Questions tagged [laplacian]

The Laplacian matrix is the representation of a graph in matrix form.

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35
votes
5answers
3k views

Explicit eigenvalues of the Laplacian

Let $(M,g)$ be a compact manifold without boundary. Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known? An important example is the $n$-sphere ...
10
votes
2answers
1k views

Characterize where the Dirichlet Problem for the Laplacian is always solvable

Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ...
6
votes
2answers
412 views

On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold

Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field ...
19
votes
3answers
1k views

Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators

EDIT: According to some comments on this post I revise the title to remove the misunderestanding. Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...
17
votes
4answers
1k views

High multiplicity eigenvalue implies symmetry?

It is well known that on any compact Riemannian symmetric space $X$, the eigenvalues of the Laplacian have very high multiplicity (comparable with the Weyl bound), and the resulting actions $\...
5
votes
2answers
697 views

Exotic spectrum of Laplace operator

Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator, it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
16
votes
3answers
686 views

Spectral properties of the Laplace operator and topological properties

Suppose that $M$ is a closed Riemannian manifold: one can construct the so called Laplace-Beltrami operator on $M$. Its spectrum contains some information of the underlying manifold: for example its ...
8
votes
1answer
331 views

Can a harmonic vector field possess a limit cycle?

Can a harmonic vector field $X$ on a Riemannian surface $(M,g)$ possess a limit cycle(An isolated periodic orbit)? Note that the Laplacian of a vector field is defined via natural correspondence ...
8
votes
2answers
460 views

Behavior of the spectrum of the Laplacian under pointed smooth convergence

The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$. On the other ...
6
votes
2answers
332 views

How to understand the combinatorial Laplacian $\Delta$ which is defined on the graph?

I have a question about the combinatorial Laplacian $\Delta$ which is defined by $$\Delta(u,v)=c(u)1_{u=v}-c(u,v)$$ where $u, v$ are some vertices in the graph $G=(V, E)$, and $c(u,v)$ is a ...
6
votes
3answers
4k views

Integral kernel for the resolvent of the laplace operator

Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of $\Delta$...
2
votes
1answer
876 views

Global Lichnerowicz Formula Proof (in the Kahler case)?

For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection ...
7
votes
3answers
475 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 ...
6
votes
0answers
208 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 $z=x+ i y \in \mathbb C$ ...
7
votes
2answers
420 views

Probabilistic Interpretation of First Dirichlet Eigenvalue?

The first Dirichlet eigenvalue of a compact domain $\Omega\subset\mathbb{R}^n$ with smooth boundary is the smallest positive number for which there exists a non-trivial solution to $$ -\Delta\psi = \...
5
votes
1answer
542 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 ...
1
vote
0answers
151 views

Lp norm estimates for the inverse of the Laplacian on a graph

I am looking at a finite connected graph and I would like to know what is the best [i.e. largest] constant $\lambda_p$ in $$ \sum_x f(x) =0 \implies \| \Delta^{-1} f\|_{\ell^p} \leq \lambda_p^{-1} \|...
3
votes
2answers
319 views

Interpolation spaces

In this paper, the authors claim that for $s\in [0,1]$, $\left [ H_0^1(\Omega), L^2(\Omega)\right ]_\theta=\text{dom}(-\Delta)^{\frac s 2}$, where $\Omega$ is a smooth and bounded domain in $\mathbb R^...