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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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For a compact Riemannian manifold $M$, $L^2(M)$ is spanned by the eigenfunctions of the Laplacian.

In some paper I read the following statement: For a compact Riemannian manifold $M$ and the corresponding Laplace-Beltrami operator $\Delta$ on $M$ we have, that $$L^2(M) = \widehat{\bigoplus_{\...
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Show that $\int_{\mathbb{S}^{N-1}}{\Delta_{\mathbb{S}^{N-1}}u(\theta)d\theta}=0$

Show that $$\int_{\mathbb{S}^{N-1}}{\Delta_{\mathbb{S}^{N-1}}u(\theta)d\theta}=0$$ Where $\Delta_{\mathbb{S}^{N-1}}$ is the Laplace operator on the sphere $\mathbb{S}^{N-1}$. My approach: Let $\...
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Question on the derivation of Laplace operator, and its application to polar coordinate system.

Let $\mathbf r = \left[ \begin{matrix} r & \phi \end{matrix} \right]^\top \; $be some curvilinear coordinates, with corresponding unit base column vectors $\hat {\mathbf h}_r \; $and $\hat {...
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REFEREENCE REQUEST for Non-Local Boundary Value problems

It would be really helpful if someone could suggest me any reference (Books or Papers) where I would find worked-out examples of Elliptic Boundary value problems (especially Laplace equation) with non-...
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Why do we need projection in the definition of the Stokes operator?

$\DeclareMathOperator{\div}{div}$ $\def\bu{\mathbf{u}}$ Let $D$ be the square $[0,1]^2$ and consider the following space: $$ V:=\{\bu: \bu\in H^2(D)^2, \div \bu=0, u|_{\partial D}=0 \}. $$ Introduce ...
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Riemannian geometry on discrete sets

We know that a discrete set (for example $S=\{a,b\}$) is a $0$-manifold. But how we can define a Riemannian metric on this set. Is this trivial and don't make sense ? Can we define the normal ...
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Laplace equation problem with numerous non-homogeneous BC(s) [Linear Superposition]

I need to solve the three-dimensional Laplace equation ($\nabla^2T = 0$) where $\nabla^2=\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$ in the domain ...
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Integration by parts using Green's formula, weak form of non-linear PDE

picture link for the problem Can anyone tell me how (8.2) is derived from (8.1) using integration by parts?
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Riemannian metric and Laplacian coming from an invariant form on the lie algebra

Let $G$ be a semisimple real Lie group. Let $\Delta \in U(\mathfrak g_{\mathbb C})$ be the Casimir element associated to the Killing form on the complexified Lie algebra $\mathfrak g_{\mathbb C}$ of $...
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Proof that $a\nabla^2 u = bu$ is the only homogenous second order 2D PDE unchanged/invariant by rotation

Looking for feedback and maybe simpler intuition for my proof of the theorem, shown below The statement of the theorem: Theorem Among all second-order homogeneous PDEs in two dimensions ...
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Inequality between infinity norm of Laplacian and Hessian

Let $M$ be a smooth compact riemannian manifold with Levi Civita connection and consider a smooth function $f: M \to \mathbb{R}$. Then the Laplacian of $f$ $$ \Delta f = \text{div} ( \text{grad} f) $$ ...
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Proof related to divergence theorem

The exercise asks me to prove that if $u:D\cup\partial D \rightarrow \mathbb{R}^{2}$ is ${C}^{2}$ on $D$ and we define $B_{\rho}$ a circle of radius $\rho < R$ such that $B_{\rho} \subset D$. Then ...
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Laplacian defined as an integration over a 3D ball.

In the book "Theory of unitary symmetry" by Rumer and Fet (see a piece of text at the bottom of this post) there is a proof which uses the following result (below I will translate as close as possible ...
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Proof that inhomogeneous solution is independent of a coordinate in Poisson equation

Suppose we have this Poisson equation $$ \nabla^{2}\phi(x,y,z)=\rho(x,y). $$ A solution would be of the form $$ \phi=\phi_0(x,y,z)+\phi_1(x,y) $$ where $\phi_0(x,y,z)$ is solution of the Laplace ...
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Two-dimensional Laplace equation with weird Robin BC

I need to solve the steady-state heat equation a.k.a. Laplace equation over a rectangle For $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$ defined on $x \in [0,a]$ and $y ...
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Integration by Parts with Laplacian over a Manifold

I am studying a text on differential geometry where it states during a proof that integration by parts is used to prove the following integral over a closed manifold: $$ \begin{split} &\int_M -\...
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Laplacian Operator + Closed graph theorem

I know that the Laplacian operator defined as $$\Delta:(L^2(\Omega),\|\cdot\|_{L^2(\Omega)}) \to (L^2(\Omega),\|\cdot\|_{L^2(\Omega)})$$ is unbounded. But under other settings like $$\Delta:(H^2(\...
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Evaluating Coefficients for a Fourier Series when Exponential terms are present [Approach needed]

On the last step of solving a three-dimensional Laplace equation,($\nabla^2T=0$) with BC(s) as $T(0,y,z) = T(L,y,z) = T_a$, $T(x,0,z) = T(x,l,z) = T_a$, $\frac{\partial T(x,y,0)}{\partial z} = p_c\...
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Green's function for $\Omega=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:x_2,x_3>0 \}$

Compute the Green's function for the Laplacian, for the region $$\Omega=\{ (x_1,x_2,x_3)\in \mathbb{R}^3:x_2,x_3>0 \}.$$ My approach is to use a reflection argument similar to the one used for ...
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Heat equation on a finite graph and computing a ratio

Let $\Delta$ denotes the Laplace operator with $-\Delta \phi = \lambda\phi$ on the compact manifold $(M,g)$. In a paper it is stated that the solution of the heat equation \begin{align} (\partial_t -...
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Separation of variables to solve Laplace's equation for the V in a cube 3D rectangular coordinates

Watch this videoLaplacian in 3D and tell me whether $C_{n,m}$ missing y in the argument of $\cosh{\sqrt{(\frac{n\pi}{a})^2+(\frac{m\pi}{a})^2}}$ in the denominator. So the final answer is $V_{(x,y,z)}=...
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Radial term of the Laplacian in polar coordinates

I am confused because I have seen two different expressions for the radial term in polar coordinates, in Wikipedia and other documents such as this one: http://ramanujan.math.trinity.edu/rdaileda/...
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What formulas of differential geometry am I missing?

I was reading a book on Riemannian analysis and the author assumes some formulas of differential geometry, which may be basic but I have a lack of knowledge on those. Specifically: $\int \delta (a(x)...
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Laplacian of $1/r$ in a tensor

As we know the $$\nabla^2(1/r) =- 4 \pi \delta^3(r).$$ However, I recently was readling an hydrodynamic book (An introduction to dynamics of colloids By J.K.G Dhont). The Oseen tensor is defined as: ...
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Two fluids flowing perpendicular in thermal contact with a Wall [Help to mathematically model]

I will try to describe briefly how I am modelling the problem. (Please bear with the length). The governing equation describing temperature for a block at steady state is $$\nabla^2 T = 0$$ where $\...
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Monotone eigenvector of the normalized Laplacian

Let $u_0 \geq u_1 \geq \cdots \geq u_{n-1}$ be positive numbers and define a matrix $n\times n$ by $M_{i,j} = u_{\left|i-j\right|}$ for all $i,j$. Let $L = I - D^{-1/2}MD^{-1/2}$ be the normalized ...
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Evaluating Fourier coefficients to complete a Laplace equation solution

While solving a PDE problem involving the Laplace equation in 3D, I arrive at the following summation relation when i substitute the only non-homogeneous boundary condition available $$ \sum_{m=1}^{\...
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Laplacian and Euler Lagrange equation

I try to find a relation between EL and Lap(EL) in polar coordinate for one variable function w(r), where Lap is laplacian and EL is Euler Lagrange equation. Please check the Maple code and help me to ...
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Laplacian with Integral BC(s)

I want to solve the three-dimensional laplacian $$\nabla^{2} T = 0$$ where $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ defined on $...
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3D Homogenous Laplace equation with integral boundary conditions

I have the 3D heat equation (Laplace equation) $$\nabla^{(3)}T_s=0$$ where $\nabla^{(3)}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ ...
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Laplace equaion with integral source terms

I have the following coupled PDEs: \begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c - ...
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On the solvability of the Dirichlet Problem $\Delta u =f$ for $f$ locally Holder continuous and $L^p$ for $p>n/2$

It's well known that if $\Omega$ is a bounded set and $f$ is locally Holder continuous on $\Omega$ and bounded, then $u = \int_{\Omega} \Gamma(x-y)f(y) \ dy$ is a classical solution to $\Delta u=f$, ...
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Example/Reference needed for Laplace equation coupled with another equation

I have been trying to solve a heat-exchanger problem where two fluids are separated by a conducting wall between them and the fluids flow perpendicular to each other. So i need to consider two ...
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Why is Laplacian ubiquitous?

What I am asking here is a moral question. Mathematically moral, don't bother physics. I mean, Euler's number is ubiquitous because, among all the exponentials, it alone is its own derivative with ...
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What is the general solution to Laplace's equation in 3d?

I've searched online but most make assumptions such as the solution can be separated into a product of functions which are each only depend on say x, y, or z. Or they assume the solution is only a ...
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Need help about references for 2D delta “function”

I am writing a paper about some numerical methods in the field of electrostatics and I remember from somewhere that the following equation is true: $$\left( \frac{\partial^2}{\partial x^2} + \frac{\...
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Solution to 2D discrete laplacian on a rectangle

I am attempting to solve the 2D discrete heat equation : Consider a function $f_{i,j}$ with $(i,j)\in[0,L+1]^2$. The values of $f_{0,j}$, $f_{i,0}$, $f_{L+1,j}$, $f_{i,L+1}$ are fixed as our boundary ...
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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, ...
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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 ...
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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 $...
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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}\...
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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!
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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 ...
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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}{...
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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 \...
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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 ...
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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 ...
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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 ...
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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....
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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 \...