order elliptic operator from spectrum
Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
- 364
5
votes
1
answer
361
views
Is this a pseudodifferential operator?
Let $M$ be a non-compact manifold and $D$ a first-order self-adjoint elliptic differential operator on $M$. Then is the bounded operator
$$A:=\sqrt{(D^2+1)^{-1}}:L^2(M)\rightarrow H^1(M)$$
a ...
- 1,903
4
votes
1
answer
156
views
approximation of a Feller semi-group with the infinitesimal generator
Let $T_t$ a Feller semigroup (see this) and let $(A,D(A))$ its infinitesimal generator.
If A is a bounded operator it is easy to show that the Feller semi-group is $e^{tA}$.
Is this formula always ...
- 293
2
votes
2
answers
141
views
Equality of spectra of products of operators
Let $A$ be a linear operator between two Hilbert spaces. Let $A^*$ be its adjoint.
Question. Under what conditions the non-zero spectra of $A^*A$ and $AA^*$ coincide counting multiplicities?
In my ...
- 21.8k
1
vote
1
answer
728
views
Elliptic regularity of Laplace-Beltrami operator on a manifold
I am currently trying to prove an elliptic regularity type result for the Laplace Beltrami operator $\Delta_g$ on a Riemannian manifold $(M^n,g)$. As a matter of convention, I will assume $\Delta_g$ ...
- 1,247
1
vote
0
answers
122
views
Series and solution of $-\Delta u + \lambda u = f(x)$
Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of
\begin{align*}
-\Delta u(x) + \lambda u(x) &= f(x), & x \in \...
- 131
1
vote
0
answers
108
views
How to define Biharmonic operator for second order sobolev spaces
I am studying an article Link of Article. There author assumes that $\Omega \subset \mathbb{R}^N$, $ N>4 $ . Some where in the paper we have
$$ \Delta^2 (\cdot) - \frac{\lambda}{|x|^4} (\cdot) : W^...
- 371