Questions tagged [laplacian]

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

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Quasimode construction on a compact Riemannian manifold

Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
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Bound on the magnitude of the entries of the Laplacian pseudo-inverse

Let $L$ be the laplacian matrix of a connected graph $G$ with real positive weights and $N$ vertices, or that can be assumed to have binary weights for simplicity.My goal is to bound $\Vert L^+\Vert_{\...
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Behaviour of higher order Laplacian in punctured domain

Bocher theorem characterize the behaviour of a positive harmonic function in punctured disc. More precisely if $\Omega$ is a domain in $\mathbb{R}^3$ and $U$ is a non negative solution of $\Delta u=0$ ...
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Sobolev regularity via Laplace spectrum

Fix a positive integer $n$ and let $\mu$ be the uniform measure on the sphere $\mathbb{S}^n$, with respect to its usual Riemannian metric $g$. Let $\nabla$ be the Laplacian on $(\mathbb{S}^n,g)$ and ...
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Embeddings of the maximal domain for the Laplacian

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function: $$D = \left\{ f \in L^2(\...
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The existence of a positive Green function for the Laplacian on $\mathbb R$

One can show explicitly and easily that the function $G(x,y) = \frac 1 2 |x-y|$ is a positive Green function for the Laplacian $\frac {\mathrm d ^2} {\mathrm d x ^2}$ on $\mathbb R$ (endowed with the ...
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Max-cut from Laplacian

(This question seems like very standard material for those well-versed in the subject. I thought I would get a quick answer from Math stackexchange, but to no avail.) Given a weighted graph with $n$ ...
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Dirichlet-to-Neumann map is analytic

Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$,...
Eduardo Longa's user avatar
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Dimension of Laplacian eigenspaces along a smooth 1-parameter family of metrics

Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $...
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Spectral characterization of complete or complete bipartite graphs

The Lemma 6 in this paper mention the following spectral characterization of complete or complete bipartite graphs: Let $G$ be a connected graph with $\ge 2$ vertices. Then $\lambda_2=...=\lambda_{n-...
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How to show that the trace of a regularized Laplacian defined on two sphere with radius $h\geq 1$ is diverging logarithmically?

Let $h,m\in[1,\infty)$. I would like to verify that the following sum diverges logarithmically \begin{equation} \sum_{d=0}^{\infty} \frac{2d+1}{2h^2(1+\frac{d(d+1)}{h^2})(1+\frac{d(d+1)}{h^2m ^2})^{2}}...
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Inequality concerning the imaginary parts of a recurrent sequence, Laplacian eigenvectors

Let $u=(u_1,\dots,x_n)\in\mathbb{C}^n$ be a sequence that satisfies the cyclic recurrence $$ \lambda+1 =a_{i-1}\frac {u_{i-1}}{u_i} + (1-a_{i+1})\frac{ u_{i+1} }{u_i } $$ with $a_i \in (0,1)$ and $\...
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Infimum of the normalized Laplacian eigenvalues

Let $(M^n,g)$ be a compact Riemannian manifold. The spectrum of the Laplacian operator $\Delta_g = -\operatorname{div} \nabla$ consists of an increasing and diverging sequence of positive eigenvalues: ...
Eduardo Longa's user avatar
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Gradient estimate of Dirichlet Heat kernel (Classical Laplacian)

Let $p^D(t,x,y)$ be the heat kernel for the Dirichlet Laplacian in an open set $D$. Do we have the following estimate and where can I find it ? $$\lvert\nabla_xp^D(t,x,y)\rvert\le C\dfrac{1}{\min (\...
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How to calculate the weights for Discrete Laplacian Operator?

I am following this paper step by step and want to build an isotropic Laplacian kernel. As shown in the following figure, I can understand until using Taylor to expand the 2D discrete Laplacian ...
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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 ...
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Sub Laplacian on the quaternion Heisenberg group $\mathbb{H}$

The sublaplacian is defined by $\mathcal{L}=-\left(X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\right)$, which is independent of the choice of the orthonormal basis of $\mathbb{H}$. It is well known that ...
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Leibniz rule for square root of Laplacian

Let $(M,g)$ be a compact Riemannian manifold (e.g. $M=S^3$ the 3-sphere) and let $\Delta$ be the metric Laplacian on $M$. Then $\Delta$ has an $L^2(M)$ basis of eigenfunctions $\pi_m$, $$ \Delta \pi_m ...
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Maximizing the first Neumann eigenvalue on disks

Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. Li and Yau proved that $$\mu_1(g) \operatorname{...
Eduardo Longa's user avatar
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Steklov eigenvalue for circle valued functions

Let $(M,g)$ be a compact Riemannian manifold with boundary. It is well known that the first positive Steklov eigenvalue $\sigma_1$ of $M$ has the following variational characterization: $$\sigma_1(M,g)...
Eduardo Longa's user avatar
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Are these the only first eigenfunctions on a hemisphere?

Let $\mathbb{S}^2_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$...
Eduardo Longa's user avatar
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A question about Gauss-Green formula - a weaker assumption

The question I have in mind is the following: how can we prove that for any $v\in H^1(\Omega)$ and for any $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ the Gauss-Green identity takes place $$\...
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Heat kernel coefficients for Laplacian in instanton background

The heat kernel coefficients $b_{2k}(x,y)$ of the covariant Laplacian in an $SU(2)$ instanton background (for simplicity let's say $q=1$ topological charge, so the 't Hooft solution) on $R^4$ is ...
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From discretized laplacian to wave equation

I am trying to establish a link between the laplacian of a function and its second derivative. Here is the method. Let $f(t;x,y)$ be a function in $\mathcal{C}^2([0,T] \times \Omega)$ with $t\in[0,T]$ ...
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Estimate value of harmonic function in the annulus

Let $D = B_{2r}(0)\backslash \overline{B}_r(0)$. Assume $Lu = 0$ in $D$ where $L$ is a uniform elliptic operator with constant coefficients $$ Lu = \sum_{i,j} a_{ij}u_{x_i}u_{x_j}, \qquad \lambda |\xi|...
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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 ...
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Upper bound for the first eigenvalue of the Laplacian on surfaces with boundary

Let $\Sigma$ be a compact smooth surface with boundary. Define $$\Lambda(\Sigma) := \sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ is a smooth Riemannian metric on $\Sigma$} \}$$ ...
Eduardo Longa's user avatar
7 votes
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Eigenvalues of the Laplacian on surfaces with boundary

Let $\Sigma$ be a compact smooth surface with boundary. Is it true that the supremum $$\sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ smooth Riemannian metric on $\Sigma$} \}$$ ...
Eduardo Longa's user avatar
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Largest eigenvalue of a Laplacian matrix to lower bound the prime counting function?

Let $L_n$ be the Laplacian matrix of the undirected graph $G_n = (V_n, E_n)$ (which is defined here: Why is this bipartite graph a partial cube, if it is? ) with sorted spectrum: $$\lambda_1 (G_n) \ge ...
mathoverflowUser's user avatar
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Entropy of eigenvectors of (traceless) laplacian of a bipartite graph

This problem is motivated by the edge states that sometimes appear (and sometimes not) at the level of Huckel hamiltonians for $\pi$-conjugated benzenoid hydrocarbons. If this sentence is cryptic, ...
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Laplace equation with bisection boundary condition on upper side

solve Laplace equation $ U_{xx} + U_{yy} = 0 $ with given boundary condition : $ (x,y) \in (0,a) \times (0,b) $ $\begin{cases} U_x(0,y) = U(x,0) = U_x(a,y) = 0 \\ U(x,b)= c= constant \hspace{3mm} ,...
math enthusiastic's user avatar
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Algorithm for efficiently calculating $(A+\sum_{i=1}^n B_i)^{-1}$ where $A^{-1}\in\mathbb S^n_+$ is known and $B_i$ are sparse matrices

Let $A\in\mathbb R^{n\times n}$ be a symmetric positive-definite matrix and $A^{-1}$ is already known. Now I want to compute the matrix $(A+\sum_{i=1}^n B_i)^{-1}$ where each $B_i$ is a sparse ...
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Laplacian on sphere after stereographic projection

How to derive that after stereo-graphical projection, $\Delta u$ in $\mathbb{R}^n$ is transformed to $$ \Delta_{\mathbb{S}^n}u - \frac{n(n-2)}{4}u\ \text{in}\ \mathbb{S}^n. $$ To be more precise, in ...
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Surprising symmetry in the Ramanujan bound

The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy $$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$ With a ...
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Prove or disprove the compactness of an operator

Consider $X=L^{2}(0,\pi, \mathbb{R})$. Let $X_{\frac{1}{2}}$ be the domain of $(\Delta)^\frac{1}{2}$ where $\Delta$ is the laplacien operator. We define the operator $K:C([0,a],X_{\frac{1}{2}})\...
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Self-ajointness of the Laplacian over a Riemannian manifold with boundary

I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf). Let $(M,g)$ be a Riemannian manifold with boundary; $E\to M$ be an hermitian fiber bundle; $\Delta$ ...
Parco Macelli's user avatar
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Interpreting the Higher-order Hodge-Laplace Operator

As an operator on functions, one intuitive way to think about the Laplacian seems to be as an operator that returns the average difference between a function's value at a point and the values of its ...
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Non-linear diffusion on networks

The diffusion equation with constant diffusion $D$ can be represented as: \begin{equation} \frac{\partial \phi(r, t)}{\partial t}=D \Delta \phi(r, t) \end{equation} where $\Delta$ is the Laplace ...
Matt's user avatar
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Fractional Laplacian of smooth cut off functions

Suppose we have a smooth compactly supported function $\phi\in C^{\infty}_c(B_\epsilon(0))$ such that $0\leq \phi \leq 1$, $\phi\equiv 1$ on the unit ball and $\phi$ vanishes outside $B_\epsilon(0).$ ...
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Asymptotics for the first eigenvalue for the Laplace-Beltrami operator on the sphere

I am trying to understand the existence of positive solutions for the following equations, $-\Delta_{\mathbb{S}^n} u + \lambda u = f(u)$ where $f$ is some non-linearity, say $f(t)=t^3.$ By considering ...
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Eigenvalues and eigenfunctions of the Laplace operator on entire plane

According to the answers in the the following questions: How to prove the spectrum of the Laplace operator? and What is spectrum for Laplacian in $\mathbb{R}^n$ , the spectrum of the Laplace operator $...
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Explicit expression for the Poisson kernel solving the Dirichlet problem for geodesic balls

Let $X$ be a Riemannian manifold with the Laplace-Beltrami operator denoted by $\mathscr L$ and we look at its geodesic balls say $B$. Let $u$ be a continuous function on the geodesic sphere which is ...
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First eigenvalue of the Laplacian on the traceless-transverse 2-forms

Let $(S^3/\Gamma, g)$ be a spherical space form with constant sectional curvature $1$, where $\Gamma$ is a finite subgroup of $SO(4)$ acting freely on $S^3$. Consider the first nonzero eigenvalue ...
Zhiqiang's user avatar
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3 votes
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Spectra of the Laplacian operator on the spherical space-form

Let $S^3/\Gamma$ be a spherical space form where $\Gamma$ is a finite subgroup of $O(4)$ acting freely on $S^3$. If $\Gamma$ is trivial, it is well-known that the spectra of the Laplacian operator on $...
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Are smooth functions with compact support a core for the Laplacian on compact manifolds with boundary?

If $M$ is a complete Riemannian manifold and $L$ is the Friedrichs extension of the Laplacian $-\Delta$, then it is known (first proven by Gaffney in the '50) that $C_0 ^\infty (M)$ is a core for $L$. ...
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2 votes
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Regularity bound

For $\Delta f_g = g$, can we prove that $f_g \in L^\infty_{\text{per}}(\Omega)$ and \begin{align*} \|f_g\|_{L^\infty_{\rm per}} \le c \|g\|_{L^\infty_{\rm per}} \end{align*} where $c$ does not ...
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5 votes
1 answer
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Smallest eigenvalue of Laplacian of periodic lattice after removing a vertex

Consider a 4-regular graph with $N^2$ vertices, which can be interpreted as a $N\times N$ lattice with periodic boundary conditions so that every vertex has degree 4. For an unweighted and undirected ...
Matt's user avatar
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1 answer
318 views

Lower bound on the first eigenvalue of the Lichnerowicz Laplacian on positive Einstein manifolds

Suppose $(M^n,g)$ is an $n$-dimensional Einstein manifold with $Ric=(n-1)g$. Let $\lambda$ be the minimal eigenvalue of the Lichnerowicz Laplacian $\Delta_L$ defined on all transverse-traceless ...
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Inequality involving the fractional Laplacian

Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \...
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Eigenvector to zero eigenvalue of general Laplacian

I was wondering what we can say about the eigenvectors of a matrix $A$ fullfilling $Ax =0$ where $A$ is symmetric with a diagonal equal to one and every row sums up to 0. Obviously this is a ...
RZA Chris's user avatar

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