# 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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### 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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### 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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### 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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### 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  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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### 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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\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 $H^2:=\left\{u\in L^... 0 votes 0 answers 35 views ### Counter example: Sobolev embedding We know that$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})$, how can I find a radial ... 0 votes 1 answer 24 views ### Does$\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 ...
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 ...