All Questions
Tagged with differential-operators laplacian
17 questions
10
votes
2
answers
1k
views
Differentiation of functions over graphs
In short: There are various ways to define differentiation over a graph. I am trying to get the big picture, like a more complete and structured bestiary.
Definitions.
Let $G=(V,E)$ be a directed ...
- 1,444
8
votes
1
answer
2k
views
Growth of Laplacian eigenvalues on a compact domain?
Let $\mathcal{M}$ be a compact Riemannian manifold and let $\Delta$ be the (scalar) Laplace-Beltrami operator on $\mathcal{M}$. Then $\Delta$ has a discrete spectrum and if we order its distinct ...
- 3,059
7
votes
2
answers
637
views
Line graphs called "graph derivatives": any intuition?
Short version: in several papers, line graphs (and closely related graphs) are called graph derivatives or derived graphs; is there any intuition for such terminologies, in connection with the ...
- 1,444
7
votes
3
answers
627
views
Criteria for Positivity of Pseudoddifferential Operators on Manifolds
Let $(M,g)$ be a Riemannian Manifold and $L^2$ the Hilbert space given by the volume form associated to the metric. Let $L_0^2$ be the subspace which is orthogonal to the constant functions. When is ...
- 225
6
votes
1
answer
1k
views
Laplace-Beltrami and the isometry group
H$\vphantom{a}$i. Consider the Laplacian on $\mathbb R^n$,
$$
\Delta=\partial_i^2
$$
It is easy to prove that the most general differential operator that commutes with rotations and translations is ...
6
votes
2
answers
2k
views
The spectrum of the Hodge Laplacian on a Riemannian manifold
The Hodge Laplacian operator on differential forms on a (compact?) Riemannian manifold carries useful information about the topology of the manifold. In particular, the multiplicity of the zero ...
- 7,746
6
votes
0
answers
306
views
How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?
The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law
$$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$
For $z=x+ i y \in \mathbb C$ ...
- 650
5
votes
1
answer
457
views
An alternative representation of the principal symbol of the Laplace operator
Assume that $(M,g)$ is a $n$ dimensional Riemannian manifold. We denote by $\Delta$, the Laplacian associated to this Riemannian structure.
Are the following two conditions equivalent?
First condition ...
- 356
4
votes
1
answer
377
views
Differential inequalities under which a flat function must be identically zero
Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $.
Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
- 356
3
votes
0
answers
129
views
Differential operators on a compact Lie group associated to bracket-generating sets
Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...
- 2,446
3
votes
0
answers
494
views
Spectrum of the Laplace-Beltrami operator on a domain of finite volume in the hyperbolic space $H^n$
What is known about the ($L^2$) spectrum of the minus Laplace-Beltrami operator ($- \Delta$) with zero boundary conditions on $B =H^n/\Gamma$, where $H^n$ is $n$-dimensional hyperbolic space ($n>1$...
- 371
2
votes
1
answer
155
views
Realization of symbol of Laplace operator via certain integral
Is there an elliptic operator $D$ on $C^{\infty}(S^2)$ whose principal symbol is not identical to thats of Laplacian but it satisfies $\int_{S^2} fDf =\int_{S^2} f\Delta (f)$ for all $f\in C^{\infty}(...
- 356
2
votes
1
answer
644
views
Reference request: inverse of differential operators
I have asked a similar question on MSE but I did not receive any replies, so I am reposting here in case it is more appropriate (though I have slightly generalized the question).
As an example ...
- 721
2
votes
1
answer
118
views
Can I characterize functions (in 2D), which will have compactly supported/support contained Poisson solution?
I have the problem of solving Poisson equation in 2D
$$
\Delta u = f
$$
Let's say for a moment I want to solve it on $\mathbb{R}^2$, for $f(x,y), x\in \mathbb{R}, y\in \mathbb{R}$.
I know however that ...
- 151
2
votes
1
answer
701
views
hyperbolic "Green function" on a product of upper half-planes
Let $\Delta_{hyp}=\Delta_{hyp,1}=-y^2(\partial_x^2+\partial_y^2)$ be the hyperbolic Laplacian acting on functions of $\mathfrak{h}$ (the Poincare upper half-plane) and consider its resolvent
$$
R(s)=(...
- 7,609
2
votes
0
answers
400
views
Spectrum of the Witten Laplacian on compact Riemannian manifolds
Below I have given what I am calling as the ${\rm Witten{-}Laplacian}_{s,p}$ on a Riemannian manifold $(M,g)$ for any constant $s >0$ and $p \in C^2(M,g)$
How generally is it true that this ${\rm ...
- 2,246
2
votes
0
answers
625
views
Is Laplacian a surjective operator?
For a closed manifold the laplacian is almost surjective operator since the index of $\Delta$ is zero and there is no a non constant harmonic function. So the codimension of the image ...
- 356