19
votes
Accepted
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 $C^{2,h}$ ...
- 6,367
10
votes
Accepted
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 $...
- 188k
10
votes
Accepted
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 ...
- 840
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
$$
b^{ij}\partial^2_{ij}u + \partial_ia^{ij}\...
- 27.5k
6
votes
Accepted
References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$
This mathoverflow question has the references you are looking for. Namely, you can look in the original papers:
Littman, W.; Stampacchia, G.; Weinberger, H. F., Regular points for elliptic equations ...
- 381
6
votes
Green's kernel estimates on finitely generated groups
Symmetric random walks on finitely generated groups of growth at most quadratic are recurrent, therefore the Green kernel
$$ \Theta(x)=\sum_{n\ge 0}p^{(n)}(x)$$
is infinite for all $x$. (The right-...
- 1,819
6
votes
Hölder continuity of Green function for simply connected domains
Scaling $\mathcal K$ by a factor $1/Cap(\mathcal K)$, we can assume $Cap(\mathcal{K})=1$. In this case the estimate
$$
G(z) \leq C \sqrt{
{\rm dist}(z,\mathcal K)}
$$
indeed follows from Koebe-...
- 2,154
5
votes
Accepted
Any formula or estimates the Green function for the Laplacian in $3D$ periodic box?
Well it depends on what you mean by "explicit". Let $(\varphi_k)_k \subset L^2(\mathbb{S}^1)$ be the eigenfunctions of the Laplacian on $\mathbb{S}^1$, these have an explicit form that comes ...
- 646
5
votes
Fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix
Let me start with a constant coefficient operator
$$
P(D)=\sum_{1\le j, k\le n} a_{jk} D_jD_k,\quad D_j=\frac{\partial }{i\partial x_j}.
$$
Note that in two dimensions, you have elliptic operators ...
- 16.2k
5
votes
Reference request for a treatment of Schwinger–Dyson equations
A quick and clear and rigorous derivation of the Schwinger-Dyson equation from BV-theory in causal perturbation theory is offered in remark 7.7 of Rejzner 16, it's spelled out at nLab:BV-operator -- ...
- 19.8k
4
votes
Accepted
How to determine the spectrum from the diagonal Green's function
For the Sturm–Liouville operators with discrete spectrum, $G(z,x,y)$ is the Stieltjes transform of the discrete measure $$\mu_{x,y}(ds) = \sum_{n = 1}^\infty \varphi_n(x) \overline{\varphi_n(y)} \...
- 17.2k
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 ...
- 7,817
4
votes
Accepted
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 ...
- 17.2k
4
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 $...
- 6,696
4
votes
Accepted
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 ...
- 91.8k
3
votes
Accepted
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), ...
- 1,263
3
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 ...
- 188k
3
votes
Accepted
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 ...
- 705
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",...
- 31
3
votes
Accepted
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 ...
- 661
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 ...
- 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 ...
- 188k
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 ...
- 14.2k
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 ...
- 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 ...
- 188k
1
vote
Heat conduction type equation in 4D
Unlike in your earlier question, the function $u(t,x)$ is analytic in $\xi$, so there are no complications arising from a nonzero real part of $\xi$.
You can simply invert the Fourier transform of $\...
- 188k
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 $...
- 253
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}...
- 188k
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 ...
- 17.2k
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