# 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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### Dirichlet conditions Laplace equation

I have a simple question. Is it possible to write some analytical solution for the two-dimensional Laplace equation $x \in [0,L]$, $t \in (-\infty,\infty)$ \begin{eqnarray} \Delta \varphi(x,\tau)=0 \...
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### Modified Laplacian in oblate spheroidal coordinates.

Let us consider a slightly modified Laplacian $L$ on the unit sphere $\mathbb{S}^2$: \begin{align*} L&=\frac {1}{\sin\theta}\partial_\theta\sin\theta\partial_\theta+\frac{k(\theta)}{\sin^2\theta}\...
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### Are solutions of $\Delta f \leq -a^2 f$ constant on a compact manifold?

Let $(M^n,g)$ be a closed (compact, without boundary) smooth Riemannian manifold and let $\Delta = -\operatorname{div} \operatorname{grad}$ be the induced Laplacian on $M$ (so that the eigenvalue ...
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### Exercise on elliptic PDE

I'm solving the following exercise: find $u\in C^2(B^\circ(0,R))\cap C(B(0,R))$ such that: \begin{equation} \begin{cases} \Delta u=|x|^\beta \ \ \text{on} \ B^\circ(0,R)\\ u(x)=0 \ \ \ \text{on} \ \...
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### Laplace-Beltrami operator of a vector field/function on an arbitrary curved surface

Given a surface $\mathcal S$, I want to compute the Laplace-Beltrami operator of a tangent vector field/function $\mathbf v: \mathcal S \to T\mathcal S$. Definitions/what I know: There exists a ...
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### Help solving this heat differential equation -u'' = 𝛿(x-1.5) ; delta

1.4 Numerical Methods Solve the boundary value problem (BVP) by modeling the temperature in a well conducting metal rod of length 3 in the presence of a point heat source in the middle of the rod in ...
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### let h be a harmonic function in a simply connected space $D$, prove that there exists an analytic $f=u+iv$ in $D$ s.t $u+v=h$

the question I'm struggling a bit with is: let h be a harmonic function in a simply connected space $D$, prove that there exists an analytic $f=u+iv$ in $D$ s.t $u+v=h$ I tried to solve by placing ...
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### Poisson equation on square with periodic boundary conditions up to first derivatives

A function $u$ on the square $\Omega=[0,L]\times[0,L]$ is said to satisfy periodic boundary conditions (PBCs) if $u(x,0)=u(x,L)$ and $u(0,y)=u(L,y)$ for all $x,y\in[0,L]$. Now consider the Poisson ...
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### Calculating the laplacian of the scalar curvature

Given a Riemannian manifold $(M, g)$, the scalar curvature of the metric $g$ is $g^{i j} Ric_{i j}$ in local coordinates. Now I want to calculate the laplacian of the scalar curvature in local ...
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