# 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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### 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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### 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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### 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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### 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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### 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\...