All Questions
7 questions
8
votes
1
answer
399
views
Spectrum of a first-order elliptic differential operator
Suppose that I have a first-order elliptic differential operator $A: \mathrm{dom}(A) \subset L^2(E) \to L^2(E)$, where $(E,h^E) \to M$ is a hermitian vector bundle and $M$ is a compact manifold.
I ...
5
votes
1
answer
222
views
Domains with discrete Laplace spectrum
Let $\Omega \subset \mathbb{R}^n$ be a domain. Assume that the Laplacian $-\Delta=-(\partial^2/\partial x_1^2+\cdots+\partial^2/\partial x_n^2)$ has a discrete spectrum on $L^2(\Omega)$ (i.e., we are ...
- 443
5
votes
1
answer
164
views
Spectrum of an elliptic operator in divergence form with a reflecting boundary condition
Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$
\begin{align}
L ...
- 53
4
votes
0
answers
410
views
Spectral Gap of Elliptic Operator
Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?
The boundary condition is that the ...
- 325
2
votes
1
answer
315
views
Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum
How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial ...
- 2,239
2
votes
0
answers
67
views
A mapping property for fractional Laplace--Beltrami operator
Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...
- 4,145
1
vote
1
answer
178
views
Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum II
This is a follow-up on a previous question. Now the parabolic PDE of $P(t,x,v)$ has two spatial dimensions.
$$
\partial_t P = L^* P \tag1
$$
$$L^*P = \frac12\left(\kappa^2\frac{\partial^2}{\partial v^...
- 2,239