Questions tagged [laplacian]
The Laplacian matrix is the representation of a graph in matrix form.
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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 ...
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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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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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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 ...
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Curvature of the boundary vs. normal derivative of the first eigenfunction
Disclaimer. I posted this question in Math.SE, but it haven't received enough attention.
Let $\varphi_1$ be the first eigenfunction of the zero Dirichlet Laplacian in a planar bounded domain $\Omega$....
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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 ...
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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 ...
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Do Laplace-Beltrami eigenfunctions vary continuously with the metric?
I'm interested in Laplace Beltrami operators $$-\Delta_g:\ \ D(-\Delta_g) \longrightarrow L^2\left(M,\sqrt{|g|}dx\right)$$
on a smooth compact Riemannian Manifold (M,g). Let us fix a unique metric $...
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Green kernel vs fundamental solution
Let $L$ being the Laplacian for a given Lie group $G$. I would like to know what is the difference between the two notions in relation to the operator $L$:
The fundamental solution $\Gamma(x)$ of $L$;...
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Optimality condition of the harmonic form representatives of a homology class
In "Hodge theory on metric spaces, Smale et al." the $d$-th harmonic forms of the Hodge Laplacian $\Delta_d=\delta^* \delta+\delta \delta^*$ satisfying $\Delta_d(f)=0$ are claimed to be ...
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Products of eigenfunctions on compact Riemann surfaces
Let $M$ be a compact Riemann surface with genus $g\geq 2$, endowed with the Riemannian metric with constant sectional curvature $-1$. Let $f_1, f_2$ be two (global) eigenfunctions for the Laplace-...
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How to show that $\Delta W \leq −2(n − 4)V$?
I am reading a preprint and trying to understand the proof of Lemma 3.5. On Pg. 19 above eqn (3.49) the authors claim that $\Delta W \leq −2(n − 4)V$ where the functions $W$ and $V$ are defined below,
...
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Is the heat semigroup on a manifold the limit of the heat semigroups associated to a compact exhaustion?
Let $M$ be a paracompact Riemannian manifold, and $E \to M$ a Hermitian vector bundle endowed with a Hermitian connection $\nabla$. Write $M$ as an exhaustion $\bigcup _{j \ge 0} U_j$ with relatively ...
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Eigenvectors of graph Laplacian for spectral clustering
I have the following questions regarding the graph Laplacian for spectral clustering:
What is the intuition behind projecting the Laplacian (D-A, where D is the degree matrix and A is the affinity ...
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Are there any known results for the spectrum of $(-\Delta)^s/V^{p-1}$?
I am interested in generalizing some results known for the $\frac{-\Delta}{U^{p-1}}$ where $U$ is a Talenti bubble to the non-local operator $\frac{(-\Delta)^s}{V^{p-1}}$ where $U$ and $V$ are bubbles ...
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Relation between the eigenvalues of the weighted laplacian and fractional laplacian?
Consider the eigenvalue problem $-\Delta u = \lambda u\rho$ for $u\in \dot{H}^{1}(\mathbb{R}^n)$ with $n\geq 3$ and weight $\rho\in L^{n/2}(\mathbb{R}^n).$ Let $(\lambda_k, \psi_k)$ be the increasing ...
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Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?
Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim_{t \to 0} t \log k_{t}(x,y)...
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What's the full assumption for Laplacian matrix $L=BB^T=\Delta-A$?
Graph with no-selfloop, no-multi-edges, unweighted.
directed
For directed graph Adjacency matrix is a non-symmetric matrix $A_{in}$ considering indegree or $A_{out}$ considering outdegree. Degree ...
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Graph Laplacian definition
Assume that we have node indices $\{1,...,p\}$ and weights between the indices $w_{ji} = w_{ij}\geq 0$ for $i,j \in \{1,...,p\}$ with $w_{ii} = 0$. Then the Laplacian matrix is defined as $L:= \...
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Limit of $e^{-t(-\Delta)^\alpha}$ when $\alpha \to 1$
Let us consider the fractional heat semigroup $\left(e^{-t(-\Delta)^\alpha}\right)_{t\ge 0}$ for $\alpha\in (0,1)$ (the fractional power is taken in whole $\mathbb R^d$). Is there any result about the ...
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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 ...
4
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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 $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
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Eigenvalues of Laplacian and eigenvalues of curvature operator
Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
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Chip-firing clocks
Let $G$ be some outdegree-regular directed graph with $n$ vertices and let $H$ be the Laplacian of $G$, so that the rows of $H$ correspond to chip-firing moves. I’m interested in linear functions $f$ ...
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Deducing elliptic regularity using spherical formula for Laplacian
It is well known that if $\Delta u = f$ and $u\in H_0^1(U),f \in L^2(U)$ for some domain $U,$ then we have $\|u\|_{H^2(U)} \leq C\|f\|_{L^2(U)}$ for some constant $C.$ One way to prove this is to ...
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How does a quasi-isometry affect Poisson or Martin boundaries?
Take two graphs (of bounded valency) or manifolds (f bounded geometry) $G$ and $G'$. Assume there is a quasi-isometry $f:G \to G'$, and assume the Poisson or Martin boundary of $G$ is known, what may ...
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Neumann/Robin Laplacian semigroup well-known estimate
Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
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Show that the Laplacian on these domains is isospectral
Let $\Omega_i\subseteq\mathbb R^d$ be bounded and open, $A_i$ denote the weak Laplacian with domain $\mathcal D(A):=\{u\in H_0^1(\Omega_i):\Delta u\in L^2(\Omega_i)\}$ on $L^2(\Omega_i)$ and $$T_i(t)f:...
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Maximum principle of fractional Laplacian
Suppose $u$ is a sign changing classical solution of the fractional Laplacian $$ (-\Delta) ^{s} u = 0 \; \text{in } \Omega;
u=g \text{ in } \mathbb R^N -\Omega .$$
(a)Is it true that ...
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Estimate of the norm of the radial part of a function
Consider a function $u\in L^2(\mathbb R^N)$, and another function $\varphi$ which is the unique solution to the Poisson equation $\Delta \varphi = u$ vanishing at $\infty.$ We know that the radial ...
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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 ...
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The contractivity of the heat semigroup in $L^p$ spaces
Let $M$ be a Riemannian manifold. By functional calculus, it is immediate to show that the heat semigroup is a contraction in $L^2(M)$. I can also show that it is a contraction in any $L^p(M)$ with $p ...
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Harmonic functions on knot complements
In Axler's Harmonic Function Theory, he and his coauthors develop the theory of harmonic functions on spheres and discs by considering the restrictions of arbitrary polynomials on the sphere $S^{n-1} =...
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What is a random eigenfunction on the hyperbolic plane?
Is there an (invariant under isometries) notion of a random eigenfunction on the hyperbolic plane, for a given eigenvalue?
It is a reference request because the answer is probably positive and I even ...
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Laplacian of Fourier-like function
The question that follows has to do with the effects of a turbulent atmosphere on wave propagation. The structure function, $D(\vec{r})$, which is defined as,
\begin{equation}
D(\vec{r}) = \left\...
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Spectrum of Laplace-Beltrami with piecewise constant coefficients
By the Laplace-Beltrami with piecewise constant coefficients I means the operator $-div (f\, \nabla .)$ in the 2-sphere. Where $f$ is a piecewise constant function that takes two values $1$ and $a>...
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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 ...
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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 ...
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2
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From one eigenvector to many, in a very local graph?
Let $\Gamma$ be an undirected graph of bounded degree $d$ with $V = \{1,2,\dotsc,N\}$ as its set of vertices, and edges only between vertices that are at a distance $\leq M$ apart (where $M$ is much ...
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Laplacian on a manifold with two boundary components
I am interested in the Laplace equation on knot complements. The full complement of a knot $K$ is in $S^3$, but for compactness, we delete an open tubular neighborhood around $K$. The Laplace PDE on $...
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Fiedler vector, what else?
In the spectral analysis of a graph with 1 connected component, the first non-trivial eigenvector (corresponding to the non-zero smallest eigenvalue) is also called the Fiedler vector. This vector is ...
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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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Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)
I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it:
...
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Is $\Delta \phi$ monotone operator on $H^1(\mathbb{R}^d)$ for monotone $\phi$
Let $H^1(\mathbb{R}^d)$ be the usual Sobolev space and let $\phi: \mathbb{R} \to \mathbb{R}$ be a non decreasing Lipschitz function with $\phi(0)=0$. Is the operator $\Delta \phi $ on $H^1(\mathbb{R}^...
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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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Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett
( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .)
Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...
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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 ...
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Positive analytic semigroup
On $L^p(\mathbb{R}^+)$, we consider the following operator:
$$
Af:= f'',\qquad D(A):=\{u\in W^{2,p}(\mathbb{R}^+),u'(0)=0\}
$$
Now I want to know if this operator is a generator of a positive analytic ...