Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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)

0
votes
0answers
6 views

Computing the 1-form eigenfunctions of the Laplace-de Rham operator on a mesh

I know that any scalar function $f$ on a 2-Riemannian manifold can be rewritten as $\sum_i a_i \phi_i$, where $a_i$ are proper coefficients and $\{\phi_i\}$ is a basis for the function space. Also, ...
0
votes
0answers
14 views

A problem regerding Laplace operator in line integral

$\bigtriangleup = \partial^2/\partial x^2 + \partial^2/\partial y^2$ denote the Laplce operator.Let $\omega =\{(x,y)\in $R$^2$$ :x^2+y^2<1\}$ denote the boundary of domain $\omega$. Consider the ...
1
vote
0answers
15 views

Laplacian equation on non-compact manifold

Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold. For the equation $$\Delta u=f,$$ for some $f\in L^2(M)$. Q How can we find a solution $...
-1
votes
0answers
13 views

What is significance of eigenvector of laplacian matrix?

The eigenvector of the Laplacian matrix is widely used in partition technique as spectral graph theory. What is the significance of 'K' eigenvector of the Laplacian matrix?
0
votes
0answers
26 views

integration by parts and laplacian operator

If -$\Delta u=F$ on $\Omega$ and I have -$e^{ax}\Delta e^{-ax}u=e^{ax}F$then I need to find $\int_{\Omega}-e^{ax}\Delta e^{-ax}u$ when I used the integration by parts then $\int_{\Omega}e^{ax}\...
0
votes
1answer
19 views

If the gradient of a vector is zero, does that imply that the laplacian of the vector is a null vector?

Suposse $\nabla \cdot \vec{u} = 0$ Does that imply that $\Delta \vec{u} = \vec{0}$ Thank you!
1
vote
1answer
15 views

Inequality of the laplacian involving the Ricci curvature

I am reading Eschenburg and Heintze's proof of the Cheeger-Gromoll splitting theorem. Lemma 1 states: Let $f\in C^\infty(M)$ with $||grad(f)||=1$. If c is an integral curve of the gradient, then it ...
0
votes
1answer
30 views

Heat equation in cylindrical coordinates at origin

I'm trying to solve a heat equation in cylindrical coordinates $$\dfrac{\partial u}{\partial t} = a \left(\dfrac{\partial^2 u}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial u}{\partial r} + \dfrac{1}{...
0
votes
1answer
46 views

Laplace's equation: separation of variables

Question: Let $(r,\theta)$ denote plane polar coordinates. Show that there are countably infinitely many $k \in \Bbb R$ for which $$\nabla^2 u=0 \qquad 1≤r≤2 \\ ku + \frac{\partial u}{\partial r}=0 \...
0
votes
0answers
18 views

Rotationally invariant Green's functions for the three-variable Laplace equation in all known coordinate systems

Green's function for the three-variable Laplace equation in Cartesian coordinates is $$\frac{1}{|\mathbf{r}-\mathbf{r'}|} = \frac{1}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}$$ It may be written in ...
0
votes
1answer
122 views

Distributional Laplacian of $\log|F(z)|$ Where F is Entire

Let $f(z) = \log|F(z)|$, where $F: \mathbb{C} \rightarrow \mathbb{C}$ is entire. Then $f$ defines a distribution on $\mathbb{R}^2$, and we want to show that its distributional Laplacian is $$\Delta f ...
1
vote
0answers
35 views

Definition of the weight $k$ hyperbolic Laplacian

I saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $\mathbb ...
0
votes
1answer
33 views

Laplacian in elliptical coordinates

I'm trying to calculate the laplacian in elliptical coordinates, just with the chain rule (because I don't know other method for doing this), but I have found difficulties to find the right expression....
0
votes
0answers
17 views

Trouble calculating the Laplace-Beltrami operator through this formula

Let $U$ be an open, bounded and connected subset of $\mathbb R^3$ with a $C^2−$regular boundary $\partial U$. For an arbitary $x_0 \in \partial U$ define the function $f:B(x_0,r) \cap \partial U \to \...
0
votes
1answer
49 views

Symmetric Boundary Conditions/Eigenvalues (PDEs)

Consider the following eigenvalue problem for the Laplacian $-\Delta u = \lambda u$ in $U$ $u + a \left(\frac{\partial u}{\partial v}\right)$ on $\partial U$ where $v$ is the outward unit normal to ...
3
votes
1answer
58 views

Hodge-$\star$ operator computation on a smooth two-dimensional manifold

Let $(x,y)$ be the local coordinates on a Riemannian manifold $M$ with $\dim(M) =2$. Let $\star$ denote the Hodge-$\star$ operator, and let $g = g_{ij}$ denote the Riemannian metric on $M$. I am ...
0
votes
1answer
28 views

Orthonormal basis for L2 (0,1) by using Laplacian's eigenfunctions.

A standard orthonormal basis for L2 (0,1) is given by the Fourier expansion, as described here, for example (Orthonormal Basis of $L^2$). On the other hand, it seems a standard result that the ...
1
vote
0answers
31 views

Reference for the relation of the Casimir element to the Laplace Beltrami operator

Wikipedia says, "If $G$ is a Lie group with Lie algebra $\mathfrak {g}$, the choice of an invariant bilinear form on $\mathfrak {g}$ corresponds to a choice of bi-invariant Riemannian metric on $G$. ...
2
votes
0answers
22 views

An upper bound of the first eigenvalue of Laplacian on a Riemannian manifold.

I'm reading the Cheng's thesis ""Eigenvalue Comparison Theorems and Its Geometric Applications," and the author obtains an estimate of eigenvalues of the Laplacian based upon his theorem: If $M$ is $...
0
votes
1answer
18 views

Neumann problem with zero average sobolev space

Let us considère the following Laplace-Neumann problem $-\Delta u=0$ with homogenuous boundary condition of type neumann, i:e $\frac{{\partial u}}{{\partial n}} = 0$. The variational formulation is ...
0
votes
0answers
24 views

Reference Request — Laplacian on manifold without boundary

I am looking for a discussion of Laplace's equation (or something similar) on asymptotically flat or asymptotically hyperbolic manifolds without boundary. In particular, I would like to see how ...
1
vote
1answer
29 views

Summability of double partial derivatives of $\frac{1}{|x|}$ in dimension $3$

I know the fact that its laplacian is equal to the Dirac delta function. However, is it true that its partial derivatives of order 2 belong to $L^{1}(\mathbb{R}^3)$? And how should I show that, in ...
0
votes
1answer
36 views

Derivation of the trace with Hessian matrix

Take the matrix $\Sigma\in\mathbb{R}^{n\times n}$ and the function $f:\mathbb{R^n}\rightarrow\mathbb{R}$ in $C^2$. 1) How can I compute the matrix derivation $$\frac{\partial (tr\left(\Sigma\Sigma^...
0
votes
1answer
21 views

Question about notation for rewriting integral of Laplace Beltrami operator

I was just reading about the Laplace Beltrami operator, which is a linear, second order, self-adjoint operator on a general Riemannian metric $g_\mu\nu$ space. $$\Delta A_{\mu}=\nabla^{\alpha}\nabla_{...
0
votes
0answers
19 views

Solving the Laplacian radial equation

I'm confused with the following bit for the radial equation derivation of the Laplacian in spherical coordinates: Once you solve the differential equation for $R$ using the Frobenius method, you note ...
5
votes
1answer
42 views

How is the Laplacian in spherical derived?

Suppose $\Phi$ is a function of $r, \theta$ and $\phi$. If I want to derive the Laplacian for this function, I would assume that.. $$\nabla ^2 \Phi = \nabla \cdot \nabla \Phi$$ And as, in spherical: ...
0
votes
0answers
24 views

Monotone eigenvector

Definition: We say that a vector $x\in\mathbb{R}^n$ is monotone if, $x_1\geq x_2 \geq \cdots \geq x_n$ or $x_1 \leq x_2 \leq \cdots \leq x_n$. Let $M=(m_{ij})_{1\leq i,j \leq n}$ be a matrix defined ...
1
vote
0answers
29 views

Question on the Laplace Beltrami operator expressed by mean curvature vector: Missing term

Disclaimer: The only course that I have seen in differential geometry is an introduction to differential geometry of manifolds and so I've never dealed with Laplace-Beltrami operator in the past. I ...
1
vote
0answers
35 views

Why is the laplacian a closed operator in $W^{2,p}(\mathbb{R}^n)$?

I have read that the laplacian is a closed operator in $W^{2,p}(\Omega)$,(that is, $\Delta : W^{2,p} \to L^p$) where $\Omega$ satisfies some conditions (I need the case $\Omega = \mathbb{R}^n$ so ...
0
votes
0answers
20 views

Using separation of variables to solve Laplace's equation without an ansatz

I'm trying to go ahead and solve Laplace's equation for a potential $V$ using separation of variables. The usual method is to write $V$ as a function of the three coordinates, so $V(x,y,z) = A(x)B(y)C(...
0
votes
1answer
18 views

Harmonic Function Analog

A function is harmonic if its non-mixed second partial derivatives with respect to each input sum to $0$. Is there a similar notion for the sum of a function's first partial derivatives, or for any ...
1
vote
1answer
35 views

Equivalence of norms in the space $H_\Delta(\Omega)$

Let $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary $\Gamma$. Consider the following space $$H_\Delta(\Omega)=\{u\in L^2(\Omega) : \Delta u \in L^2(\Omega)\},$$ with the ...
1
vote
0answers
16 views

Neumann Laplace eigenfunctions

Let $u_k, u_m$ be two Neumann Laplace eigenfunctions on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, corresponding to eigenvalues $\mu_k, \mu_m$ respectively. ...
1
vote
0answers
18 views

Eigenfunction of Dirichlet Laplacian on smooth domain in $\mathbb{R}^n$

I was reading about eigenfunctions of the Dirichlet Laplacian on bounded domains $\Omega \subset \mathbb{R}^n$. It seems that such eigenfunctions are real analytic in the interior of $\Omega$ and ...
0
votes
0answers
31 views

Pulling the Laplacian into an Integral (Poisson's equation)

So I'm reading a book about PDE and I have following question: For Poisson's equation in $\mathbb{R}^2: -\bigtriangleup u=f$ The solution is $u(x)=-\frac{1}{2\pi}\int_{\mathbb{R}^2} log(|x-y|)f(y)dy$ ...
1
vote
0answers
40 views

Do ground states of the higher-dimensional Schrodinger equation admit nodes?

Neglecting some irrelevant physical constants, the one-dimensional Schrodinger equation for a single particle is the eigenvalue equation for a particular second-order linear ordinary differential ...
0
votes
3answers
34 views

Partial derivative of nabla operation

Given that $$\nabla^2f = \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = \frac{\partial^2f}{\partial x'^2} + \frac{\partial^2f}{\partial y'^2}$$ $$ x = x' \cos \theta - y'\sin \...
5
votes
0answers
46 views

Using Separation of Variables to Solve a Laplace Eigenproblem

Let $r,\theta$ be the usual polar coordinates in $\mathbb{R^2}$, let $\Omega$ be the unit disc $r<1$ and recall that the Laplacian is given by $$\nabla^2u=\frac{1}{r}\frac{\partial}{\partial r}\...
0
votes
1answer
29 views

Derivative of nabla operation

$$ x = x' \cos \theta - y'\sin \theta \\ y = x' \sin \theta + y'\cos \theta $$ $$ \frac{\partial f}{\partial x'} = \frac{\partial f}{\partial x} \cos \theta + \frac{\partial f}{\partial y} \sin \...
1
vote
0answers
38 views

Weitzenbock formula for manifold on boundary

Let $(M,g)$ be a Riemannian manifold with boundary, we know that for one-form $A$ we have $$\Delta A=\nabla^*\nabla A+Ric(A).$$ Q Assume $A(\nu)=0$, for the normal vector field $\nu$. How to show ...
0
votes
0answers
33 views

Solving a 2D Laplace's Equation by boundary condition analyses

I have two main questions on this subject. 1 Firstly, I am asked to note the following boundary conditions for some Laplace equation: $$\Phi(x,0) = 0$$ $$\Phi(x,a) = V_0$$ $$\Phi(0,y) = 0$$ $$\Phi(...
0
votes
1answer
39 views

Nabla operation to cos and sin replacement

There are given equations, $$ x = x' \cos \theta - y'\sin \theta \\ y = x' \sin \theta + y'\cos \theta $$ $$ \frac{\partial f}{\partial x'} = \frac{\partial f}{\partial x}\frac{\partial x}{\...
3
votes
1answer
22 views

Eigenfunctions Laplacian - Bounding the Fourier Coefficients

Let $\Omega \subset \mathbb{R}^{N}$ be an open set with boundary of class $C^{\infty}$ and let $\{\lambda_{k}\}$ and $\{v_{k}\}$ be the eigenvalues and eigenvectors of -$\Delta$ with Dirichlet ...
0
votes
0answers
20 views

Laplacian of the position vector

Let $M$ be a $n$-dimensional Riemannian manifold (hypersurface) in $\mathbb{R}^{n+1}$, $X=(x_1, ..., x_{n+1})$ is position coordinate vector and $H$ the mean curvature of $M$. I'm having trouble ...
2
votes
0answers
40 views

Harmonic functions interpolation

Denote by $B = \{(x,y) \in \mathbb{R}^2|x^2 + y^2 < 1\}$ the open unit ball in $\mathbb{R}^2$, and by $S$ it's boundary, i.e. the unit sphere. For some $n > 0$ let $x_1,...,x_n \in B$, $y_1,...,...
2
votes
0answers
27 views

Correct form of the Laplacian on a 1-D ellipse, and it's solutions

I wanted to derive the Laplacian operator for a 1-D ellipse, and it seemed to me that there are two equivalent approaches: 1) Start with 2-D elliptic coordinates $$ x = a \cosh(\mu) \cos(\nu)$$ $$ y ...
2
votes
0answers
64 views

What is the necessary and sufficient condion for a laplacian to be zero?

Let $F$ be a function of $x,y$, and $z$ such that: $F(x,y,z)$. What is the necessary and sufficient condition for $\triangledown$$^2$$F(x,y,z)$=$0$, what does it signify? I am aware that if $d$ is a ...
0
votes
0answers
29 views

Laplace transform of Laplacian

I have seen the laplacian being used in the partial differential equations involving quantum mechanics, particularly some form of the Schrodinger equation. My question is simple: What is the Laplace ...
0
votes
0answers
21 views

Superposition of solutions to Boundary Value problem with multiple boundaries

I have a Laplace equation $\bigtriangleup u = 0$ with Dirichlet boundary condition at three separate segments $u_{\partial \Omega_1}=c_1$, $u_{\partial \Omega_2}=c_2$, and $u_{\partial \Omega_3}=c_3$....
1
vote
0answers
45 views

Second highest order coefficient of ordinary differential operator

Let $n$ be a natural number, $a_k(x)$ functions defined on an interval and $$Df(x)=\frac{d}{dx}(a_1(x)\frac{d}{dx}(a_2(x)\dots\frac{d}{dx}(a_{n}(x)\frac{d}{dx}f(x))\dots)$$ be an ordinary differential ...