The laplacian tag has no usage guidance.

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### 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 $z=x+ i y \in \mathbb C$ ...

**2**

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**1**answer

87 views

### The inverse of Laplacian operator for different orders

I post this question in MSE couple of days before and get no response. So I repost it here for better luck. Thank you!
Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open ...

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85 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 ...

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114 views

### Harmonic and Primitive Forms on a Kaehler Manifold

Let $M$ be a compact Kaehler manifold, and $p$ a primitive form, which is to say it is contained in the kernel of the adjoint of the Lefschetz operator $L$ associated to the Kaehler form. If $p$ is ...

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52 views

### Laplace Beltrami of the Gauss map

Let $M$ be a surface in $\mathbb{R}^3$ given by a regular chart, say $X:M \longrightarrow \mathbb{R}^3$, with its first fundamental form $g$, Gauss map $N$, Gaussian curvature $K$ and mean curvature ...

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votes

**3**answers

150 views

### Show that the Laplacian operator on the Heisenberg group is negative

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 $z=x+ i y \in \mathbb C$ ...

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votes

**0**answers

99 views

### Estimate for the first eigenvalue of the Laplacian

I was studying the paper of S. T. Yau - Seminar on Differential Geometry - and there asks if the first eigenvalue is equal to $ n $, if we have a embedded oriented Riemannian manifold and closed ...

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83 views

### Spectral theory of differential forms over a circle bundle

Here is the set up : I consider the unitary tangent bundle of a surface $(S,g)$ endowed with the Sasakian metric ; $(T^1S, g_s)$, in fact we have the following fibration :
\begin{equation*}
...

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66 views

### Laplace Equation with Tangential Derivative Prescribed on the Boundary [closed]

I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :)
Consider the following Laplace boundary value problem (BVP)
$$\matrix{
{{\nabla ^2}\Phi ...

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317 views

### Fundamental solution of Discrete Laplace in the plane

We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator
Then, it is natural to consider its fundamental solution $u$, i.e. ...

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**1**answer

135 views

### Connection between the p and q Laplacians

I'm just looking for some quick and dirty intuition(and/or reading material) about the following:
I read that Hodge duality provides a way to interchange the p-Laplacian $ \Delta_p = \nabla\cdot( ...

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69 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} ...

**1**

vote

**1**answer

63 views

### Negativity of a quadratic form on $L^2(M)$

Let $M$ be a compact Riemannian manifold and $V=L^2(M)$. Let $\Delta$ be the negative-definite Laplacian. Let $f \in V$ and $x \in M$ be arbitrary, but fixed.
Is it true that ${\rm Re} \ (\Delta f) ...

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**1**answer

115 views

### The Laplacian of an expression involving the Ricci tensor

While doing some computations on a compact Riemannian manifold I have reached the following expression:
$$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$
where $\Delta_y$ is the ...

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**1**answer

167 views

### Laplace-Beltrami and averaging

For a Riemannian manifold $M$ with metric $g$ and Laplace-Beltrami operator $-\Delta_{g}$, what conditions on $M$ guarantee that $-\Delta_{g} u(x)$ measures the difference between $u(x)$ and the ...

**3**

votes

**1**answer

158 views

### Gagliardo Nirenberg inequality for the laplacian

It is a classical result due to Gagliardo and Nirenberg that there exists a constant C such that it holds
$$
||\nabla \psi|| _{L ^\infty (\mathbb{R}^2)} ^2
\le ||D ^2 \psi|| _{L ^\infty ...

**2**

votes

**1**answer

105 views

### Limited expansion of mean curvature of geodesic spheres

I am working with the Laplacian on a Riemannian manifold $(M,g)$ (compact, without boundary). In spherical geodesic coordinates $(r, \sigma)$ around some arbitrary $x \in M$ (where $\sigma$ denotes ...

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**3**answers

616 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 ...

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**2**answers

127 views

### Connectivity of weighted graph and zero Laplacian eigenvalues

Given an undirected graph $G$, and let $V$ denote its set of vertices and $E$ its set of edges. Suppose that there are no edges connecting the same vertex, and no more than one edge connecting any ...

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**1**answer

106 views

### Analysis of the Laplacian of a random bipartite graph

My analysis of an engineering problem reduced to analysis of the Laplacian of a (random) bipartite graph. There are a few particular questions I am interested in, but not sure which direction to take ...

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181 views

### Limits for eigenvalues for the Dirichlet Laplacian

If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem
$$
\begin{cases}
-\Delta u=\lambda u & \mbox{in }\Omega\\
u=0 & \mbox{on ...

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**1**answer

180 views

### Does the Laplacian commutes with elements of the basis of the Lie algebra?

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. I know that if $g$ is semi-simple then the Laplace-Beltrami operator on $G$ agrees with the Casimir element and therefore commutes with ...

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**2**answers

297 views

### First eigenvalue of the Laplacian on a regular polygon

Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:
...

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361 views

### Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold

It has been proved by Li-Yau and Zhong-Yang that if $M$ is a closed Riemannian manifold of dimension $n$ with nonnegative Ricci curvature, then the first nonzero eigenvalue $\lambda_1(M)$ of the ...

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**2**answers

94 views

### Resolvent operator of fractional Laplacian

For $0<\alpha<2$, we define the fractional Laplacian with Fourier transform
\begin{align}
\widehat{(-\Delta)^{\frac{\alpha}{2}} u}(\xi) = |\xi|^\alpha \widehat u(\xi).
\end{align}
Consider the ...

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**1**answer

378 views

### Stochastic interpretation of heat kernel on fiber bundle

I'm looking for a stochastic interpretation of the heat equation for vector valued function.
The classical set up is the following :
If $(M,g)$ is a riemannian manifold then we could consider the ...

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**1**answer

163 views

### On fundamental solutions to Poisson equation on 3-dimensional manifolds

I am interesting in solutions to Poisson equation
$$\triangle \varphi = 4 \pi \rho \qquad (1)$$
defined on 3-dimensional oriented Riemannian manifold $(M,g)$,
where $g$ is metric and ...

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**1**answer

39 views

### Analytical value for the first eigenvalue of a certain spherical triangle

I am testing some numerical algorithms for computing the Laplace-Beltrami eigenvalues on the sphere. One thing that came up was computing the first eigenvalue of the "equilateral" spherical triangle ...

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**1**answer

84 views

### Does the green kernel converge as a series of functions?

Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following ...

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**1**answer

119 views

### Functional equations for spectral zeta function

Functional equation in the theory of zeta functions is one of the important components of this theory.
I am interested to know whether the similar property, having functional equation, for the ...

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320 views

### Decompose the Laplacian

Is there a way to write the negative Laplacian on the 2-sphere as a decomposition of an operator $A$ and its adjoint $A^*$? I am interested in finding such a decomposition, but I could not get one by ...

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146 views

### Solving gradient of an especial heat equation

In my research I came up with a gradient of heat equation on a edge-weighted graph as:
\begin{equation*}
\nabla_w T_t(t,w) + T(t,w) . \nabla_w L_w + L_w . \nabla_w T(t,w) = 0
\end{equation*}
where ...

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**1**answer

304 views

### Fractional Laplacian and stereographic projection

The ordinary Laplacian on $\mathbb{R}^N$ behaves nicely under a stereographic projection onto $\mathbb{S}^N\setminus\{P\}$. (Here $P$ is either the north or south pole of the unit sphere ...

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168 views

### Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix

I'm trying to find upper boundaries on the smallest Eigenvalue $\lambda_1$ of $L + E$, where $L$ is a standard Laplacian of an unweighted digraph, with $\lambda_1(L) = 0$ and $E \in \{0,1\}^{n \times ...

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426 views

### Are these three different notions of a graph Laplacian?

I seem to see three different things that are being called the Laplacian of a graph,
One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ ...

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60 views

### Obstructions for existence of a Riemannian metric such that a given function is harmonic

Let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a smooth function. What type of obstructions exist for existence of a Riemannian metric $g$ on $\mathbb{R}^{n}$ such that $f$ is a harmonic ...

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123 views

### Worst-Case Solution to (Stochastic) Matrix Inequality

EDIT: Some specific conjectures added.
This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...

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**2**answers

127 views

### Does the Laplacian commutes with the indicator function [closed]

We define the laplacian operator $\Delta$ with the Neumann boundary conditions on the space $H^2(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, and ...

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92 views

### What is the significance of the median eigenvalue?

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...

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198 views

### The complex heat kernel on a Riemann manifold

There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial ...

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**1**answer

302 views

### Paper by Mumford

In the paper of "The spectrum of difference operators and algebraic curves",
by P. van Moerbeke and D. Mumford, Acta Mathematics, vol.143, 1979,
(link: ...

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363 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 ...

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**1**answer

296 views

### Laplace-Beltrami operator on a Lie group

For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f = \delta^{i j} X_i X_j f$$
for ...

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**1**answer

310 views

### heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...

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**1**answer

136 views

### How to model a time-discrete heat equation on a graph?

I would like to know how one can set up a time-discrete model for the heat equation on a graph?
Is this problem using the heat kernel equation on a graph?

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215 views

### Heat equation with graph laplacian

I would like to start with considering the time-dependent heat equation on a connected graph and consider its Laplacian matrix.
Suppose we have a connected graph with unknown temperature on vertices. ...

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215 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 = ...

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**1**answer

424 views

### Calculating a generalized inverse (Moore–Penrose pseudoinverse)

I am considering a graph with $n$ edges with the following nicely structured adjacency matrix:
\begin{equation}
A_n=
\begin{pmatrix}
0 & 0 & 0 &\cdots & 0 & 0 & 1\\
0 & 0 ...

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102 views

### Basic doubt in a free boundary problem for the Laplacian

I am studying the following article : http://hal.archives-ouvertes.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf
In this article the authors considers $K \subset \{ x \in R^n ; x_1 = 0 \}$ a smooth, ...

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406 views

### Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...