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Questions tagged [poissons-equation]

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics. (Def: https://en.wikipedia.org/wiki/Poisson%27s_equation)

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics. It is given by $\nabla^2\varphi=f$ where $\varphi,f$ are real- or complex-valued functions on a manifold. Reference: Wikipedia.

It is used, for instance, to describe the potential energy field caused by a given charge or mass density distribution.

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Green's Function for 2D Poisson Equation

In two dimensions, Poisson's equation has the fundamental solution, $$G(\mathbf{r},\mathbf{r'}) = \frac{\log|\mathbf{r}-\mathbf{r'}|}{2\pi}. $$ I was trying to derive this using the Fourier transformed equation, and the process encountered an…
ClassicStyle
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Suppose $\phi$ is a weak solution of $\Delta \phi = f \in \mathcal{H}^1$. Then $\phi\in W^{2,1}$

I'm trying to prove the statement in the title in as simple a way as possible. It is Theorem 3.2.9 in Helein's book "Harmonic maps, conservation laws, and moving frames", although it is not proved there. The statement is as follows. Suppose…
Glen Wheeler
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Solve Poisson Equation Using FFT

I am trying to solve Poisson equation using FFT. The issue appears at wavenumber $k = 0$ when I want to get inverse Laplacian which means division by zero. We have ${\nabla ^2}\phi = f$ Taking FFT from both side we get: $-k^2\hat\phi = \hat f…
FFTNewbie
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Semi-group theory and Poisson equation on the upper half plane

We first look at the 2D Laplace equation , say on the upper half plane: $$\Delta u=0,\quad -\infty0$$ $$u(x,0)=g(x),$$ where $g\in L^p(\mathbb{R})$ for some $1\leq p<\infty$. Then the general solution can be represented using the…
Closure
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Existence and Uniqueness of Poisson Equation with Robin Boundary Condition using First Variation Methods

I'm currently stuck on the following exercise from Evans PDE Chapter 8 Exercise 11. Let $\beta: \mathbb{R} \rightarrow \mathbb{R}$ be smooth with \begin{equation} 0 < a \leq \beta'(z) \leq b, \text{ } z \in \mathbb{R} \end{equation} for constants…
Story123
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A continuous function for which Poisson's equation has no C^2 solutions

I am trying to solve exercise 4.9 of Gilbarg and Trudinger, and in particular need to show that for the function $f(x)=\sum_{k=0}^{\infty}\frac{1}{k}\Delta(\eta{P})(2^kx)$ the problem $\Delta{u}=f$ has no $C^2$ solution in any neighbourhood of the…
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Why can we pass limit under integral sign in proof of solving Poisson's equation? (Evans PDE)

On page 23 of Lawrence Evans' Partial Differential Equations text (2nd edition) he claims that $$\frac{ f( x + he_i - y) - f( x-y)}{h} \to \frac{ \partial f}{ \partial x_i} ( x-y)$$ uniformly on $\mathbb{R}^n$ as $h \to 0$. So $$\frac{ \partial u}{…
kam
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What are the differences between Heat equations and Poisson Equations?

Am fairly new into heat equations and wanted to have some clarifications. What are the distinguishing features between the heat equation and the Poisson equation?
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Prove the uniqueness of poisson equation with robin boundary condition

We have $\Delta u=f$ in $D$, and $\dfrac{\partial u}{\partial n}+au=h$ on boundary of D, where $D$ is a domain in three dimension and $a$ is a positive constant. $\dfrac{\partial u}{\partial n}=\triangledown u\cdot n$ ($n$ is normal vector). My…
jack
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Solving Poisson's equation for a point charge in 1-D

I Apologize that this is a continuation of a question that I just asked. Anyway here is where I am: Ok so I was trying to solve the Poisson's equation for a point charge with a Fourier transform to get the familiar equation. This is what I did so…
sTr8_Struggin
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Solving the 2D Poisson equation with variable boundary location

I am trying to find $z(r,\phi)$ from the 2D Poisson equation in polar coordinates: $$\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial z}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2z}{\partial \phi^2}=C \tag{1}$$ where $C$ is a…
Michiel
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Value of $u(0)$ of the Dirichlet problem for the Poisson equation

Pick an integer $n\geq 3$, a constant $r>0$ and write $B_r = \{x \in \mathbb{R}^n : |x|
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Estimate solution of Poisson equation on unit ball

Consider the following boundary value problem where $U=\{x \in \mathbb{R}^3 \mid |x|<1\}$ and $g$ is some nice bounded function, $$\Delta u = g ~~~ \text{on}~U\\ u=0 ~~~\text{on} ~\partial U.$$ Assume that $x_0 \in U$ with $|x_0|=r$ for some $0
mathematikos
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Fundamental solution of Poisson equation on the torus

The potential of a point charge placed at $y\in\mathbb{R}^N$, for $N=1,2,3$, is (see e.g. this MathSE post): \begin{align} &\mathbb{R}^3: \qquad \nabla^2 \phi(x) = \delta^3(x-y) \quad \Rightarrow \quad \phi(x) = \frac{-1}{4 \pi |x-y|} \qquad…
Quillo
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Soft enforcement of boundary constraints in linear PDE

Suppose I have a region $\Omega\subset\mathbb R^2$ with smooth boundary curve $\partial\Omega$. Consider the PDE \begin{cases} \Delta u(x) = 0\ \forall x\in\mathrm{int}\ \Omega \\ au(x)+b\frac{\partial u}{\partial n}(x)=g(x)\ \forall…
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