# Questions tagged [green-function]

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53
questions

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135 views

### 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,\...

**4**

votes

**0**answers

75 views

### 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

43 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 ...

**3**

votes

**1**answer

206 views

### 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**

vote

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107 views

### 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

56 views

### 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 $f$. ...

**5**

votes

**3**answers

667 views

### 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 ...

**2**

votes

**0**answers

134 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

55 views

### 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 ...

**1**

vote

**0**answers

53 views

### 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

78 views

### 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

54 views

### 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

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210 views

### 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{...

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99 views

### 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

111 views

### 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^...

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vote

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70 views

### 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

224 views

### 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 ...

**5**

votes

**1**answer

392 views

### Green's function for fourth order equation

I know the D'Alembert operator ${\frac {1}{c^{2}}}\partial _{t}^{2}-\Delta _{\text{3D}}$ has a well-known Green's function $\frac{\delta(t-\frac{r}{c})}{4 \pi r}$. This is very useful for studying 3D ...

**2**

votes

**1**answer

329 views

### hyperbolic “Green function” on a product of upper half-planes

Let $\Delta_{hyp}=\Delta_{hyp,1}=-y^2(\partial_x^2+\partial_y^2)$ be the hyperbolic Laplacian acting on functions of $\mathfrak{h}$ (the Poincare upper half-plane) and consider its resolvent
$$
R(s)=(...

**1**

vote

**1**answer

158 views

### How to determine the spectrum from the diagonal Green's function

Let $L: L^2(\mathbb{R}) \supseteq Dom(L) \rightarrow L^2(\mathbb{R})$ be a densely defined closed operator. Assume that the resolvent admits an integral kernel (Greens function) $G$, i.e. for $z\in \...

**1**

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**0**answers

91 views

### singular integral operators

Let $(\Omega,g)$ be a compact domain with smooth boundary and suppose that $g$ is smooth. Let $g_D$ and $g_N$ denote the Dirichlet and Neumann green functions for the Laplace-Beltrami operator.
My ...

**7**

votes

**1**answer

335 views

### Variation of the Green function with respect to the metric

Consider a (closed) Riemann surface and let $G(x,y)$ be the Green function of the Laplace-Beltrami operator. We can informally identify $G$ with the two-point correlation function for the Gaussian ...

**2**

votes

**1**answer

234 views

### Singularity of the heat kernel

The heat kernel in one dimension for the real line is given by the usual gaussian density function:
$$g(t,x,y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{(x-y)^2}{2t}}\, .$$
In particular, by differentiating ...

**3**

votes

**1**answer

146 views

### Methods to compute the Green's function for the 1D wave equation with nonsmooth coefficient?

I am seeking advice on the best available numerical methods to compute the Green's function for a 1D wave equation with rough coefficient.
Suppose that the coefficient $c(x)$ in the 1D wave equation ...

**3**

votes

**0**answers

508 views

### Convexity of the electrostatic energy on a Riemann surface

Let $M$ be a compact Riemann surface.
Let $\Lambda$ be a differentiable real $2$-form of integral one.
Let $G$ be the Green function associated to $\Lambda$, i.e.
$G: M \times M \to \mathbb R \cup \{...

**1**

vote

**0**answers

202 views

### Diagonal of Green's Function

I am looking to numerically calulate the diagonal of Green's function. I am interested in Green's functions of elliptic PDEs and in those that arise from stochastic processes (discrete and continuous)....

**1**

vote

**0**answers

67 views

### Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g \,...

**5**

votes

**1**answer

136 views

### Modified mean value property

Let $L=\Delta + c$ in 3 dimensions, where $c$ is a positive constant.
I met this modified mean value property of a solution $u$ of $Lu=0$ as
$$u(\xi)=\frac{\sqrt{c}\rho}{sin(\sqrt{c}\rho)}\frac{1}{4\...

**2**

votes

**0**answers

127 views

### Green's functions on linear subspaces and relations to boundary conditions

Consider the Laplacian $-\Delta$ on (in a suitable sense) twice differentiable functions subject to homogeneous Dirichlet boundary conditions $\mathscr{H}=\{f : f(0)=f(1)=0\}$. We can identify the ...

**1**

vote

**0**answers

607 views

### Green's function for fractional Laplacian

Consider the fractional differential equation
\begin{align}
D_{|x|}^\alpha u(x) +bu(x)=f(x)
\end{align}
with $0<\alpha<2$ on an unbounded domain. Instead of $D_{|x|}^\alpha$ one also often sees ...

**0**

votes

**0**answers

123 views

### smoothness of green's function in wave equation

I have a linear acoustic wave propagation originated from a monopole source, written as
\begin{align}
\mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega
\end{align}
where the ...

**2**

votes

**1**answer

117 views

### Does the green kernel converge as a series of functions?

Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following series,...

**1**

vote

**0**answers

278 views

### Existence of Green's functions for PDEs

Here is what I think I know: Given a symmetric linear differential operator $\mathcal{L}$ that is positive definite on a function space(/space of distributions) $\mathcal{H}$, we can find its inverse, ...

**4**

votes

**1**answer

468 views

### Analytical solution of diffusion PDE with Robin boundary condition

I need to find the analytical solution of the time-independent diffusion equation with constant coefficients on the unit disk $\Omega$ with subject to Robin boundary conditions. The formulation is as ...

**10**

votes

**2**answers

1k views

### Green's function of the Ornstein-Uhlenbeck operator

The Ornstein-Uhlenbeck operator $L$ is given by
$$
Lu = \Delta u- \frac{1}{2}x\cdot \nabla u.
$$
Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ ...

**9**

votes

**5**answers

771 views

### Reference request for a treatment of Schwinger–Dyson equations

Is there a treatment of Schwinger–Dyson equations with no mention of Green's functions? Is there perhaps a purely algebraic analog?

**1**

vote

**0**answers

291 views

### What's Known About the Green's Function to the 1D Diffusion Equation with Position-dependent Diffusion Coefficient?

Consider a one-dimensional diffusion equation
$$
C(x) \partial_t \Phi(t,x) = \partial_x^2 \Phi(t,x),
$$
on the interval $[0,1]$. The function $C(x)$ has a pole of order 1 at $x=0$ and a pole of finite ...

**3**

votes

**1**answer

200 views

### What can we say about the left inverse of the Green's function?

Let $\mathbb{D}$ be an self-adjoint elliptic operator of a compact manifold and $G(x,y)$ the Green's function of $\mathbb{D}$. By definition $G(x,y)$ is the right inverse of $\mathbb{D}$ in the sense ...

**1**

vote

**1**answer

312 views

### Is Green's function of an elliptic operator always symmetric?

Let $D$ be an elliptic operator of a compact Riemannian manifold and $G(x_0,x_1)$ the Green's function of $D$. Is $G$ always symmetric in variables $x_0$ and $x_1$, i.e. $G(x_0,x_1)=G(x_1,x_0)$? If ...

**0**

votes

**0**answers

165 views

### Green function and translational symmetry

I have met this problem in solving the classical field theory of a scalar field with a cubic term. I am able to solve exactly each equation, given in a form of odes, but this question escapes my ...

**5**

votes

**1**answer

308 views

### Green's function for *GJMS* operator

Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the ...

**0**

votes

**0**answers

214 views

### Solvable PDEs and their Green's functions

I have a class of PDEs of the form
$$
-\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0
$$
with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...

**3**

votes

**1**answer

365 views

### Is Poisson's kernel integrable?

Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic ...

**0**

votes

**1**answer

886 views

### direct proof that schrodinger's equation kernel corresponds to delta-function initial value [closed]

I want to show directly, that the kernel for the n-dimensional free linear schrodinger equation, if taken to time t=0, is dirac's $\delta $ function. I can show that the integral is constant, but it ...

**0**

votes

**1**answer

2k views

### Existence of Green's function and the Dirichlet problem

Let $U \subset \mathbb{R}^n$ be a bounded domain, and consider the following problem :
$$\left\{ \begin{array}{lcr} -\Delta u = 0 & & \text{in } U, \\ u = g & & \text{on } \partial U, \...

**2**

votes

**0**answers

1k views

### Nonlinear PDE and Green functions

This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like
$$
\partial^2\phi+V(\phi)=\delta^D(x).
$$
I do not know if a real ...

**0**

votes

**2**answers

435 views

### Estimates for Green's function

Let $n$ - dimension $\geq 3$.
Consider a compact manifold (M,g). Let $\epsilon_0$ denote the injectivity radius of $(M,g)$. Let $B_\epsilon(0)$ denote a geodesic ball of radius $\epsilon < \...

**1**

vote

**2**answers

616 views

### Green's function - Hyperbolic Riemann surface

A Riemann surface is said to be:
-Potential-theoretically hyperbolic if it has a non-constant bounded subharmonic function.
-Poincaré hyperbolic if it is covered by the unid disk.
Are this ...

**3**

votes

**3**answers

665 views

### Hyperbolic Riemann Surface

Let $X$ be a compact Riemann surface and $x\in X$.
Is $X - \overline{D(x,r_x)}$ hyperbolic?

**0**

votes

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684 views

### Green's function of coupled ODEs

For functions $a(x)$ and $b(x)$ and "sources" $S_1(f,g)$, $S_2(f,g)$ and $S_3(f,g)$ lets say one has the differential equations for functions $f(x)$ and $g(x)$,
$f' + af + bg = S_1(f,g) + S_2(f,g)$
...