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Let $\mathcal{D}\subset\mathbb{R}^n$ be a bounded Euclidean domain with Lipschitz (possibly smoother) boundary and consider an Elliptic Dirichlet problem of the form \begin{align} \mathcal{L} u + \...
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Let $Q$ be unbounded convex domain in $\mathrm R^n$ and $G(x,y,t)$ be the Green function of the first (or second or third) boundary value problem for the heat equation $u_t-\Delta u=0$ in the cylinder ...
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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 \...
benny's user avatar
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
CLR's user avatar
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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 \...
Dorian's user avatar
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $$\tag{1} G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
Elio Li's user avatar
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For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem $$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{...
Abhishek Halder's user avatar
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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 ...
Alex M.'s user avatar
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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 $G_{\lambda}(x,p)$ ...
Davidi Cone's user avatar
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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/(...
Alexander Pruss's user avatar
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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) = - \...
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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 ...
Isaac's user avatar
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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) =...
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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+...
Tree23's user avatar
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[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 ...
Victor Ramos's user avatar
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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 ...
Victor Ramos's user avatar
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Let's have a generic elliptic second order PDE in $n$-dimensions with a Dirac delta on the right hand side $$\left( a_{ij}(x) \partial_i \partial_j + b_j(x) \partial_j + c(x) \right) G(x) = \delta(x)$$...
Victor Ramos's user avatar
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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. ...
Bettina's user avatar
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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) \ \ ...
Carlos Adir's user avatar
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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 ...
Paul's user avatar
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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 $...
Student's user avatar
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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 ...
Goulifet's user avatar
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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 ...
Manuel Pena's user avatar
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$\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 ...
Joshua Isralowitz's user avatar
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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-...
Frederik Ravn Klausen's user avatar
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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 ...
Luis Yanka Annalisc's user avatar
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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 ...
T. Huynh's user avatar
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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 + ...
Joshua Isralowitz's user avatar
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1 answer
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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 ...
MathqA's user avatar
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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$;...
Z. Alfata's user avatar
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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 ...
Kernel's user avatar
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3 answers
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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\...
username's user avatar
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1 answer
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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 ...
Vera's user avatar
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1 answer
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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 ...
phryas's user avatar
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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 ...
user158773's user avatar
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What is the green function of the triangular kernel $K$: $$ K(x,y)=1-|x-y| $$ where $x,y\in R$ such that $|x-y|<1$?
Fabio's user avatar
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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^{...
MathMath's user avatar
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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,\...
Desura's user avatar
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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 ...
Amir Sagiv's user avatar
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