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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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help me understand the Spectral theorem for the laplacian

Let $u:Ω→R$ be the solution of: $∆u=λu$ and $u=0$ on $∂Ω$ Let $S=$ The spectrum of $∆ =$ all the values of $λ$ for which there is a solution. If I understand correctly if $Ω$ is bounded, we have the ...
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Laplacian of arbitrary power of arbitrary norm

So I have the function $f : \mathbb R^d \to \mathbb R$ given by $f(x) = \lVert x \rVert_p^q$, where $\lVert \cdot \rVert_p$ denotes the $p$-norm on $\mathbb R^d$, given by $$ \lVert x \rVert_p = \left(...
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Determining weak solution for Dirichlet problem

Let $D$ be the unit disk in the plane and let $\Omega= D\setminus\{0\}$. The Dirichlet problem \begin{cases} Δu = 1 & \text{in } \Omega \newline u=0 & \text{on } \partial \Omega \end{cases} ...
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Solving the fundamental solution of the origin of $\Delta u=\Delta_x u+\frac{a}{r}u_r+u_{rr}=0$

The discussion starts from introducing a function $u(x,y):\mathbb{R}^n\times\mathbb{R}^{1+a}\to\mathbb{R}$ is radially symmetric, i.e. for $|y|=|y'|=r$, we have $u(x,y)=u(x,y')$. I am working on ...
Christy's user avatar
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What boundary for Laplace's equation on a square gives the roughest interior?

Consider Laplace's_equation on an $N \times N$ grid of squares with Dirichlet boundary conditions boundary $B \to$ interior $X,\ (4N - 4)$ boundary points $ \to (N-2)^2 $ interior points. As a ...
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Inconsistency in the Fundamental Solution Constant for the Laplacian in Higher Dimensions

I am working through a problem involving the extension problem for fractional Laplacians, and I've encountered some inconsistencies in the derivation of the fundamental solution and the associated ...
PowerPoint Trenton's user avatar
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Neumann Greens function for the exterior of a ball.

I would like some clarification on the Neumann Green's function for the following Poisson problem: $\nabla^2\phi(x) = f(x) \;\;\;\; x\in R^2/ B(0,1)$ $\hat{n} \cdot\nabla\phi = 0 \;\;\;\; x\in \...
antoniosgeme's user avatar
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Inverse of the Dirichlet Laplacian?

I have read somewhere that $(-\Delta)^{-1}u$ for $\Delta :H_0^1 \rightarrow H^{-1}$ is defined as the unique weak solution to $-\Delta v= u$ with $v=0$ on the boundary (assuming $U$ is some sufficient ...
Perelman's user avatar
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Equivalence of solutions to PDE in different coordinate systems

Wave equation in two dimensions can be solved either in Cartesian or polar coordinates. One can derive the expressions for Laplacian by putting in the coordinate transformations directly. However, ...
Sanjana's user avatar
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Sign of the first eigenfunction of the Laplacian

I am trying to prove that the first eigenfunction of the Laplacian operator in an open domain $\Omega$ does not change sign and that the first eigenvalue $\lambda_1$ is simple (with Dirichlet-boundary ...
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Dirichlet Problem with $L^p$ Boundary Data

I am seeking a proof of the following result related to the Dirichlet problem with $L^p$ boundary data. I am not quite sure how to approach the proof. Does anyone know where I might find such a proof ...
RiXaTorAgu's user avatar
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Checking 'No-resonance' condition for the eigenvalues of a discrete Laplacian matrix with Dirichlet boundary condition

1-D discrete Laplacian matrix (finite difference scheme) has eigenvalues as (page-2 in ref.): $$\lambda_j = sin^2(\frac{j\pi}{2(N+1)});\ j\in \{1, 2, ..., N\}$$ Where $N$ is the number of ...
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$\nabla^2 \phi - m^2 \phi =0$ and $\frac{\partial\phi}{\partial n} = g$ prove that $\phi$ is unique

Let $V$ be a region in $\mathbb{R}^3$ with a closed surface $S$. Consider a function $\phi$ in $V$ that satisfies $$\nabla^2 \phi - m^2 \phi =0$$ where $m\ge 0$. If $\frac{\partial\phi}{\partial n} = ...
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2D Laplace equation analytical solution

I am trying to solve a simple Poiseulle Flow in 2D in Cartesian coordinates numerically and analytically. For the analytic part, I am stuck at the following: Suppose we have a 2D Laplace equation $$ \...
kirkos73's user avatar
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Laplace equation in 2D with all 0 boundary conditions

I have a pretty basic questions for which I can't find the answer elsewhere. Suppose we have a 2D Laplace equation $$ \frac{\partial^2 u(x,y)}{\partial x^2}+\frac{\partial^2 u(x,y)}{\partial y^2}=0$$ ...
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Invariance of the Laplacian under orthogonal transformations

I am struggling to solve the following question. Consider the function $u(x, y)$ which is twice-differentiable in its arguments and in arguments $\xi$ and $\eta$ obtained by the following linear ...
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How does the Laplace operator act on a functional determinant?

Consider the functional $n \times n$ matrix $$ D_n = \begin{pmatrix} f_{1,1} & f_{1,2} & \ldots & f_{1,n}\\ f_{2,1} & f_{2,2} & \ldots & f_{2,n}\\ \vdots & \vdots & \...
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Why $0$ must be an eigenvalue of any Laplacian matrix?

I asked my teacher one question: Why $0$ be an eigenvalue of any Laplacian matrix, $L$? He tells me below the text: Since the sum of entries along a row/column of $L$ is $0$, $\mathrm{rank}(L)\leq n$...
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Weak convergence in $L^p$ spaces their dual

It is well known that the dual space of $L^p$ is $L^{p^\prime}$ for $p\neq \infty$ and that these spaces are reflexive. Now assume you have an operator $L^{p^\prime} \rightarrow (L^{p^\prime})^*, x\...
Perelman's user avatar
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Solution of $V'(t)=AV(t)+V(t)A^T+\sigma^2I_m$, where $A$ is the discretized Neumann-Laplacian

I'm considering a PDE involving the Laplacian on $[0,1)^2$. I'm discretizing the problem using the finite difference approach. The resolution discretized Laplcian opertor $A$ with Neumann boundary ...
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A uniform bounded in a bounded $\Omega\subset \mathbb{R}$

I’m in trouble with some inequality. I want to prove that $$\|\Delta u-\frac{c_1}{\epsilon^3}\int_{\mathbb{R}}J(\frac{x-y}{\epsilon})(u(y)-u(x))dy\|_{L^{\infty}(\Omega)}\leq C\epsilon^\alpha,$$ where $...
Luiza Camile's user avatar
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Laplacian of a function as the limit over the ball

I've been trying to prove the following identity $$ \lim_{r \to 0}\frac{1}{r^2}\left(\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)} u(y)dy - u(x)\right) = \frac{1}{2(N+2)} \Delta u(x), $$ where this is done in $\...
Thomas Petit's user avatar
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What is the Laplacian of the Gradient? Is $\boldsymbol{u}(\nabla\cdot\nabla p) = \nabla (\boldsymbol{u}\cdot\nabla p)$?

I am supposed to find out whether for a scalar function $p$ and a divergence-free vector function $\boldsymbol{u}$ we have that $$\nabla\cdot\Big [\boldsymbol{u}(\nabla\cdot\nabla p) - \nabla (\...
user1313292's user avatar
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Are there isospectrally equivalent exotic spheres?

Let $X$ and $Y$ be two different exotic spheres. Are there metrics $g$ and $h$ on $X$ and $Y$, respectively, such that the laplacians of $(X,g)$ and $(Y,h)$ have the same spectrum?
discretephenom's user avatar
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Hyperbolic paraboloids as solutions to Laplace's equation

I decided to try my hand at solving Laplace's equations in as many ways as I could, and I've come across the result that non-rotated conic sections (those which do not depend on $xy$ arise naturally ...
Lagrangiano's user avatar
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Boundedness of the Laplacian Eigenfunctions

I have a doubt regarding the Laplacian eigenfunctions $\left\{\phi_n\right\}_{n=1}^\infty$ with Dirichlet boundary conditions. I know that the functions form an orthonormal basis in $L^2$ and an ...
mathmath's user avatar
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Correlation Between Laplacian And Convexity Around Point In R^D Euclidean Space [closed]

So, in 1D Euclidean space, the Laplacian of a function is simply the second derivative of that function. In this case, the sign of the Laplacian at extremum points will tell us the convexity of that ...
BurgerMan's user avatar
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Principal eigenfunction of $\Delta$ on rectangular patch of $S^d$

Let $S^d \subset \mathbb{R}^{d+1}$ be the unit sphere. Using $d+1$-dimensional spherical coordinates, fix $$\begin{align*} 0 \leq \varphi_{1,1} &< \varphi_{1,2} \leq 2\pi \\0 \leq \varphi_{2,1} ...
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Linearized equation of PDE

I am finding some trouble in calculating the Linearized equation for even Laplacian. The $p$th Laplacian is defined as $-\Delta_pv= \text{div}(|Dv|^{p−2}Dv)$. I know the linearized $p$ th Laplacian ...
Mayank's user avatar
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Adjoint of the identity is the fractional Laplacian?

I want to show that the adjoint of the identity $i: H_0^s(\Omega)\rightarrow L^2(\Omega), x\mapsto x$ where $s>0$ is arbitrary is the fractional Laplacian i.e. $(-\Delta)^{s}$ where $(-\Delta)^s$ ...
Perelman's user avatar
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The positive Laplacian is indeed the negative Laplacian

I know this question sounds like a joke. And it probably is:). I found it kind of annoying, but also interesting, to call $-\Delta=-\sum_{j=1}^n\partial^2_{jj}$ "the positive Laplacian" as ...
Liding Yao's user avatar
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Global $L^{2}$ solution of Laplace equation in $\mathbb{R}^{3}$.

Does there exists a global and non-trivial solution $u\in L^{2}(\mathbb{R}^{3})$ to the Laplace equation $\Delta u=0$? Using the spherical symmetry of the problem, one can consider the usual solution ...
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Two Neumann eigenfunctions with parallel gradients along the boundary of a manifold

Let $(M, g)$ be a compact Riemannian $3$-manifold with boundary. Is it possible to exist two independent Neumann eigenfunctions $u, v \in C^{\infty}(M)$ associated to the same eigenvalue $\mu > 0$ ...
Eduardo Longa's user avatar
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Notational Ambiguity: Covariant Derivative

Let $M$ be a smooth manifold and $\nabla$ the Levi-Civita connection. Now, I am a bit puzzled by a serious notational ambiguity, namely for the second covariant derivative. To explain myself, let us ...
B.Hueber's user avatar
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Method to produce polyharmonic functions from harmonic ones

Let $F:\mathbb{R}^2\to\mathbb{R}$ be a harmonic function (it can be thought as one component of a holomorphic function), take $m$ odd and let $f:\mathbb{R}^{m+1}\to\mathbb{R}$, $$f(a,x_1,\dots,x_m)=\...
Giulio Binosi's user avatar
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Proving two local coordinate formulae of Laplace Beltrami-operator are equivelant

I am trying to prove two formulations of the Laplace-Beltrami operator $\Delta$ are equivelant: $$\Delta f = \frac{1}{\sqrt{|g|}}\frac{\partial}{\partial x^i}\left(\sqrt{|g|} g^{ij} \frac{\partial f}{\...
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Question about proof of fundamental solution for Laplacian - Teorem 2.17 Folland, G. Introducction to PDE

I'm reading Introduction to PDE of Folland, G. but I'm stuck in the following theorem: My question is about the $n=2$ case. I tried to do the same argument of $n>2$ but since $N$ and $|\log|x||+1$ ...
matdlara's user avatar
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Integration of laplace operator

Given two functions $f(x,y)$ and $g(x,y)$ defined on $D =]0,a[ \times ]0,b[ \subset \mathbb{R}^2$. $$$$ And given that $$\int _D f(x,y).g(x,y) dxdy =0$$ I have to show the following : $$\int _D \...
Ada Az's user avatar
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Taking the laplacian out of the integral

Just under equation 3.40 in the book The Boundary Element Method for Engineers and Scientists (Second Edition) by John T. Katsikadelis, in the context of solving a Poisson Equation $\nabla^2 u(P) = f(...
Cedric Martens's user avatar
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Question about the fomula of Green function of Laplacian on closed manifold.

I'm reading a paper which said that the Green function for $\left(-\Delta_g\right)^m$ on $2m-dimension$ closed manifold is of the form $$\tag{1} G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
Elio Li's user avatar
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Notation for the d'Alembert operator

From the Wikipedia page on the d'Alembert operator it is stated that equivalent ways of writing the d'Alembert operator are as follows, $$\begin{align}\Box =\eta^{\mu\nu} \partial_\nu\partial_\mu = \...
Sirius Black's user avatar
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Spectrum of discretized Laplace operator with homogeneous Neumann B.C.s

Consider the Laplace operator over a 1D domain with homogeneous Neumann B.C.s. The cell-centered Finite Volumes discretization of this operator on a uniform grid looks as follows, which is a well-...
Nicola's user avatar
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1 answer
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Spectrum of 4th order discretized Laplace operator

Consider the Laplace operator over a 1D domain, with homogeneous Neumann boundary conditions. I have discretized this operator using the cell-centered Finite Volume method on a uniform grid of size $h$...
Nicola's user avatar
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Why are the eigenfunctions of the Laplacian on a region with spherically symmetric boundary condition are not spherically symmetric?

For example the eigenfunctions of a spherically symmetric membrane can be found in https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane. Then again I sometimes see people in Physics saying ...
TheFibonacciEffect's user avatar
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Constant Laplacian with Dirichlet condition

I start saying that it's clearly an easy problem, but I'm stupidly stuck, so be kind please. The problem is: $ \mbox{Given}\ \Omega \in \mathbb{R}^n\ \mbox{bounded and }\ v \in C^2(\Omega)\cap C^1(\...
ManneredPizza's user avatar
2 votes
1 answer
115 views

Is $\text{d}^{\dagger}\text{d}$ or $\text{d}\text{d}^{\dagger}$ by itself a valid operator?

If I simply consider just one of the combinations $\text{d}^{\dagger}\text{d}$ or $\text{d}\text{d}^{\dagger}$ both of them take from $\Omega^r(M)\to \Omega^r(M)$ for some manifold $M$. But do they ...
Dr. user44690's user avatar
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1 answer
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Analytical Solution of a Laplace Equation with Given Boundary and Initial Conditions

I'm trying to solve the following Laplace equation analytically: \begin{align*} \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} &= 0 \end{align*} Subject to the boundary ...
AM production's user avatar
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Dimensional analysis of Laplacian

Given a function $$f(x, y): \mathbb{R} \left[kg \right] \times \mathbb{R} \left[K \right] \mapsto \mathbb{R} \left[m \right]$$ where the units of the variables $x, y$ and of the function $f(x, y)$ are ...
jordi's user avatar
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Terminology: functions proportional to own Laplacian

Is there a generic term for functions that are proportional to their own Laplacians, and/or for basis sets composed of such functions? Simple examples would include the well-known Fourier ...
G_B's user avatar
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4 votes
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Why does my solution to $\Delta u = \lambda u$ contradict the regularity theorem?

I am confused about the regularity theorem for Laplacian. It states that if we take a weak solutions of $\Delta u = \lambda u$, then $u$ must be a smooth function. But I cannot understand this ...
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