Questions tagged [green-function]
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99 questions
4 votes
0 answers
93 views
Green function of the Dirac operator on spinor bundle
I am working with the following definition for the Green function of the Dirac operator $D$ on a spinor bundle $\mathcal{S}$ over a closed Riemannian manifold $(M^{n},g)$. Let $\pi_{1},\pi_{2}:M \...
- 41
0 votes
0 answers
148 views
Integral representation for sections satisfying the Dirichlet problem for the Dirac Laplacian
Let $(X, g)$ be a compact Riemannian manifold with smooth boundary $\partial X \neq \emptyset$, and let $(V, \langle \cdot, \cdot \rangle, \nabla, \gamma)$ be a Dirac bundle over $X$ in the sense of ...
1 vote
0 answers
82 views
On the tangent derivative of the Neumann's Green's function
It is well known that the Poisson kernel for the Laplace equation on the exterior of a disk can be obtained as the normal derivative at the boundary of the Dirichlet Green's function. Correspondingly, ...
- 41
1 vote
0 answers
100 views
Trotter formula for heat kernel on $\Omega\neq\mathbb R^n$
I was reading Brascamp and Lieb's paper 'On Extensions of the Brunn-Minkowski and Prekopa-Leindler Theorems, Including Inequalities for Log Concave Functions, and with an Application to the Diffusion ...
- 41
0 votes
0 answers
38 views
Decay of fundamental solution of differential operator
How can we find the decay of the fundamental solution of the 1D operator $L=D_x^{\alpha}+1$ for a real number $2>\alpha\geq 1$ where the differential operator $D_x^{\alpha}$ is defined by the ...
- 101
1 vote
0 answers
108 views
Connection between the Poisson kernel and the Dirac delta function for generalized Laplacians
It is well known that the Poisson kernel for the Laplace equation reduces to a Dirac delta function (an approximation of unity) at the boundary. Is there a similar relationship for generalized ...
- 41
1 vote
0 answers
91 views
Identity for the Green function with Neumann boundary conditions
Consider $\Omega\subset\mathbb{R}^n$ a smooth bounded open set. Let $G$ be the Green function for the Dirichlet boundary condition: for any $y\in\Omega$, the map $G(\cdot,y):\Omega\setminus \{y\}\to \...
- 605
1 vote
0 answers
90 views
How is the time decay of the heat semigroup in sectorial domains obtained?
I am having trouble understanding Lemma 20.10 from Souplet's book Superlinear Parabolic Problems, which provides a time decay estimate for the semigroup in a sectorial domain. I will first write the ...
- 687
4 votes
0 answers
139 views
Can someone explain the definition of time-ordered Wightman function on the IAS lectures?
The following question was asked in this post of physics stackexhange. Perhaps here an answer can be given. For clarity let me say that I am not asking for the general definition of time-ordered ...
- 93
7 votes
2 answers
276 views
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
239 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 ...
- 321
3 votes
0 answers
248 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 ...
- 109
2 votes
0 answers
84 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
251 views
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
267 views
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 $$\tag{1} G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
- 999
2 votes
1 answer
283 views
Green's function for a linear PDE initial value problem
For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem $$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{...
- 1,538
1 vote
0 answers
253 views
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 ...
- 5,477
0 votes
0 answers
137 views
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 $G_{\lambda}(x,p)$ ...
- 491
2 votes
0 answers
261 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/(...
- 2,855
1 vote
1 answer
298 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) = - \...
- 67
3 votes
1 answer
625 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 ...
- 3,745
1 vote
0 answers
138 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
695 views
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
348 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 ...
- 1,426
4 votes
1 answer
444 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 ...
- 1,426
1 vote
0 answers
78 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 $$\left( a_{ij}(x) \partial_i \partial_j + b_j(x) \partial_j + c(x) \right) G(x) = \delta(x)$$...
- 1,426
10 votes
1 answer
508 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
3 votes
1 answer
494 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) \ \ ...
- 273
1 vote
0 answers
131 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 ...
- 742
2 votes
0 answers
91 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 $...
- 541
0 votes
0 answers
102 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 ...
- 2,602
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 ...
- 279
2 votes
0 answers
232 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
80 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-...
- 1,086
2 votes
0 answers
106 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 ...
- 1,229
3 votes
1 answer
340 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
87 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
315 views
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
154 views
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$;...
- 322
3 votes
1 answer
438 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
628 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
193 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
245 views
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
76 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
250 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 $|x-y|<1$?
- 359
3 votes
0 answers
337 views
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^{...
- 1,465
3 votes
0 answers
217 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,\...
- 223
4 votes
0 answers
151 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 ...
- 3,644
5 votes
0 answers
149 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
1k 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