Questions tagged [laplacian]
The Laplacian matrix is the representation of a graph in matrix form.
283 questions
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Different definitions of fractional sobolev spaces
Let $\Omega$ be a bounded and smooth domain in $\mathbb R^d$. For any $s\in (0,1)$ we can define $H_s(\Omega)$ to be the space of functions $u\in L^2(\Omega)$ such that $$(x,y)\mapsto \frac{|u(x)-u(y)|...
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Fractional sobolev spaces
On the whole space $\mathbb R^d$, the fractional Sobolev space
$H_s(\mathbb R^d)$ of order $s\in \mathbb R$ can be defined as the subspace of tempered distributions $T$ such that $\mathcal F T \in L^...
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Error term in the Euclidean Weyl law
Let $\Omega\subset \mathbb R^n$ be an open bounded set with smooth boundary. The Laplacian on $\Omega$ with Dirichlet boundary conditions has discrete spectrum $\lambda_1\le \lambda_2\le \ldots$ that ...
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is it possible to have two non-isomorphic non-regular graphs with the same adjacent spectrum and the same laplacian spectrum?
For two regular graphs $G$ and $H$, it is possible for them to share the same adjacent spectrum and the same laplacian spectrum. While, on the other hand, is it possible to have two non-regular graphs ...
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If $A=BC$, where $A$ and $C$ are given Laplacian matrices, how to calculate $B$?
Let me be more specific: If $A=BC$, where $A$ and $C$ are given Laplacian matrices, how to calculate $B$? The graph corresponding to $A$ is a directed ring, which is strongly connected and $1_n$ and $...
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For what potentials is the heat operator with a potential term hypoelliptic?
If $(M,g)$ is a Riemannian manifold and $\Delta$ is the Laplace-Beltrami (negatively defined) operator, is it possible to describe the class of smooth potentials $V :M \to \Bbb R$ that make the heat ...
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How is the Jacobian or Sandpile group of a graph computed?
From what I understand, given a graph, the Jacobian group and the Sandpile group refer to the same object. Until now, I have been computing this group in the way detailed in Chapter 1 of this ...
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Boundary regularity of the solution of a Poisson equation in a polyhedron
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be bounded and open
$f\in L^2(\Lambda,\mathbb R^d)$
$u\in H_0^1(\Lambda,\mathbb R^d)$ with $$-\langle\nabla\phi,\nabla u\rangle_{L^2(\Lambda,\:\...
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Locally symmetric spaces: spectrum of the Laplacian
Let $M = \Gamma\backslash X$ denote a locally symmetric space of non-compact type and $\Delta$ the Laplacian on $L^2(M)$.
It is known that the spectrum of $\Delta$ decomposes into finitely many
...
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Hilbert Modular Surface: Eigenfunctions of The Laplacian
The spectrum of the Laplacian on $L^2$ of a Hilbert modular surface decomposes into a discrete part and a continuous part $[1/4,\infty)$. The continuous part contains eigenvalues $\geq 1/4$.
I would ...
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Gradient of spectral function on noncompact homogeneous space
Let $(M,g)$ be a noncompact Riemannian manifold whose isometry group acts transitively on $M$, i.e. a (not necessarily normal) homogeneous space. Let $e_{\lambda}(x,y)$ be the integral kernel of
$f \...
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"Designing" Nodal sets of Laplacians in 2 or 3 dimensional domains
The properties of nodal sets (i.e. zero level sets of eigenfunctions) for the first non-trivial eigenfunction for Laplacians have been studied extensively.
My rough understanding is that one could ...
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heat kernel on closed manifolds - error in Chavel's book?
first of all, I am not sure if this question fits here. I asked this question on math.stackexchange also but didn't get an answer so far.
In Isaac Chavel's book Eigenvalues in Riemannian Geometry, ...
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certain smoothness of principal eigenvalue of Dirichlet Laplacian on polygons
For a given polygon $P_N$, with side lengths $x_1,\cdots,x_N$ and interior angles $\theta_1,\cdots,\theta_N$ let $\lambda(x_1,\cdots,x_N,\theta_1,\cdots,\theta_N)$ denote the least eigenvalue of ...
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Laplacian spectrum of directed network (digraph) and its complement
There is a well-known relation between the spectrum of graph laplacian and its complement's laplacian, namely
$$λ_j (G^c) + λ_{n+2−j} (G) = n\;,$$
where the eigenvalues $λ_j$ are sorted in ...
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Universal constant for reverse inequality between first eigenvalues of Neumann and Dirichlet problems
I finally decided to post the following naive question but will if consensus is that it is out of the scope of this site , it will be immediately deleted.
Suppose $\Omega\subset\mathbb R^2$ is a ...
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Inequalities concerning principal eigenvalues of Laplacian with different boundary conditions
Suppose $\Omega\subset\mathbb R^2$ is a bounded simply connected domain with sufficiently smooth boundary. Consider the following three BVPs (respectively with Dirchlet, Neumann and certain non-local ...
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A conjecture on shape optimization for Dirichlet-Laplacian
For a given domain $G$, with sufficiently smooth boundary, in the plane we denote the first two eigenvalues of Dirichlet-Laplacian on $G$ of by $\lambda_1(G)$ and $\lambda_2(G)$.
$\textbf{Open(?) ...
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Laplacian on two Lie groups have the same Lie algebra
I know that if $G$ is a Lie group and $\mathfrak g = span\{X_i, 1\leq i \leq n\}$ be its Lie algebra, where $\{X_i\}$ are the vector fields of $G$. Then, the Casimir-Laplacian of $G$ is given by
$$\...
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First eigenvalue for strictly convex domains
Let $M^n$ be a compact Riemannian manifold with boundary, suppose 1). $Ric(M)\ge (n-1)$ and 2). the principle curvatures of the boundary is bounded from below by $h\ge 0$. Is there any results on the ...
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Ellipticity of Bott-Chern Laplacian
I want to prove that Bott-Chern Laplacian
$$\tilde{\Delta}_{BC}^{p,q}=(\partial\bar\partial)(\partial\bar\partial)^*+(\partial\bar\partial)^*(\partial\bar\partial)+(\bar\partial^*\partial)(\bar\...
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Hodge decomposition for Bott-Chern cohomology
$\DeclareMathOperator{im}{im}$
I want to prove that Bott-Chern cohomology group $H^{p,q}_{BC}=\frac{\ker\partial\cap \ker\bar{\partial}}{\im\partial\bar{\partial}}$ has finite dimension via Hodge ...
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Create a graph with a specified number of spanning trees
I read that one of the current challenging problems in mathematics is constructing a minimal graph with a specified number of spanning trees (say, $k$).
However, is there a quick way to create some ...
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Eigenspaces of the Laplace operator on a unit ball
I am interested in structures of the eigenspaces of the Laplace operator on the $n$-dimensional unit ball with Neumann or Dirichlet boundary conditions as representations of the special orthogonal ...
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Are harmonic mappings non-singular outside a set of measure zero?
Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb D^n$.
Let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving immersion, and let $\omega :\...
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The square root of Laplacian with nonconstant coefficent
I am still a newbie to $\Psi$DO-Operators. As far as i understood, one can easily compute the square root of the Laplace operator $\Delta$ by
$$(-\Delta)^{1/2} \ u=\mathcal{F}^{-1}(\|\xi\| \widehat{u}...
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non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?
We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e., the matrix is not only weak diagonal-dominant, but ...
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Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3 $ is sub-elliptic but not elliptic?
I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant ...
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Laplace eigenvalue and floer theory
I'm trying to study the spectrum operator via Floer Theory. The idea is to consider the Rayleigh quotient as an operator on the Hilbert space $W^{1,2}$ and to study it's negative gradient flow.
Any ...
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An H2 estimate for Helmholtz equation
How to show the following statement?
Let $\Omega$ be a bounded Lipschitz convex domain. If $u$ satisfies the following equation,
$$
-\Delta u - k^2 u = f \quad\mbox{ in }\Omega \\ \nabla u \...
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If the two smallest eigenvalues of the Laplacian matrix of a network are equal to zero, then does it mean that the network is not connected? [closed]
What does it mean if the two smallest eigenvalues of the Laplacian matrix of a graph are equal to zero?
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Difference between the Laplacian and the sub-Laplacian of a Lie group
Given a Lie group $G$, what is the difference between the Laplacian $\Delta$ and the sub-Laplacian $\Delta_{sub}$ of $G$. And what are the properties that we lose when going from sub-Laplace to ...
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Zero set of eigenfunction along a sub manifold
Let $M$ be a 2-dimensional closed Riemannian manifold and let $$\phi:M\rightarrow M$$ be an isometry with $\phi^2=Id_M$. Consider the fixed point set $$F:=\lbrace x\in M: \phi(x)=x \rbrace\subset M,$$ ...
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Why decompose a function with eigenvectors of Laplace operator? [closed]
On periodic domain, people always use Fourier basis, which eigenvectors of Laplace operator. On sphere, people use spherical harmonics, which also are eigenvectors of Laplace operator. In applied ...
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Invertibility of group Laplacian in $\ell^1$
Let $G$ be a discrete group and let $S$ be a generating set for $G$; assume that $S$ is symmetric (i.e., $g\in S$ iff $g^{-1}\in S$). Let $L=L_S=\frac{1}{|S|}(\sum_{g\in S} g-1)$ be an element of the ...
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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 $z=x+ i y \in \mathbb C$ ...
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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 bounded ...
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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 ($n>1$...
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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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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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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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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*}
\mathbb{...
user50806
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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 ...
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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)
$$\matrix{
{{\nabla ^2}\Phi (x,y)...
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Fundamental solution of Discrete Laplace in the plane
We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator
Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)...
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Why should the map $-\Delta^{-1}$ be continuous?
I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$
in the unit ball $\Omega$ in dimension $>2$. I'm looking for ...
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Explicit eigenvalues of the Laplacian
Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...
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Connection between the p and q Laplacians
I'm just looking for some quick and dirty intuition(and/or reading material) about the following:
I read that Hodge duality provides a way to interchange the p-Laplacian $ \Delta_p = \nabla\cdot( |\...
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Lp norm estimates for the inverse of the Laplacian on a graph
I am looking at a finite connected graph and I would like to know what is the best [i.e. largest] constant $\lambda_p$ in
$$
\sum_x f(x) =0 \implies \| \Delta^{-1} f\|_{\ell^p} \leq \lambda_p^{-1} \|...
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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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