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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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1-D Laplacian on an ellipse

I would like to know if there is a simple formula for the 1-D Laplacian $\Delta$ on an Ellipse $$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1.$$ So far I've considered the following ...
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Nth-order Laplacian

Mind that I'm coming from a mostly physics background, so this may in fact be common mathematical notation that I simply haven't come across in my own field. I've seen the symbol $\nabla^2$ "applied" ...
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Dirichlet Spectrum of Laplacian In 2D Annulus

What are the eigenvalues and eigenfunctions of the Euclidean Laplacian for the annulus in 2D bound by radii r_1 < 1 < r_2 if we consider zero Dirichlet boundary conditions? I’m aware that this ...
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What does the superficial part of the Laplacian in spherical coordinates mean?

I am reading Alan Turing's "The Chemical Basis of Morphogenesis" article, http://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf, and I can't understand where the last expression of the ...
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34 views

Laplacian commutes with covering maps

I believe that this should hold true: Result. Let $\pi: M \to N$ be a Riemannian covering ($M$,$N$ closed). Then $\Delta_M \circ\pi^* = \pi^* \circ \Delta_N$. This should be true, but I could not ...
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Boundary regularity for classical solutions of Poisson's equation

I have seen results concerning the uniqueness of solutions of Poissons equation $\Delta u = f$ with Dirichlet boundary conditions, but they always seem to ignore existence, with the excuse that it is ...
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35 views

Prove an equation concerns about $(\Delta f)\circ g$.

Suppose $g:[0,\infty)\times\mathbb{R}\to\mathbb{R}^2,\;(r,\varphi)\mapsto(r\cos\varphi,r\sin\varphi)$. $X$ is open in $\mathbb{R}^2\backslash H$ where $H=\{(x,y)\in\mathbb{R}^2\;\big|\;x\leq0,\;y=0\}$....
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Under what conditions the operator $-\sum \partial_{x_i} a_{ij} \partial_{x_j}$ is self-adjoint?

Define: $$ \mathcal{L} = -\sum_{i,j} a_{ij} \partial_{x_i} \partial_{x_j}, $$ where $a_{i,j}=a_{j,i}$. If $a_{i,j}=\delta_{i,j}$ the operator is the Laplacian which is known to be self-adjoint in the ...
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$\Delta \mathbf n = -2 \mathbf n$ on the Euclidean sphere

Let us consider the Euclidean two-sphere, defined by the embedding in the three dimensional Euclidean space as $$ \mathbf n \cdot \mathbf n = 1\,, $$ where $\cdot$ denotes the standard scalar product. ...
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Computation of the laplacian of an isometric immersion

Say that $X:M \to \mathbb R^3$ is an isometric immersion of an oriented riemannian surface (oriented $2$ dimensional riemannian manifold). I understand there holds a vector-valued equation, namely $$ ...
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Graph Fourier Transform - Intuition of Eigenvalues of the Laplacian

first of all I don't have a solid math background, so a less general, but easier answer is preferable. In "The Emerging Field of Signal Processing on Graphs" by Shuman et. al. they describe how the ...
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1answer
41 views

What is the change of Laplacian operator under small coordinate transformation? [closed]

Consider the Laplacian operator $\nabla^{2}=\frac{1}{\sqrt{g}}\frac{\partial}{\partial q^{i}}\left(\sqrt{g}g^{ij}\frac{\partial}{\partial q^{j}}\right)$. How does the Laplacian operator transform ...
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Inferring the Lipschitz constant of a vector Laplacian, given Lipschitz constants of the second and third derivative

I have a function $f: \mathbb{R}^d \to \mathbb{R}$, which is three times continuously differentiable. I know that the Lipschitz constants for the second and third derivative of this function are $\...
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Adjoint of the Coupled Covariant Derivative on Spinors

I want to understand the proof for a $C^0$ bound of solutions to the Seiberg-Witten equations. Among other places, it can be found in Kronheimer, Mrowka: "The genus of embedded surfaces in the ...
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Hodge theory: $\Delta \alpha = 0$ iff $d\alpha = d^* \alpha = 0$ on a noncompact manifold?

Let $M$ be a Riemannian manifold (connected, oriented). One can define the co-differential $d^* : \Omega^k(M, \mathbb{R}) \to \Omega^{k-1}(M, \mathbb{R})$ even if $M$ is not compact (for example use ...
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Limit of Laplacian of the distance at the origin

Let $p$ be a point in a Riemannian manifold $M$ and $d_p$ be the distance from the point $p$. Prove that $\lim_{x\rightarrow p}\Delta d_p(x)=\infty$ I can easily prove it in $\mathbb{R}^n$. But for a ...
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How to find the laplace transform of $\cos(\sqrt t)$?

I tried solving for the transform using the same method the book uses to find laplace transform for $\sin(\sqrt t)$ which is, by writing the Maclaurin's expansion for $\sin(\sqrt t)$ and then using ...
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44 views

When do the eigenvectors of a Laplacian matrix form a basis?

Eigenvectors do not always form a basis. When do the eigenvectors of a Laplacian matrix form a basis? When the associated adjacency matrix is symmetric? Why?
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Proving that two variational problems are equivalent

Let $\Omega$ be an open set of finite measure. Let $\lambda_1(\Omega)$ be the first Dirichlet eigenvalue for the Laplace operator, i.e. $$- \Delta u = \lambda_1(\Omega) u, \ \ \ \ \ \text{in} \ \...
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Gaussian curvature with Laplacian

In a lot of papers and books, I have seen the following expression of Gauss Curvature in $2$-dimensional surfaces with a conformal metric $$\overline{g} = e^{2u}g$$ $$K - \overline{K} e^{2u} = \...
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Weitzenböck identity for $TM$-valued differential forms

Let $M$ be a Riemannian manifold, and let $\nabla$ denote its Levi-Civita connection. We have two second order differential operators $\Gamma(T^*M \otimes TM) \to \Gamma(T^*M \otimes TM)$: The ...
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Trace and Det of Laplacian on the rectangle

I consider the eigenvalue problem $\Delta \varphi = \lambda \varphi$ with the Dirichlet boundary condition $\varphi|_{ \partial \Omega}=0$ on the rectangle $\Omega= [0,l] \times [0,m]$. By using ...
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Show that $\int_{S_r}{y_{i}y_{j}d\sigma(y)}=0$ on the sphere $S_{r}(x)$.

Let $S_r(x)$ the sphere of radius $r>0$ centered at the point $x\in\mathbb{R}^{n}$, that is $$S_{r}(x)=\{y\in\mathbb{R}^n : |x −y| = r\} $$ Let $\sigma$ be the $(n-1)$-volume on $S_r(x)$, and ...
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Periodic boundary conditions, general dimension, sets and spectral properties of $-\Delta$ - reference recommendation

Let's consider the eigenvalue problem $-\Delta u = \lambda u$ on the interval $[0,1]$ with periodic boundary conditions: $u(0)=u(1),$ $\frac{du}{dx}(0) = \frac{du}{dx}(1).$ Similar conditions could be ...
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$u=f􏰀(\frac{x}{y})$ as a solution to $\Delta u = 0$

Find the solutions of $\Delta u=0$ of the form $u=f􏰀(\frac{x}{y})$ Since $\Delta u = u_{xx} + u_{yy} = 0$, I differentiated and got the following. $$f'' + \frac{2xy}{y^2 + x^2}f' = 0$$ I'm a little ...
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Solutions to $\Delta u = u_{xx} + u_{yy} + u_{zz} = 0$ that only depend on r

Find all the solutions of $\Delta u = u_{xx} + u_{yy} + u_{zz} = 0$ in three dimensions that depend only on $r = 􏰃x^2 + y^2 + z^2$, the radial variable in polar coordinates. Use the following ...
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Constructing a tridiagonal Laplacian matrix

Is it possible to create a Laplacian matrix of the form: $$\left(\matrix{ & a&0&0&0&0\\a&&b&0&0&0\\0&b&&c&0&0\\0&0&c&&d&...
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Laplace Equation For Infinite Plate

I initially put it into the form of \begin{align} X'' + \lambda^2 X &= 0 \\ Y'' - \lambda^2 Y &= 0 \end{align} which has solutions \begin{align} X &= a_n \cos(\lambda x) + b_n \sin(\...
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Bounded Laplacian

Let $M$ be a Riemannian manifold and fix a point $p_0 \in M$. Denote by $d : M \to [0, \infty)$ the "distance to $p_0$" function. It is well known that $d$ is not smooth at $p_0$. Can we say that $\...
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If $u(r, \theta)$ is a solution of Laplace’s equation show that $u(\frac{1}{r}, \theta)$ is also a solution.

Suppose that $u(r, θ)$ is a solution of Laplace’s equation. Show that $u(\frac{1}{r}, θ)$ is also a solution. So far, I know that if $u$ satisfies Laplace's equation, then $$\Delta u = u_{rr} + \...
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Green's Equation for $\Delta u=0$ with specified conditions.

I need to find the Green's function for $$\Delta u=0$$ in the upper half plane subject to $$G(x,0;x_0,y_0)=0=\lim_{y\rightarrow\infty} G.$$ I know that $\Delta G=0$ and I have tried solving this with ...
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Eigenvalues of 1D laplacian discretized matrix

I have the matrix resulting form the finite difference discretization and now I should find its eigenvalues. the text of the exercise is Hint: Write out a typical equation of the system $Aw = λw$ ...
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$ W_{0}^{2}(\Omega)=\{ f\in W_{0}^{1}(\Omega):\Delta f\in L^{2}(\Omega)\}? $

Let $\Omega\subset\mathbb{R}^{n}$ be an open bounded domain. Let $W^{2}\left(\Omega\right)$ be the usual Sobolev space $$ W^{2}\left(\Omega\right)=\left\{ f\in L^{2}\left(\Omega\right):f,\partial_{i}...
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Three dimensional Laplace equation with constant Temp. on one face. [Solution not satisfying BC]

The governing differential equation is $$\nabla^2 T=0 \tag A$$ The boundary conditions for this problem are as foll0ws: $$T(0,y,z)=T_{hi} \tag {1A}$$ $$T(L,y,z) = T(x,0,z) = T(x,l,z) = T(x,y,0)= ...
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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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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 in 3D with numerous Non-Homogeneous BC(s) [Strategy Check]

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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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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37 views

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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1answer
48 views

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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31 views

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\...