# Questions tagged [green-function]

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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 ...
3answers
197 views

0answers
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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 $u(x,0)=g(x), \forall x\in \Bbb R$...
0answers
80 views

### 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 $f$. ...
3answers
1k 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 ...
0answers
156 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 ...
0answers
56 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 ...
1answer
137 views

0answers
57 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 ...
0answers
258 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{...
0answers
103 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 ...
0answers
118 views

1answer
242 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 ...
1answer
479 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 ...
1answer
419 views

1answer
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1answer
387 views

### Is Poisson's kernel integrable?

Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic ...
1answer
971 views

### direct proof that schrodinger's equation kernel corresponds to delta-function initial value [closed]

I want to show directly, that the kernel for the n-dimensional free linear schrodinger equation, if taken to time t=0, is dirac's $\delta$ function. I can show that the integral is constant, but it ...
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