Questions tagged [green-function]

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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 ...
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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 $...
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  • 443
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65 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 ...
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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 ...
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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 ...
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1 vote
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44 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-...
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2 votes
0 answers
43 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 ...
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Existence of Green function for an elliptic equation with a singularity

I am working in the hyperbolic space of constant curvature $-\kappa^2$ and dimension $d$, say $\mathbb{H}^d(\kappa)$ with the metric $g:=\frac{1}{1+\kappa^2r^2}dr^2+r^2g_{\mathbb{S}^{d-1}}$, $r$ is ...
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1 answer
99 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 ...
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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 + ...
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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 ...
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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 $L$;...
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1 answer
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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 ...
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3 answers
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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\...
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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 ...
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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 ...
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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 $|x-y|<1$?
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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^{...
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  • 1,037
3 votes
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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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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 ...
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5 votes
0 answers
57 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 ...
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3 votes
1 answer
422 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)\...
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1 vote
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 $u(x,0)=g(x), \forall x\in \Bbb R$...
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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 $f$. ...
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9 votes
3 answers
2k 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 ...
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2 votes
0 answers
180 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 ...
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2 votes
0 answers
57 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 ...
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3 votes
1 answer
204 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(...
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5 votes
0 answers
88 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_{...
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1 vote
0 answers
61 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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6 votes
0 answers
300 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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0 answers
108 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 ...
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4 votes
0 answers
133 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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1 vote
0 answers
79 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(...
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2 votes
1 answer
264 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 ...
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5 votes
1 answer
589 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 ...
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2 votes
1 answer
551 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)=(...
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1 vote
1 answer
236 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 \...
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1 vote
0 answers
95 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 ...
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9 votes
1 answer
617 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 ...
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  • 307
3 votes
1 answer
294 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 ...
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  • 590
3 votes
1 answer
214 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 ...
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4 votes
0 answers
523 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 \{...
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1 vote
0 answers
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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)....
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1 vote
0 answers
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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 \,...
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5 votes
1 answer
161 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\...
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  • 1,208
2 votes
0 answers
134 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 ...
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1 vote
0 answers
711 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 ...
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0 votes
0 answers
128 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 ...
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2 votes
1 answer
123 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,...
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