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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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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differential-privacy: show $\epsilon$ -differentially privacy
In this problem we consider a sensitive dataset $x \in \{−1, 1\}^n$. We consider the bounded setting where neighboring n-dimensional datasets differ in one coordinate. $A$ mechanism is available that ...
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Laplacian of 1/r^n in the distributional sense
Let $\Omega = \mathbb{R}^3\setminus\{0\}$. Consider the function
$$
f_n \colon \Omega \to \mathbb{R},\quad \vec{x} \mapsto \frac{1}{\|\vec{x}\|^n}
$$
with $n \in \mathbb{Z}^{+}$.
I want to calculate ...
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Sharp constant in the $L^p$ regularity estimate?
Problem: Let us denote $\mathbb{W}^{2,p}(\mathbb{R}^2)$ the space of Sobolev functions in the plane. Let us denote with $\Delta$ the classic Laplacian operator. We know that there exists a constant $C&...
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Representation formula for the $\Delta^2 u =f$
I already know the representation formula of $\Delta u =f$ is the that $u(x)$ equal to the $\phi * f(x)$, where $\phi$ is fandamental solution.
Now I have wonder konw the representation formula of $\...
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Laplacian Operator for 2D Skew Coordinates [closed]
It is common to define a new coordinate system to help with mathematical manipulation of various kinds. But if the original system was subject to the Laplace Equation, how would that be maintained in ...
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L2 gradient solution time
There is something I don't understand.
Imagine I want to solve :
$$Min_{u}\int_{B(0,1)} \left| \nabla u(x) \right|^{2}+F(u(x))dx$$
It's L2 gradient flow is given by :
$$\partial_{t} u =2 \Delta u - F^{...
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Does the Dirichlet eigenvalue problem make sense on a sphere?
The Dirichlet eigenvalue problem on a given suitable domain $\Omega\subset\mathbb{R}^n$ asks one to find such function(s) $u$ and eigenvalue(s) $\lambda$ that
$$\begin{cases}\Delta u &= \lambda u\...
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Write the Laplacian operator in a particular coordinate system
I have to write the differential operator the Laplacian, divergence, gradient in the following curvilinear coordinate system:
$$\left \{ \begin{array}{rl}
x=(R_0+r\cos\theta )\cos\phi\\
y=(R_0+r\cos\...
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Coefficient conditions for square root of the Laplacian
I am confused about what's going on in the attached picture (from the introduction of Friedrich's Dirac Operators in Geometry). The author claims that for an operator $P$ to satisfy $P = \sqrt{\...
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Sum of derivatives of christoffels, vector laplacian on basis
One can define the connection laplacian on $TM$ as
$$
\Delta X := \sum_i \nabla_i \nabla_i X - \nabla_{\nabla_i \partial_i} X
$$
where $\{\partial_i\}$ give an orthonormal basis at a point, induce by ...
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Distributional Laplacian of Logarithmic function.
Here is the exercise:
Compute the distributional Laplacian $\left(\text{in }\mathbb{R}^2\right)$ of $d(x,y)=\ln\left(\|(x,y)\|\right)=\ln\left(\sqrt{x^2+y^2}\right)$. Relate your answer to $\delta$ ...
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Existence of a solution to the Dirichlet problem
Let $\Omega\subseteq \mathbb{R}^N$ be a regular domain (a bounded open set with $C^1$ boundary). Let $g\in H^{\frac12}(\Omega)$ and let
$$\DeclareMathOperator{\Dm}{d\!}
H^1_g(\Omega):=\{u\in H^1(\...
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solution of $\nabla^2 \phi = K\phi \nabla^2 \frac{1}{\phi}$
Is there any known analytical solution to the below equation?
$\nabla^2 \phi = K\phi \nabla^2 \frac{1}{\phi}$, where $K$ is a constant.
Assume spherical co-ordinates and spherical symmetry, i.e.,
$\...
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Is the kernel of the Laplacian fractional operator positive and us a Schwartz function?
Let $p_t(x):=\int_{\mathbb{R}^n} \mathrm{e}^{ix\cdot \xi}\mathrm{e}^{-t|\xi|^2}\,d\xi$ be the heat's kernel. within the properties of the kernel $p_t$, are fulfilled that
$p_t(x)>0$ for all $t\geq ...
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If $p$ is bounded and bounded away from $0$, can we find upper and lower bound for $\|\nabla\hat p\|^2+\Delta\hat p$?
Let $d\in\mathbb R^d$ and $p:\mathbb R^d\to(0,\infty)$. Moreover, let $\sigma>0$, $$\tilde p(x):=p(\sigma x)\;\;\;\text{for }x\in\mathbb R^d$$ and $$\hat p:=\frac12\ln\tilde p.$$
Question: Can we ...
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About the notation of $2$D Laplace operator
I'm reading a paper on $2$D discrete Laplace operator, and perhaps because it's an old paper, the notation in it really bothers me a lot. So can someone please explain it to me? For example, the ...
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Is $\sigma(-\Delta)=\sigma_{\mathrm{ess}}(-\Delta)$? Or under which conditions do we have this?
Let $\Delta: H^2(\mathbb{R}^n)\subseteq L^2(\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n)$
be the Laplace operator in the weak sense.
A Lemma in the book of Borthwick (Spectral Theory) says:
It is ...
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Parameter range below critical exponent
it is known that the critical exponent $\frac{2n}{n+2}$ in the theory of pde often times poses issues, e.g. when considering the $p$-Laplacian. What actually is the meaning of the exponent $p$ in the $...
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Connection laplacian and abstract index notation
Let $(M,g)$ be a (pseudo-)Riemannian manifold. I am having some struggles to relate two different approaches, one based on the abstract index notation, and the other one based on global, coordinate-...
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$L^{\infty}$ norms of Robin eigenfunctions
Let $\Omega\subseteq \mathbb{R}^2$ be a bounded Lipschitz planar domain. Suppose that $u$ is a Robin eigenfunction of the (negative) Laplacian on $\Omega$: $-\Delta u=\lambda u$ with $\partial_{\nu}u+\...
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Non-standard finite difference for a reaction diffusion system
I want to discretize the following reaction diffusion system:
$\frac{\partial u(x,y,t)}{dt}=\nabla ^2u+ u(1-u)-\frac{uv}{u+\alpha v}$,
$\frac{\partial v(x,y,t)}{dt}=d\nabla ^2v+ \delta v\left(\beta-\...
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What determines the coefficients of the laplacian filter?
I am building an approximately isotropic Laplace kernel based on the guidance in enter link description here. What I don't understand is the derivation process from (2) to (3). I don't know why the ...
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How to calculate the weights for Discrete Laplacian Operator?
I am following this paper enter link description here 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 ...
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What does $F$ stand for in this equation $\Delta f = \Delta_0 f + n \frac{\partial F}{\partial r} + \frac{\partial^2 F}{\partial^2 r}$?
I am reading Riemannian Geometry and Geometric Analysis by Jost, and this is one of the exercises concering Laplace operators:
3.1.a)
The Laplace operator $\Delta$ on $S^n$ on functions $f: S^n \to \...
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Properties of the inverse Laplacian operator
The inverse of the Laplacian is given by
$$(-\Delta)^{-1} u(x) = C \int_{\mathbb{R}^n} u(x-y) \frac{1}{|y|^{n-2}} dy$$
where $n$ is the dimension of $\mathbb{R}^n$.
I would like to learn more about ...
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Solving the 2D Poisson equation relying on 1D formulations on the x- and the y- axis
When solving the Poisson equation by means of centered finite difference, one usually ends up with a formulation of the type
$$
Au = f,
$$
where $A$ encodes the finite difference coefficients and the ...
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Is there a closed form for the $m$th Laplacian of a radial function?
It is well known the Laplacian of a radial function $f(r)$ on $\mathbb{R}^n$ is itself a radial function given by $ (\Delta f)(r) = f''(r) + \frac{n-1}{r}f'(r)$. One can iterate this to compute $\...
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Understanding why the Laplace operator is bounded with a domain $H^2(X)\cap H_0^1(X)$ when $X\subset\mathbb{R}^n$ is compact?
As of writing I have not managed to found a single literature reference nor mathexchange post which proves explicitly why the Laplacian can be interpreted as bounded linera operator in some Sobolev ...
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Are there harmonic 1-forms which induce harmonic circle-valued maps?
I was reading this question and it got me thinking more about harmonic maps.
A smooth circle-valued map $\varphi : M \to S^1$ from a Riemannian manifold is harmonic if $\varphi^*(d \theta)$ is in the ...
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Laplacian of a curve in a Riemannian manifold.
I have a Riemannian manifold $\mathcal{M}$ and a closed curve in such manifold (say $\mathcal{C}$) what I want to workout is the Laplace Beltrami operator of $\mathcal{C}$ (with the metric restricted ...
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How to do finite differentiation on diagonal values using 9-point stencil Laplacian filer.
When I was comparing the Laplacian kernels of 5-point and 9-point, I found that the 9-point kernel (such as the kernel with a value of 8 in the center and -1 in all 8 neighboring points) simply added ...
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Laplace-Beltrami operator in coordinates
I know that on a Lorentzian manifold the Laplace-Beltrami operator of a function $\phi\in C^\infty(M)$ is defined as
$$
\triangle_g \phi = \text{div}(\text{grad}\phi).
$$
Now I've come across the ...
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Computing the image of $f$ under the Laplace-Beltrami operator with metric tensor $g_{x,y}=(c + \alpha(x))dx^2 + \beta(x)dy^2$
Remark: This will be one an attempt at a "fail fast" post and it is likely that a standard textbook contains answer to this question. The problem is that I am so new to the Laplace-Beltrami ...
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How to solve Laplace equation on $\Omega$ with Robin boundary condition?
Let $\Omega$ be a "nice (say $C^1$)" open domain in $\mathbb{R}^n$.
Consider the equation $$ \Delta u = 0 \text{ in } \Omega$$ $$\alpha u + \frac{du}{dn} = f(x) \text{ on the boundary} .$$
I ...
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$L^2$ Correspondence Between Dirichlet and Neumann Domains
Let $\Omega\subset \mathbb R^k$ be compact with smooth boundary, $T$ denote the trace operator to its boundary (i.e. a continuous extension of the restriction operator), and define the following ...
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Inversion with respect to circle maps the quarter plane onto the lens-shaped domain
There is a problem on the book Introduction to Partial Differential Equations with Applications by E. C. Zachmanoglou and Dale W. Thoe on page 240:
Show that inversion with respect to $ S(0,1) $ maps ...
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Intuition on the cotangent weights in the discrete Laplacian
The discrete Laplacian matrix:
$L_{ij} = $
$
\begin{cases}
w_{ij} = \frac{1}{2} \left(cot\; \alpha_{ij} + cot \;\beta_{ij}\right) \text{if $j$ is adjacent to $i$}\\
-\sum_{j \in \mathcal{N(i)}} w_{...
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The divergence of gradient of an integral
Let given a $ C^2(\bar{\Omega}) $ function
$$u(r)=\int_{\Omega}(\nabla f).(\nabla g))dv$$
Then how to find $ \nabla^2 u(r)? $ Can i pass the Laplace operator inside the integral? If so how could I do ...
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Solution of Neumann problem for Laplace equation.
I have the following problem:
Let $ u $ be in $ C^2(\Omega) $ and in $ C^1(\overline{\Omega}) $, where $ \Omega $ is a normal bounded domain in $ R^n $, and suppose that
\begin{equation*}
\begin{split}...
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Laplace equation's solution as a "convex combination" of the Dirichlet data
I was playing around with numerical solutions of the Laplace equation with mixed boundary conditions:
\begin{alignat}{3}
\Delta u(x) &= 0, &\quad &x \in \Omega, \\\\
u(x) &= g(x), &...
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Algebra of quadratic elements of $\mathcal{U}(\mathfrak{so}(4,1))$
In [1] it is stated that the algebra of first-order symmetries of the Laplacian operator $\Delta$ on $\mathbb{R}^3$ is isomorphic to $\mathfrak{so}(4,1)$, spanned by 10 elements: three momentum ...
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Why is determinant present in the definition of the metric tensor version of the Laplace-Beltrami operator?
Suppose that our ambient space if $\mathbb{R}^n$. Then the metric tensor version for the Laplace-Beltrami operator is given by
$$
\begin{align}\tag{$\ast$}
\Delta_{LB}\, u = \dfrac{1}{\sqrt{\left\...
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Eigenvalues and Eigenfunctions of the Laplace operator an ellipsoid
I am currently trying to find the spectrum of the Laplace operator for ellipsoids in $\mathbb{R}^{3}$ with Dirichlet boundary conditions, i.e., I am looking for solutions to the following PDE,
$$
\...
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What means "Harmonic function is radially symmetric?"
I was wonder about following statement. "Harmonic function is radially symmetric."
What is correct meaning about above statement? I think it means 'Harmonic function is invariant under ...
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Associating a non-local differential operator to its integral representation
It is known that, in $\mathbb{R}^2$, we can define the non-local operator $\frac{1}{\Delta}$ with the Green function of the Laplace operator $\Delta$. This provides the non-local operator with an ...
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Is $-(I-\Delta)$ an dissipative operator in the context of $L^p$ spaces?
\begin{align}
\frac{d}{dt}u(t)&=-(I-\Delta)u,\quad t>0\\
u(0)&=f
\end{align}
with initial condition $f\in L^2$.
If $P:=-(I-\Delta):D(P):={H}^{2}\subset L^2\to L^2$ with $H^2:=\left\{u\in L^...
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Counter example: Sobolev embedding
We know that $H^1(\mathbb{R^2})$ is not embedded in $L^\infty(\mathbb{R^2})$. Using the fact that $u \in H^2(\mathbb{R^4})$ if and only if $u, \Delta u \in L^2(\mathbb{R^4})$, how can I find a radial ...
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Does $\Delta (g(t, \cdot) *f) = \Delta g(t, \cdot) *f$ hold if $f$ is bounded continuous without compact support?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
I would like to verify that
...
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Are boundary value problems with homogeneous time dependent boundary conditions never separable?
Consider the following initial-boundary value problem for the heat equation:
$$u_t(x,t)=u_{xx}(x,t),\ \ \ x\in[0,1] \\
u(x,0)=u_0\\
a(t)u(0,t)+b(t)u'(0,t)=c(t)u(1,t)+d(t)u'(1,t)=0$$
Meaning, our ...
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