# 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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### If $\Delta \varphi =0$ then $\nabla \times \nabla \varphi =0$.

I was looking into a question about vector functions $\varphi$ satisfying the Laplace equation $\Delta \varphi =0$. I found that this answer to the question was given in most places I looked for, but ...
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### Multivariable Calculus, Laplacian problem

$u$ is a function of $x,y,z,$ and $t,$ but the Laplacian Equation does't involve $\frac{\partial^2 u}{\partial t^2}.$ Why is that? One explanation I have seen before is that the gradient is ...
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### What is the interpretation of the normalized adjacency matrix raised to a power?

What the interpretation of the normalized adjacency matrix raised to a power $K$? If we take the exponent of an adjacency matrix we get the number of walks, but what about if we do that for the ...
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### Solving this Laplacian Equation?

How do I solve the following? $\frac{∂^2\phi}{∂x^2}$ + $\frac{∂^2\phi}{∂y^2}$ + $\lambda^2\phi$ = $0$ To start of with, is it right to consider using the method of separation of variables, or is ...
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### Generalization of Gradient Using Jacobian, Hessian, Wronskian, and Laplacian?

I know there is a lot of topic regarding this on the internet, and trust me, I've googled it. But things are getting more and more confused for me. From my understanding, The gradient is the slope of ...
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### prove that a solution of Laplace equation is correct

Prove that $$U(x,y,z)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}A(k_1,k_2)exp(i(xk_1+yk_2)-zk_3)d{k_2}d{k_1}$$ is a solution for the Laplace equation in cartesian coordiantion by plugging it into ...
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### Is $\Delta \phi$ monotone operator on $H^1(\mathbb{R}^d)$ for monotone $\phi$ [migrated]

Let $H^1(\mathbb{R}^d)$ be the usual Sobolev space and let $\phi: \mathbb{R} \to \mathbb{R}$ be a non decreasing Lipschitz function. Is the operator $\Delta \phi$ on $H^1(\mathbb{R}^d)$ monotone? i....
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### Construction of purely radial harmonic function

So my question is In a spherical shell with $1<r<2$ construct a purely radial harmonic function v such that it takes the values $5$ and $4$ at $r=1$ and $r=2$ , respectively I know that I ...
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### Problem of separation of variables for Dirichlet boundary data of Laplace's equation in polar coordinates

Need help here with figuring out boundary conditions for this problem. Also, for (i), I do know a general way or method but here I am confused since from both equations how do I find out my desired ...
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### Find a bounded but non-constant solution of Poisson's equation

Consider Poisson's equation with homogenous Dirichlet boundary conditions \begin{equation} \begin{split} \Delta u&=g \text{ in } B_R(0)\\ u&=0 \text{ on } \partial B_R(0), \end{split} \...
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### Vortex solution of Laplace equation (XY model)

The hamiltonian of XY model, which is closely connected with BKT - transition is following: \begin{equation} H=\frac{J}{2} \int \text{d}^2 r \, \nabla \varphi \cdot \nabla \varphi, \quad \...
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### Approach to solve a Coupled system of PDE [Heat transfer in cylindrical coordinates]

I have the following two PDEs, which describe steady-state coupled heat transport between a externally heated axi-symmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it: ...
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### Solving $\Delta u=f(u)$

I am currently reading Partial Differential Equations on Riemannian manifold without boundary. I want to consider a PDE of the form $$\Delta u=f(u).$$ Can anyone tell me a reference which contains ...
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### Laplacian of Kelvin Transform - Confusion about notation

There is an exercise in Evans' book (chapter 2, problem #11). Suppose $u:\mathbb{R}^n\to\mathbb{R}$ is harmonic. Show that $\overline{u}(x)=u(\overline{x})|\overline{x}|^{n-2}=u(x/|x|^2)|x|^{2-n}$ is ...
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### Definition of Tensor Laplacian

Let $(M,g)$ be an orientable Riemannian manifold, $\nabla$ its Levi-Civita connection and $\epsilon$ its volume form. Let $f\in C^\infty(M)$ be a scalar field. Then we know that we can define its ...
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### A continuous function for which Poisson's equation has no C^2 solutions

I am trying to solve exercise 4.9 of Gilbarg and Trudinger, and in particular need to show that for the function $f(x)=\sum_{k=0}^{\infty}\frac{1}{k}\Delta(\eta{P})(2^kx)$ the problem $\Delta{u}=f$ ...
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### Rotational invariance of Laplacian operator

I was reading in Wikipedia about Rotational invariance and noticed that the two-dimensional Laplacian operator $\nabla^2 = \frac{\partial^2 }{\partial x^2} + \frac{\partial^2 }{\partial y^2}$ is ...
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### Are there any bounds on the decay rate of the spectrum of graph laplacians?

I am trying to analyze graph laplacians and in that context, I want to know about the spectral decay of graph laplacians i.e. how fast does the tail sum of its eigenvalues decays? Any research answers/...
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### Where can I find a proof of the fact that the Gaussian semigroup is generated by the Dirichlet Laplacian?

Let $H_0$ denote the Dirichlet Laplacian on $\mathbb{R}^d$ given by $H_0u=\Delta u$ for $u \in \text{dom}(H_0):= \{f \in H^1(\mathbb{R}^d): \Delta f \in L_2(\mathbb{R}^d) \}$. Further, for $t>0$ ...
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### Spectrum of the bi-Laplacian

I have the following operator: \begin{align*}D(A)&=\{f \in C[0,1]:f''''\in C[0,1] \text{ and } f^{(k)}(0)=0 \ \forall\ 0\leq k\leq 3\}\\Af&=-f''''.\end{align*} I'm trying to find the ...
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I am trying to solve the following Boundary Value Problem - a particle is moving on the surface of a sphere: $$\triangle y(\theta)=\frac{d²y(\theta)}{d \theta²} + \cot{\theta } \frac{dy(\theta)}{d\... 0answers 24 views ### Poisson equation on M\times \mathbb{R} with exponential asymptotic condition Let M be a closed Riemannian manifold, g be a smooth function with compact support on M\times\mathbb{R}. Equip M\times \mathbb{R} with the standard product metric. My problem : Determine ... 1answer 28 views ### Laplacian of complex function proof If f \in C^2(as a two variable real function) and \Delta f = \frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2} then$$\Delta f = 4 \frac{\partial^2f}{\partial z \partial \overline{...
Define the Fractional Laplacian of a Schwartz class funtion $u$ for $\lambda \in (0,1)$ as $$\mathscr{L}_\lambda[u](x):= PV \int_{|z|>0} \frac{u(x)-u(x+z)}{|z|^{d+2\lambda}}\,dz$$ I wish to prove ...