Questions tagged [laplacian]

The properties of the Laplace differential operator, denoted $\Delta$ or $\nabla^2$, and defined as the divergence of the gradient. For Laplace equation, see (harmonic-functions)

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Dirichlet conditions Laplace equation

I have a simple question. Is it possible to write some analytical solution for the two-dimensional Laplace equation $x \in [0,L]$, $t \in (-\infty,\infty)$ \begin{eqnarray} \Delta \varphi(x,\tau)=0 \...
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Instantonic solution with zero boundary conditions

Let us consider the scalar field $\varphi(x,\tau)$, obeys Laplace equation \begin{eqnarray} \Delta \varphi=0, \end{eqnarray} and also a vortex condition (two instantonic configurations with charges $\...
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calculating partial derivative for the proof of mean-value property for function with laplacian =0

On page 152 of "Fourier Analysis" by ELIAS M.STEIN Lemma $2.8$ (Mean-value property) Suppose $\Omega$ is an open set in $\mathbb{R}$ and let $u$ be a function of class $C^{2}$ with $\...
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Modified Laplacian in oblate spheroidal coordinates.

Let us consider a slightly modified Laplacian $L$ on the unit sphere $\mathbb{S}^2$: \begin{align*} L&=\frac {1}{\sin\theta}\partial_\theta\sin\theta\partial_\theta+\frac{k(\theta)}{\sin^2\theta}\...
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Are solutions of $\Delta f \leq -a^2 f$ constant on a compact manifold?

Let $(M^n,g)$ be a closed (compact, without boundary) smooth Riemannian manifold and let $\Delta = -\operatorname{div} \operatorname{grad}$ be the induced Laplacian on $M$ (so that the eigenvalue ...
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Exercise on elliptic PDE

I'm solving the following exercise: find $u\in C^2(B^\circ(0,R))\cap C(B(0,R))$ such that: \begin{equation} \begin{cases} \Delta u=|x|^\beta \ \ \text{on} \ B^\circ(0,R)\\ u(x)=0 \ \ \ \text{on} \ \...
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Laplace-Beltrami operator of a vector field/function on an arbitrary curved surface

Given a surface $\mathcal S$, I want to compute the Laplace-Beltrami operator of a tangent vector field/function $\mathbf v: \mathcal S \to T\mathcal S$. Definitions/what I know: There exists a ...
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Help solving this heat differential equation -u'' = 𝛿(x-1.5) ; delta

1.4 Numerical Methods Solve the boundary value problem (BVP) by modeling the temperature in a well conducting metal rod of length 3 in the presence of a point heat source in the middle of the rod in ...
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Domain of 1D distributional laplacian

Let $\Delta_0$ be the operator $$ \Delta_0\colon f\in C_C^\infty(0,1)\mapsto -f''\in C_C^\infty(0,1)\,. $$ $\Delta_0$ is a symmetric operator of $L^2([0,1])$ and it's closure is $\Delta_{\min}\colon H^...
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The Laplacian in $L^{2}(\mathbb{R}^{n})$

$\DeclareMathOperator\Dom{Dom}$I'm trying to prove that $f \in L^{2}(\mathbb{R}^{n})$ and $z \in \mathbb{C} \setminus[0,\infty)$, then $G_{z}(H_{0})f \in \Dom(H_{0})$ and $$ (H_{0} -z)G_{z}(H_{0})f = ...
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Why Laplacian algebra instead of Adjacency algebra?

I'm reading the paper `New bounds for the max-$k$-cut and chromatic number of a graph' by E.R. van Dam and R. Sotirov. In this paper, the authors attempt to find a new bound for the max-$k$-cut ...
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Simplification of an integral involving a linear combination of tensor products of Legendre polynomials

I have this expression: $$W(x,y)=\sum_{i=0}^{I}\sum_{j=0}^{J}W_{ij}P_i(x)P_j(y)$$ Where $I, J, W_{ij}$ are constants and $P_i , P_j$ are Legendre polynomials. I want to compute the following equation: ...
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Laplace operator singularity at the poles

I have the following mixed boundary value problem $$ \begin{array}{l} \Delta \tau(\theta) = -1, \quad 0\leq\theta <\theta_{\max}, \\ \tau(\theta_{\max}) = 0, \\ \left. \dfrac{\partial \tau(\theta)}...
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Approximation of the $\Delta$ operator with harmonic functions

Let $D\subset\mathbb{R}^d$ be a bounded domain, and a fixed function $f\in C^2(D)$. I define $u_f^B$ as the solution of the Dirichlet problem on the ball $B$ of boundary conditions given by $f$, ie I ...
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A question an estimate for eigenvalues of eigenvalue of Laplace

Let $(M,g)$ be a closed Riemannnian manifold with dimension $m$, $\Delta$ be the Laplace operator and $\phi_i$ be the $i$-th eigenfunction of $\Delta$ with eigenvalue $\lambda_i$ and $\|\phi_i\|_{L^2}=...
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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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Laplacian in terms of integral of Hessian over unit sphere [duplicate]

Let $u \in C^2$. Somehow I tend to believe that the following identity $$ \int_{\partial B_1 (x)} \big\langle (\nabla^2 u) (x) y, y \big\rangle d\sigma_y = C \Delta u(x) $$ holds for some constant $C&...
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Prove that $\Delta u=F$ with these conditions has at most one solution

Let $\alpha >0$, and let $\Omega\subset \mathbb{R}^N$ be and open domain. I want to prove that the following problem has at most one solution. $$\Delta u=F \quad \text{in } \Omega$$ $$u=f \quad \...
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Deficiency indices of the Laplace operator on the unit disk

Let $\mathbb{D}=\{(x,y)\in \mathbb R^2: x^2+y^2\le 1\}$ be the unit disk. Let consider the Laplace operator $\Delta_0$ defined on $C_0^\infty(\mathbb D)$, the space of smooth functions $\mathbb D\to \...
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Elliptic regularity with right-hand side in $H^{-1/2}$

If $\mathbf{f} \in H^{1/2}(\Omega)$ for a bounded domain $\Omega$ with smooth boundary, does the elliptic regularity for the Laplacian guarantee that the solution to the elliptic problem in divergence ...
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What does $\nabla\left(\nabla \phi_{1}\right)$ mean?

I found this notation in a paper: $\nabla\left(\nabla \phi_{1}\right)$ where $\phi_{1}$ is a scalar. I can understand $\nabla \times \left(\nabla \phi_{1}\right)$ or $\nabla \cdot \left(\nabla \phi_{1}...
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Verify Fundamental Solution of the 3 Dimensional Laplace Operator

I would like to verify that $\frac{1}{4\pi |x|}$ is the fundamental solution of the 3 dimensional laplace operator so that $$ \triangle \frac{1}{4 \pi |x|} = \delta(x) $$ What I have tried: I think ...
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Laplacian of spherical coordinate

In Strauss PDE textbook, he derived the formula of Laplaican of spherical coordinate as follows: changing $(x,y,z)$ to $(r,\theta,\phi)$: Use te notation $r = \sqrt{x^2+y^2+z^2} = \sqrt{s^2+z^2}$, $s ...
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How to see that the Laplacian of a positive vector field results in a negative vector field (or negative component).

In a x,y,z cartesian coordinate system with the z-axis positive downwards, equations (1) and (2) are "equivalent", in that they both describe how the vector field $\mathbf{v}$ is governed by ...
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Norm on $H^2 \cap H^1_0$

Let $\Omega$ a smooth domain in $\mathbb{R}^3$. Is it true that the norm $$ |||u||| := \sqrt{||u||_2^2 + ||\nabla u||_2^2} $$ makes $H^2(\Omega) \cap H^{1}_0(\Omega)$ a Banach space? Context: I need ...
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Express Laplacian in polar coordinates

Part of this problem is asking to express $u$ in polar coordinates and express the domain and BCs to those coordinates. The PDE is the Laplacian on disc with BC $u=0$: $\Delta u+\lambda u=0, \quad$ ...
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Weak Laplacian is a closed map

Consider the map $\Delta : H^{2}(\Omega) \rightarrow L^{2}(\Omega)$ the weak laplacian. My professor has commented that this map is a closed map, that is, the graph $G = \{(u, \Delta u) : u \in H^{2}\}...
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Eigenvalues of symetric matrix and eigenvalues of symetric matrix plus rank one matrix

If $M$ is a symetric matrix with eigenvalues $m_1 \geq m_2 ... \geq m_n$ is there any connection between those eigenvalues and the eigenvalues $m'_1 \geq m'_2 ... \geq m'_n$ of the matrix $M' = M + xx^...
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Eigenvalues of graph laplacian question

I need some guidance on solving the following question I am stuck on: Let L be a graph Laplacian matrix of a graph G = (V, E) with n vertices. Denote the eigenpairs of L by {λi,ψi} for every i=1..n, ...
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Maximum principle for a strong solution to non-homogenous Laplace equation

I am searching for a reference for this apparently well known fact (the part below Theorem 1.1 in the picture i.e. the equation $(6)$): This screenshot is from https://math.aalto.fi/~astalak2/files/...
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Differential operator has changed into another one of laplacian operator. How this change occured?

$$ \begin{align} \text{Given equation}~:\nabla^2\mathbf E&={\partial\mathbf E\over\partial\mathrm{t}}+{\partial^2\mathbf E\over\partial\mathrm{t^2}}~~\text{where}~~E_z=0\\ \text{Equation which I ...
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let h be a harmonic function in a simply connected space $D$, prove that there exists an analytic $f=u+iv$ in $D$ s.t $u+v=h$

the question I'm struggling a bit with is: let h be a harmonic function in a simply connected space $D$, prove that there exists an analytic $f=u+iv$ in $D$ s.t $u+v=h$ I tried to solve by placing ...
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How do I show a function is the separable solution to a partial differential equation (The Laplacian)?

In the context of fluid dynamics I am given a partial differential equation (in polar coordinates) $$\frac{1}{r} \frac{\partial}{\partial r}\left( r \frac{\partial \phi}{\partial r}\right) + \frac{1}{...
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Separation of variables on a polar Laplace IVBP

I am puzzled by the generation of the two ODEs for a Laplace equation in polar coordinates. I have checked around on SE, and found some different and thus confusing answers. I list my attempt, The ...
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Interpretation of the Higher Order Laplace-Hodge Operator

As an operator on functions, one intuitive way to think about the Laplacian is 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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Integration by parts on compact, non-orientable Riemannian manifold with boundary

Let $(M,g)$ be a compact Riemannian manifold, not necessarily orientable or without boundary. Let $\mu$ be a normalized volume measure on $M$ and $u$ be a smooth function on $M$. In some notes that I ...
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Eigenvalues ​decreasing to zero

Let $(H, (\cdot,\cdot))$ be infinit dimensional separable Hilbert space. Also considerer $T : H \rightarrow H$ a non-null compact, self-adjoint operator such that $$ (T(v),v) \geq 0, \forall v \in H.\...
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Poisson equation on square with periodic boundary conditions up to first derivatives

A function $u$ on the square $\Omega=[0,L]\times[0,L]$ is said to satisfy periodic boundary conditions (PBCs) if $u(x,0)=u(x,L)$ and $u(0,y)=u(L,y)$ for all $x,y\in[0,L]$. Now consider the Poisson ...
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Calculating the laplacian of the scalar curvature

Given a Riemannian manifold $(M, g)$, the scalar curvature of the metric $g$ is $g^{i j} Ric_{i j}$ in local coordinates. Now I want to calculate the laplacian of the scalar curvature in local ...
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Spectral theory over 2D torus

I want to write a python simulation that requires the eigenfunctions and eigenvalues of the Laplace-Beltrami operator over the torus: $$ x=(R-rcos\theta)cos\phi , y=(R-rcos\theta)sin\phi , z=rsin\...
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How to find evolution equation of certain geometric quantities under Ricci flow

I want to find evolution of certain geometric quantities, like Weighted Laplacian $\Delta_\phi:=\Delta-\nabla\phi\nabla$, where $\Delta$ is Laplacian operator and $\nabla$ is gradient operator, under ...
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The Laplacian of the function $1/\|x\|$ in $\mathbb{R}^d$

Put $r_d(x)=\|x\|=(\sum_{i=1}^dx_i^2)^{1/2}$. The Laplacian of $1/r_d$ in $\mathbb{R}^d$ is given by $$\Delta(\frac{1}{r_d})=-\frac{d-3}{r_d^3}$$ as a direct calculation shows. Thus, $1/\|x\|$ is ...
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Laplacian on sphere after stereographic projection

I'm reading a paper about Yamabe equation. Consider the equation $$-\Delta u=\frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u\quad\text{on }\mathbb{R}^n.$$ After stereographic projection, the above equation can ...
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Show that this is a solution of this wave equation

Given the wave equation $\frac{1}{c^2} \frac{\partial^2}{\partial t^2}u=\Delta u$ where $\Delta u$ is the Laplacian operator and a function $g$ that's two times continuously differentiable, show that $...
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Is there an integral transform formula for $(-\nabla^2 + m^2)^{\frac{1}{2}}$ in three dimensions? What about its one sided inverse?

I have come across the following formula for the positive square root of the (negative) 3D Laplacian $$(-\nabla^2)^{\frac{1}{2}}[u](y) = C \text{ p.v. }\int_{\mathbb{R}^3}\frac{u(y)-u(x)}{\|y-x\|^4}dx$...
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Kernel of Laplacian plus a function

It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified ...
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Is $\nabla^2 f/f\in\mathbb{R}$ for a complex-valued $f\in L^2$?

Given a complex-valued function $f: \mathbb{R}^N\rightarrow\mathbb{C}$ with $f\in L^2$, which vanishes at the domain boundaries, is twice differentiable, and antisymmetric upon two-coordinate exchange—...
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Is the conformal Killing factor always an eigenvalue of the Laplacian?

Suppose $(M, g)$ is a pseudo-Riemannian manifold and $\xi_{\mu}$ is a conformal Killing field, i.e. $$ \nabla_{\nu} \xi_{\mu} + \nabla_{\mu} \xi_{\nu} = \kappa g_{\mu \nu} $$ for some smooth ...
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Explicit form of Harnack's inequality

PDE text Evans defines Harnack's inequality for non-negative harmonic functions as $$\sup_{B_{R}(0)}u\leq c \inf_{B_{R}(0)}u$$ where $c$ is a constant that only depends on the dimension $n$ such that ...
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How to calculate Laplacian matrix

I'm a newbie to this Laplacian matrix. We can have a Laplacian matrix for graph data. But can we convert any matrix to a laplacian matrix? Is it possible to have a Laplacian matrix for nongraph. data.
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