Questions tagged [green-function]
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90 questions
7
votes
2
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Hölder continuity of Green function for simply connected domains
I read in a paper by E. A. Rakhmanov "Orthogonal Polynomials and S-curves" the following statement, in Lemma 9.9 therein, which I state here in my words.
Under the standard hypotheses for ...
user528012
1
vote
0
answers
126
views
Green's function of the conformal Laplacian
I am reading T. Parker, S. Rosenberg, "Invariants of conformal Laplacians", J. Differential Geom. 25(2): 199-222 (1987). I would like to understand how Green function changes if the metric ...
- 311
3
votes
0
answers
179
views
Green function of an elliptic operator
Let $L$ be an elliptic operator on $\Bbb C$ with
$$\DeclareMathOperator{\Img}{Im}
L^{-1}f(z)=\int_{\Bbb {C}}K_1(|z-w|^2)f(w) e^{i\cdot\Img\langle z,\overline{w}\rangle} \, dw
$$ where $K_1$ is the ...
- 113
2
votes
0
answers
60
views
Semigroup property in SPDEs
In fact, we know that a bounded linear operators on a Banach space $X$ satisfies the semigroup property, i.e. $$S(t+s)=S(t)S(s), \text{for every}\ t,s\geq 0.$$
However, in various literatures, I ...
- 57
4
votes
1
answer
202
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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 ...
- 131
3
votes
0
answers
202
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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
$$\tag{1}
G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
- 809
2
votes
1
answer
203
views
Green's function for a linear PDE initial value problem
For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem
$$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{...
- 1,538
2
votes
0
answers
162
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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 ...
- 5,407
0
votes
0
answers
115
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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 $G_{\lambda}(x,p)$ ...
- 471
2
votes
0
answers
110
views
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/(...
- 2,555
0
votes
1
answer
177
views
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) = - \...
- 57
3
votes
1
answer
426
views
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 ...
- 3,477
1
vote
0
answers
113
views
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) =...
- 91
7
votes
1
answer
506
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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+...
- 217
2
votes
1
answer
274
views
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 ...
- 1,150
4
votes
1
answer
394
views
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 ...
- 1,150
1
vote
0
answers
63
views
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
$$\left( a_{ij}(x) \partial_i \partial_j + b_j(x) \partial_j + c(x) \right) G(x) = \delta(x)$$...
- 1,150
10
votes
1
answer
436
views
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. ...
- 113
2
votes
1
answer
421
views
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) \ \ ...
- 253
1
vote
0
answers
103
views
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 ...
- 914
2
votes
0
answers
70
views
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 $...
- 537
0
votes
0
answers
82
views
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 ...
- 2,306
3
votes
1
answer
2k
views
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 ...
- 269
2
votes
0
answers
166
views
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
vote
0
answers
65
views
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-...
- 1,054
2
votes
0
answers
83
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 ...
- 1,219
3
votes
1
answer
290
views
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 ...
- 41
1
vote
0
answers
64
views
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
231
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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 ...
- 313
0
votes
0
answers
131
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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$;...
- 650
3
votes
1
answer
333
views
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 ...
- 446
4
votes
3
answers
394
views
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\...
- 43
3
votes
1
answer
156
views
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 ...
- 49
1
vote
1
answer
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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 ...
- 11
2
votes
0
answers
62
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 ...
- 163
0
votes
1
answer
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views
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 $|x-y|<1$?
- 329
3
votes
0
answers
265
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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^{...
- 1,305
3
votes
0
answers
193
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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,\...
- 233
4
votes
0
answers
109
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 ...
- 3,574
5
votes
0
answers
107
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
847
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 $c=1$)
$$[\partial_t^2 + 2\gamma\partial_t -\Delta_{3D}] u(t,x) = \delta(t,x) = \delta(t)\...
- 131
1
vote
0
answers
228
views
Diffusion equation solution using Laplace transform [closed]
Consider the operator
$$
L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t}
$$ with domain $D(L)={u} \in \Bbb R \times [0,+\infty )$, initial value $u(x,0)=g(x), \forall x\in \Bbb R$...
- 111
0
votes
0
answers
141
views
Green's Function for Fractional Laplacian on the Union of Two Balls
I have two disjoint open intervals $B_1, B_2 \subset \mathbb{R}$, and variables $0 < s < 1$ and $t \in B_1 \cup B_2$. I want to solve:
$$r_{B_1 \cup B_2}(\Delta^{s} f) = \delta_t$$ for $f$. ...
- 93
11
votes
3
answers
3k
views
Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix
I am looking for the fundamental solution of the following PDE
$$\partial_i (a^{ij}\partial_j u)=f$$
where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients.
I could find a ...
- 1,303
2
votes
0
answers
291
views
Fundamental solution of parabolic PDE with variable coefficients
Let us consider the parabolic operator
$$
\mathcal{L} = \partial_t - \nabla_x \cdot(a(x)\nabla_x)
$$
over a bounded domain $\Omega\subset\mathbb{R}^d$. The coefficient matrix $a(x)$ is elliptic and ...
- 21
2
votes
0
answers
74
views
Bessel decay for nonhomogeneous PDE
I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
- 53
3
votes
1
answer
268
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(...
- 197
5
votes
0
answers
105
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
88
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 ...
- 2,239
6
votes
0
answers
390
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{...
- 61