# Questions tagged [green-function]

The green-function tag has no usage guidance.

86
questions

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### Green's kernel estimates on finitely generated groups

I was reading a paper by W. Hebisch and L. Saloff-Coste titled "Gaussian Estimates for Markov Chains and Random Walks on Groups" where I came to know about certain bounds on convolution ...

2
votes

0
answers

177
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### Question about the formula of Green function of Laplacian on sphere

I'm reading a paper which said that
the Green function for $\left(-\Delta_g\right)^m$ on $2m$-dimensional closed manifold is of the form
$$\tag{1}
G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...

1
vote

1
answer

171
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### Green's function for a linear PDE initial value problem

For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem
$$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{...

2
votes

0
answers

137
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### The existence of a positive Green function for the Laplacian on $\mathbb R$

One can show explicitly and easily that the function $G(x,y) = \frac 1 2 |x-y|$ is a positive Green function for the Laplacian $\frac {\mathrm d ^2} {\mathrm d x ^2}$ on $\mathbb R$ (endowed with the ...

0
votes

0
answers

111
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### Existence of Green functions and some properties

Let $\Omega$ be a smooth domain in $\mathbb{R}^N$， $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ ...

2
votes

0
answers

92
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### Defining a metric on $\mathbb Z^n$ using Green's function for the simple random walk

Let $G$ be Green's function for the simple random walk on $\mathbb Z^n$ for $n\ge 3$, i.e., $G(x)$ is the expected number of visits to $x$ when the walk starts at the origin.
Define $d(x,y)=G(x-y)^{1/(...

0
votes

1
answer

138
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### Green's function in terms of logarithmic potential and energy of a measure

Let $\mu$ be a finite (Borel) measure on $\mathbb{C}$ with compact support $K := \mbox{supp } \mu$.
The logarithmic potential associated to the measure $\mu$ is
\begin{equation}
\Phi_{\mu}(z) = - \...

3
votes

1
answer

303
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### Any formula or estimates the Green function for the Laplacian in $3D$ periodic box?

Let $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ be the three-dimensional torus with sides identified. That is, I am considering the unit box $[0,1]^3$ with periodic boundary conditions.
In this case, I ...

1
vote

0
answers

103
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### Construct the square root of Green's function

The boundary value problem
\begin{align}
&\frac{\mathrm{d} }{\mathrm{d}x } \left( p(x) \frac{\mathrm{d} y(x)}{\mathrm{d}x } \right) + q(x) y(x) = f(x), \quad a \leq x \leq b \nonumber\\
&y(a) =...

7
votes

1
answer

352
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### Existence and estimates of Green's function on Riemannian manifold

In Yau and Schoen's differential geometry,in Ch5 before Thm 3.5,the author says
When $R$(scalar curvature of a manifold M)$>0$,there exists a unique Green's function $G$ to the operator $L=-\Delta+...

2
votes

1
answer

253
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### Heat conduction type equation in 4D

[I asked a similar question, Linear PDE, analytic continuation, Green's function and boundary conditions, and was told that a follow-up question should be a separate post.]
I'm interested in a ...

3
votes

1
answer

379
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### Linear PDE, analytic continuation, Green's function and boundary conditions

I'm looking at the linear PDE in 3+1 dimensions,
$$
\left[ -(\partial_t - \xi)^2 - \partial_k \partial_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1}
$$
Where $\xi$ is generally a ...

1
vote

0
answers

63
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### Behavior of Green's function $G(x)$ for $x\to 0$ for general second order PDE

Let's have a generic elliptic second order PDE in $n$-dimensions with a Dirac delta on the right hand side
$$\left( a_{ij}(x) \partial_i \partial_j + b_j(x) \partial_j + c(x) \right) G(x) = \delta(x)$$...

9
votes

1
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394
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### Propagators and PDEs

I have already asked this at MSE but did not get an answer.
In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. ...

2
votes

1
answer

360
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### Double integral in a polygon domain

I want to compute a integral of a polynomial $f(x, y)$ over a polygon domain $D$ of $n$ sides.
$$
I(f) = \int_{D} f(x, \ y) \ dx \ dy
$$
The vertex of this polygon are
$$\vec{p}_{i} = (x_i, \ y_i) \ \ ...

1
vote

0
answers

100
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### Regularity of the Robin function

I consider an analytic bounded domain $\Omega\subset \mathbb R^3$ and an the operator $L_a=-\Delta +a$ where $a$ is a function from $\Omega$ to $\mathbb R$. I assume the operator to be coercive, in ...

2
votes

0
answers

68
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### Fundamental solutions for weighted laplace equation

Consider the equation $L_w u = \frac{1}{w}\operatorname{div}(w\nabla u) =f(x)$, on $\mathbb{R}^n$ with radial weights $w(x)=w(|x|).$ Then I am interested in the fundamental solutions for the operator $...

0
votes

0
answers

81
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### Discontinuity of the Fourier transform of $ x \mapsto (1+ x^2)^{- \gamma/2}$ for $\gamma \leq 1$

Fix $\gamma > 0$. Let $\mathcal{F}$ be the Fourier transform and consider the function
$f(x) = (1+ x^2)^{- \gamma/2}$ for $x \in \mathbb{R}$. This function is in $\mathcal{S}'(\mathbb{R})$ and its ...

3
votes

1
answer

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### What's going on with the two-dimensional Helmholtz equation?

I've come to realize that its somehow harder to find results for this equation than for the three-dimensional one.
For example the wikipedia article on Green's functions has a list of green functions ...

2
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0
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### Green's function for elliptic PDE with potential

$\newcommand{\div}{\operatorname{div}}$Suppose I have an elliptic operator $\mathcal{L} u = -\div (A \nabla u) $ on some open set $\Omega \subseteq \mathbb{R}^d$ where here $A$ is uniformly elliptic ...

1
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0
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58
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### Intuition behind bound of second moment of Greens function by fractional moment

Consider the Hilbert space $ \mathcal{H} = l^2(\mathbb{Z}^d)$ for some dimension $d$ with basis given by the basisvectors $\{ \vert {x} \rangle \}_{x \in \mathbb{Z}^d} $.
Let $A$ be an either self-...

2
votes

0
answers

72
views

### The Green function for elliptic systems in two dimensions

I am reading some papers on Green functions of elliptic equations. Here the elliptic systems is stated as $ Lu=-\operatorname{div}(A\nabla u) $ where $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix ...

3
votes

1
answer

277
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### Definition of Martin kernels

Let $\Omega \subset \mathbb{R}^n$ $(n \ge 3)$ be a bounded $C^{1,1}$ domain and let $X$ be a Markov process in $\Omega$. My question is regarding the existence of the Green function and Martin kernel ...

1
vote

0
answers

63
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### Positive semidefinite fundamental solution to Schrodinger operator

Lets say $V : \mathbb{R}^n \rightarrow \mathbb{M}_d (\mathbb{R})$ is a $d \times d$ symmetric, positive semidefinite matrix function on $\mathbb{R}^n$ and consider the Schrodinger operator $- \Delta + ...

4
votes

1
answer

193
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### Elliptic equations in asymptotically hyperbolic manifolds

I am interested in reading about existence and regularity theorems for elliptic equations on manifolds with negative (constant) curvature outside a compact subset. I am aware of some results in this ...

0
votes

0
answers

125
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### Green kernel vs fundamental solution

Let $L$ being the Laplacian for a given Lie group $G$. I would like to know what is the difference between the two notions in relation to the operator $L$:
The fundamental solution $\Gamma(x)$ of $L$;...

3
votes

1
answer

291
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### References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$

I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a_{ij}= a_{ji}$ are only assumed to ...

4
votes

3
answers

368
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### Does the Green's function of the simple random walk on $\mathbb Z^d$ always vary locally?

Let $G_0(x)=G(x,0)$ be the Green's function of the simple symmetric random walk on $\mathbb Z^d$, $d\geq 3$. The question is whether $G_0$ must always vary locally, i.e., whether
$$
\sum_{\substack{y\...

3
votes

1
answer

149
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### Green potential and Hölder continuity

Assume that $U$ is the unit disk and $g\in L^{3/2}(U)$. Define $$f(z) = \int_{U} \log\left|\frac{z-w}{1-z\bar w}\right|g(w)\frac{du \, dv}{\pi}, \ \ w=u+iv.$$ Is there an elementary proof of the fact ...

0
votes

1
answer

184
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### Numerical methods for evaluating singular integrals

The Helmholtz decomposition for a vector field B contains both volume integrals and two boundary integrals (https://en.wikipedia.org/wiki/Helmholtz_decomposition). For brevity I show just one of the ...

2
votes

0
answers

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views

### Singularity of reproducing kernel for elliptic operator

Let $(M,g)$ be a smooth compact Riemannian manifold and dimension $2$, $\Gamma$ a smooth vector bundle over $M$, and suppose $L: W^{k,2}(\Gamma)\to W^{k-2,2}(\Gamma)$ is a second order strongly ...

0
votes

1
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211
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### Green function of the triangular kernel?

What is the green function of the triangular kernel $K$:
$$
K(x,y)=1-|x-y|
$$
where $x,y\in R$ such that $|x-y|<1$?

3
votes

0
answers

232
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### Proving the exponential decay of Green's function for the lattice $-\Delta+p$

The Green function $G(x,y) =G(x-y)$ of the discrete Klein-Gordon operator $-\Delta+p$ on $\mathbb{Z}^{d}$ is given by:
\begin{eqnarray}
G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}\frac{e^{...

3
votes

0
answers

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### Fourier transform of Green function and its derivative

Consider a real Sturm-Liouville operator $L$ on $[0,+\infty)$ and use the following notations : https://www.encyclopediaofmath.org/index.php/Titchmarsh-Weyl_m-function
Assume $a = 0$, $\alpha \in [0,\...

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0
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### Biharmonic heat flow on compact manifolds

Consider $\partial _t u (t,x) = -\partial _x ^4 u$ on a compact manifold, or even a special specific one like the torus.
Are there any estimates on the Green function (bihamornic heat kernel), for ...

5
votes

0
answers

98
views

### Expression for the (1+1)-dimensional retarded Dirac propagator in position space

Where an expression for the (1+1)-dimensional retarded Dirac propagator in position space can be found, especially including the generalized funcion supported on the light-cone?
In particular, is it ...

5
votes

1
answer

757
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### Green's Function for 3D Relativistic Heat Equation

On the Wikipedia page here , it states that the Green's function for 3D relativistic heat conduction (with $c=1$)
$$[\partial_t^2 + 2\gamma\partial_t -\Delta_{3D}] u(t,x) = \delta(t,x) = \delta(t)\...

1
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0
answers

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### Diffusion equation solution using Laplace transform [closed]

Consider the operator
$$
L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t}
$$ with domain $D(L)={u} \in \Bbb R \times [0,+\infty )$, initial value $u(x,0)=g(x), \forall x\in \Bbb R$...

0
votes

0
answers

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### Green's Function for Fractional Laplacian on the Union of Two Balls

I have two disjoint open intervals $B_1, B_2 \subset \mathbb{R}$, and variables $0 < s < 1$ and $t \in B_1 \cup B_2$. I want to solve:
$$r_{B_1 \cup B_2}(\Delta^{s} f) = \delta_t$$ for $f$. ...

9
votes

3
answers

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### Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix

I am looking for the fundamental solution of the following PDE
$$\partial_i (a^{ij}\partial_j u)=f$$
where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients.
I could find a ...

2
votes

0
answers

276
views

### Fundamental solution of parabolic PDE with variable coefficients

Let us consider the parabolic operator
$$
\mathcal{L} = \partial_t - \nabla_x \cdot(a(x)\nabla_x)
$$
over a bounded domain $\Omega\subset\mathbb{R}^d$. The coefficient matrix $a(x)$ is elliptic and ...

2
votes

0
answers

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### Bessel decay for nonhomogeneous PDE

I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...

3
votes

1
answer

262
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### Reconstructing the Green's function of an initial-value problem of partial differential equation

Consider a partial differential equation that is of the following form:
\begin{equation}
(-\partial_x^2+g(x))f(x, t)=i\partial_tf(x, t)
\end{equation}
where $g(x)$ is a real function. Suppose that $f(...

5
votes

0
answers

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### Paving property

In their famed paper (https://arxiv.org/abs/math-ph/0011053), Bourgain and Goldstein conjecture what they call the paving property:
Let $H_{jk}=\delta_{j,k+1}+\delta_{j,k-1}+v(\theta+j\omega)\delta_{...

1
vote

0
answers

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### Parabolic (heat) PDE Green's function spatial asymptote at infinity

Consider a general parabolic partial differential equation with its spatial dimensions on $R^n$, such as a heat equation, with the diffusion coefficient dependent on the spacial variables. Does its ...

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votes

0
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377
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### Linear PDE with non constant coefficients and properties of Green's Function

Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case
\begin{...

0
votes

0
answers

118
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### Green's third identity potential massive object

Consider a massive object occupying a volume $U$ with boundary $\partial U$. Let the gravitational potential inside be $V_{in}$ and outside $V_{out}$
Normally the gravitational field of a massive ...

4
votes

0
answers

145
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### Limit of Green's function as metric changes (S^2 -> R^2)

The Laplace-Beltrami operator is invertible on the space of 1-forms on $S^2$ (since $S^2$ has zero first betti number). Therefore it has an inverse, the Green's function. Now let the radius $r$ of $S^...

2
votes

0
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143
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### When are Green's functions causal convolution kernels

Let $L$ be a linear differential operarator acting on distributions over $\mathbb{R}$ and $G(t, s)$ be a Green's function, i.e., a solution to $LG(t, s) =\delta(t-s)$.
$G$ is said to be causal if $G(...

2
votes

1
answer

317
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### The study of dynamics of a polynomial vector field via Green's function methods

In the litterature, in particular in the papers on dynamical investigation of polynomial vector fields on the plane, are there some research devoting to study the Green's function for the PDE which is ...