15 votes
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Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix

This question requires an articulated answer, since the topic dealt is complex and ramified. A fundamental solution for a not necessarily divergence form $2$nd order elliptic system with $C^{2,h}$ ...
Daniele Tampieri's user avatar
9 votes
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Propagators and PDEs

You'll find some more info about the fundamental solutions of the wave equation in chapter 5.D of Folland and chapter I.7 of Trèves. The trick you use is the idea that (tempered) distributions, even ...
Daniel Shapero's user avatar
8 votes
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Linear PDE, analytic continuation, Green's function and boundary conditions

Q: Do I have to consider both problems (real $\xi$ or imaginary $\xi$) totally independently and work hard twice?. A: A single calculation suffices, you could just do the inverse Fourier transform of $...
Carlo Beenakker's user avatar
8 votes

Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix

If you assume that the coefficients $a^{ij}$ are smooth functions and let $$b^{ij} = \frac{1}{2}(a^{ij} + a^{ji}),$$ then the PDE can be written as $$ b^{ij}\partial^2_{ij}u + \partial_ia^{ij}\...
Deane Yang's user avatar
  • 26.9k
6 votes
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References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$

This mathoverflow question has the references you are looking for. Namely, you can look in the original papers: Littman, W.; Stampacchia, G.; Weinberger, H. F., Regular points for elliptic equations ...
sharpend's user avatar
  • 381
5 votes
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Any formula or estimates the Green function for the Laplacian in $3D$ periodic box?

Well it depends on what you mean by "explicit". Let $(\varphi_k)_k \subset L^2(\mathbb{S}^1)$ be the eigenfunctions of the Laplacian on $\mathbb{S}^1$, these have an explicit form that comes ...
Michele Caselli's user avatar
5 votes

Green's kernel estimates on finitely generated groups

Symmetric random walks on finitely generated groups of growth at most quadratic are recurrent, therefore the Green kernel $$ \Theta(x)=\sum_{n\ge 0}p^{(n)}(x)$$ is infinite for all $x$. (The right-...
Corentin B's user avatar
5 votes

Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix

Let me start with a constant coefficient operator $$ P(D)=\sum_{1\le j, k\le n} a_{jk} D_jD_k,\quad D_j=\frac{\partial }{i\partial x_j}. $$ Note that in two dimensions, you have elliptic operators ...
Bazin's user avatar
  • 15.1k
5 votes

Reference request for a treatment of Schwinger–Dyson equations

A quick and clear and rigorous derivation of the Schwinger-Dyson equation from BV-theory in causal perturbation theory is offered in remark 7.7 of Rejzner 16, it's spelled out at nLab:BV-operator -- ...
Urs Schreiber's user avatar
4 votes
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How to determine the spectrum from the diagonal Green's function

For the Sturm–Liouville operators with discrete spectrum, $G(z,x,y)$ is the Stieltjes transform of the discrete measure $$\mu_{x,y}(ds) = \sum_{n = 1}^\infty \varphi_n(x) \overline{\varphi_n(y)} \...
Mateusz Kwaśnicki's user avatar
4 votes

Reference request for a treatment of Schwinger–Dyson equations

In the formulation of QFT using formal functional integrals, as mentioned by Igor in his answer, the Schwinger-Dyson equation becomes an infinite-dimensional differential equation for the partition ...
Pedro Lauridsen Ribeiro's user avatar
4 votes
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Singularity of the heat kernel

Yes, away from the boundary: the heat kernel for the interval is given by $$\tag{1}g(t,x,y)=(2\pi t)^{-1/2}\sum_{n\in\mathbb{Z}} (-1)^n \exp\left(-\frac{(x-y-n\pi)^2}{2t}\right),$$ and it is not ...
Mateusz Kwaśnicki's user avatar
4 votes
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Green's function in terms of logarithmic potential and energy of a measure

What you stated cannot be true: Green function depends only on $K$, but your $\Phi_\mu$ is the potential of an arbitrary measure on $K$. These formulas become true when $\mu$ is the EQUILIBRIUM ...
Alexandre Eremenko's user avatar
3 votes

Green's function for a linear PDE initial value problem

The time-dependent Schrödinger equation with Coulomb potential (hydrogen atom) has the form $$ i\frac{\partial u}{\partial t} = \left( -a^{\prime } \Delta -\frac{b^{\prime } }{|x|} \right) u $$ with $...
Michael Engelhardt's user avatar
3 votes
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Existence and estimates of Green's function on Riemannian manifold

For further details on the existence argument, see Chapter 4 of Aubin's book, "Nonlinear Analysis on Manifolds: Monge-Ampère Equations" [1] for the harmonic case, and [2] for the ...
sorrymaker's user avatar
3 votes
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Elliptic equations in asymptotically hyperbolic manifolds

The definite reference for this is the monograph by John Lee "Fredholm operators and Einstein metrics on conformally compact manifolds", Mem. Am. Math. Soc. Series Profile 864, 83 p. (2006), ...
Romain Gicquaud's user avatar
3 votes

Analytical solution of diffusion PDE with Robin boundary condition

Yes, there is an associated Green's function for your problem, which is explicitly known (at least for complex $\alpha$, $\beta$ with positive real parts). More precisely, the "Robin Green's function",...
Denny Otten's user avatar
3 votes
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Variation of the Green function with respect to the metric

It seems that the naive derivation from the path integral only picks up the term coming from quasiconformal variations. Combining the known result for the quasiconformal variation with the much more ...
level1807's user avatar
  • 469
3 votes

The study of dynamics of a polynomial vector field via Green's function methods

This is not a answer,but some comments.At first,I want to say the Hilbert 16th problem is very difficult,and there is a lot of way to approach the problem.when I learn the ordinary differential ...
Hu xiyu's user avatar
  • 697
3 votes

Green's function for fourth order equation

The time-independent equation is the biharmonic equation, $$-\Delta^2f({\bf r},{\bf r}')=\delta({\bf r}-{\bf r}'),$$ with 3D solution $$f({\bf r},{\bf r}')=\frac{1}{8\pi}|{\bf r}-{\bf r}'|,$$ see ...
Carlo Beenakker's user avatar
3 votes
Accepted

Does the Green's function of the simple random walk on $\mathbb Z^d$ always vary locally?

We will use the elementary fact that for $m \ge k \ge m/2$, the binomial coefficients satisfy $${m\choose k+1} <{m \choose k}. \quad (\#)$$ The case $x=0$ is obvious so we may assume $x$ has ...
Yuval Peres's user avatar
2 votes

Reconstructing the Green's function of an initial-value problem of partial differential equation

One can regard your problem as probing a medium by a known source $f_1$ generated at time $t=0$, recording the medium response $f_2$ at time $t=T$ (called the data) and the goal is the recovery of ...
Rakesh's user avatar
  • 21
2 votes

Numerical methods for evaluating singular integrals

The equation in the OP does not quite make sense (why does the same field appear on both sides of the equation?), but I understand the question as inquiring how to numerically compute the Helmholtz ...
Carlo Beenakker's user avatar
2 votes
Accepted

Green's Function for 3D Relativistic Heat Equation

The "relativistic" heat equation is more generally known as the Telegrapher's equation, $$\frac{\partial f}{\partial t}+\tau\frac{\partial^2 f}{\partial t^2}=\kappa\nabla^2 f.$$ The Green's function ...
Carlo Beenakker's user avatar
1 vote
Accepted

Double integral in a polygon domain

In the example, I got that $$ I(1) = \sum_{i=1}^{n} \dfrac{(x_{i+1}+x_{i})(y_{i+1}-y_{i})}{2} = \sum_{i=1}^{n} \dfrac{(x_{i+1}-x_{i})(y_{i+1}+y_{i})}{2} $$ And it's indeed correct. If I expand around $...
Carlos Adir's user avatar
1 vote

What's going on with the two-dimensional Helmholtz equation?

You seek the solution of $$(\nabla^2+\kappa^2+i\epsilon)G(\mathbf{r})=\delta(\mathbf{r}),$$ in the limit $\epsilon\rightarrow 0^+$, which is given by a Hankel function of the first kind, $$G(\mathbf{r}...
Carlo Beenakker's user avatar
1 vote

Definition of Martin kernels

I do know a reference for this particular result — although I am completely sure it is true! The original Bogdan–Burdzy–Chen paper does not even provide a complete proof of the existence of the ...
Mateusz Kwaśnicki's user avatar
1 vote

Green potential and Hölder continuity

An elementary proof (~ 1 page) for the more general case of Riesz potentials in $\mathbb{R}^n$ (hence for the logarithmic potential or Green potential in $\mathbb{C}$), and for more general exponents (...
user111's user avatar
  • 3,761
1 vote

Green function of the triangular kernel?

This is probably not unique. Due to translational invariance, we can restrict the space on which we're acting to functions $f$ on the interval $[-1/2,1/2]$. If we specify those to be odd and with ...
Michael Engelhardt's user avatar
1 vote

hyperbolic "Green function" on a product of upper half-planes

The answer depends on what you call "explicit", and the passage from the heat kernel to the Green function is not that straightforward from this point of view. There is a quite efficient uniform ...
R W's user avatar
  • 16.6k

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