Skip to main content

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)

Filter by
Sorted by
Tagged with
1 vote
1 answer
25 views

What is wrong with using the $H^1_0$ inner product here?

This question is a problem I am toying with. Consider the Poisson equation, $$-\Delta u =f \ \text{in} \ \Omega \times (0,\infty).$$ $$u=0 \ \text{on} \ \partial \Omega \times (0,\infty)$$ Suppose the ...
ali's user avatar
  • 194
0 votes
0 answers
16 views

Solving the Radially Symmetric Poisson Equation with Exponential Source Term

I want to solve the poisson equation $$ -\Delta u(\mathbf{x})=\rho(\mathbf{x})=\frac{e^{-|\mathbf{x}|^2}}{|\mathbf{x}|^2-1} $$ The problem want me to use the fundamental solution of laplace operator, ...
Gao Minghao's user avatar
1 vote
0 answers
19 views

Singularity in Poisson's Equation

Consider an instance of Poisson's equation in spherical coordinates for the radial dimension: $$ \nabla \cdot \nabla \phi(r) = \frac{1}{r^2} \frac{d}{dr}\left(r^2 \frac{d\phi}{dr} \right) = -\sin(r). $...
SeanBrooks's user avatar
0 votes
1 answer
44 views

How to use Fourier method to solve the Poisson equation $-\Delta v=x y$?

Suppose that $B_{1}$ is a ball centered at $0$ with radius $1$, consider the equation $-\Delta v=x y$ and $v=0$ on boundary, I want to use Fourier method to solve it, but the Fourier transform is ...
YuerCauchy's user avatar
0 votes
1 answer
34 views

Proving a bound on $|\nabla w(0)|$ for a solution to $\Delta w = f(w)$

Let $f \in C_c^{\infty}(\mathbb R)$ with $0 \leq f \leq 1$ on $\mathbb R$. I am trying to prove the following: Suppose $w \in C^\infty(B_3(0))$, $w \geq 0$, and solves $\Delta w = f(w)$ on $B_3(0)$. ...
Luke's user avatar
  • 765
0 votes
1 answer
56 views

What justifies the $\epsilon \rightarrow 0$ limit in the domain of this integral?

I am following these notes on Green's function for Poisson's equation, which are based on Evan's PDE book. Let $\Omega \subset \mathbb{R}^n$ be open and bounded. Let $u \in C^2(\overline{\Omega})$ be ...
CBBAM's user avatar
  • 6,159
0 votes
0 answers
21 views

Solution formula for Poisson equation on upper half space

Consider $G$ be the upper half space, i.e. all points $x\in \mathbb{R}^2$ so that $x_2>0$. Let $f\in C_c^{\infty}(G)$. I want to understand how the solution of the following Poisson equation looks ...
user99432's user avatar
  • 900
1 vote
1 answer
32 views

Help me intuit Equal Probabilities in Poisson Distribution for $k = λ$ and $k = λ-1$

I was trying to understand Poisson distribution and I'm confused as to why the likelihood of $k=λ-1$ is equal to $k=λ$. Here is my understanding and where my confusion is: For a given time interval, ...
Tyler Short's user avatar
1 vote
1 answer
40 views

Solving the Poisson equation $-\Delta u=f$ on a domain $G$

Let $G$ be a domain and $f\in C_c^1(\mathbb{R}^n)$. We want to solve $$-\Delta u =f \text{ in } G$$ without any boundary condition. Can we just take $\Delta^{-1}$ the inverse Laplacian on $\mathbb{R}^...
user99432's user avatar
  • 900
0 votes
0 answers
80 views

Solution of $-\Delta u=f$

Fact 1. The function $1/|\xi|^{s}$ locally integrable (in the unit ball) if and only if $s<n$. Fact 2. If $f\in L^1(\mathbb{R}^n)$, then $\widehat{f}\in\mathcal{C}^{\infty}(\mathbb{R}^n)$ (bounded)....
eraldcoil's user avatar
  • 3,586
0 votes
0 answers
56 views

Kronecker product and finite difference discretization for poisson equation

In this notebook from MIT's Intro to Linear PDEs course, it is unclear to me why the Kronecker product is used to formulate the coefficients matrix $A$ for solving the linear system of equations $A u =...
Jared Frazier's user avatar
0 votes
0 answers
41 views

Maximum principle to get $C^0$ estimates in terms of $L^1$ norms

I'm reading this paper https://arxiv.org/abs/1401.7366 and trying to prove Corollary 4.6. The result essentially says the following. Let $B_r \subseteq \mathbb R^4$ be the ball of radius $r$ with the ...
Holmes's user avatar
  • 906
0 votes
0 answers
54 views

Navier Stokes Eqns: Boundary Conditions and Pressure Coupling

I have made it a personal long-term goal of mine to numerically solve the 1D transient Navier-Stokes equations for simple incompressible and compressible flow scenarios. I want to do it for a few ...
UserHandel's user avatar
2 votes
1 answer
159 views

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 ...
Quillo's user avatar
  • 2,083
1 vote
1 answer
82 views

Application of weak maximum principle.

Fix any open, bounded set $U \subset \mathbb{R}^n$ and suppose that $u \in C^2(U) \cap C(\bar{U})$ is a solution of $$-\Delta u = f \,\,\,\text{in}\,U$$ $$u=g \,\,\,\,\,\,\text{on}\,\,\,\partial U,$$ ...
Lilili123's user avatar
  • 139
1 vote
0 answers
24 views

custom FFT library vs FFTW library produce different results for Poisson Solution Spectral Gradient

I am trying out different FFT solvers, with two of them being FFTW (fftw version 3.3.3.8, https://www.fftw.org/index.html) and FAST FFT ( https://www.kurims.kyoto-u.ac.jp/~ooura/fft.html). I am using ...
Christos's user avatar
0 votes
1 answer
40 views

The relationship between the norm of the gradient solution of Poisson's equation

Question: Consider the following two special Poisson equations: $$\Delta u=-2 \quad \text{in} \quad L$$ $$u=0 \quad \text{on} \quad \partial L$$ and $$\Delta u=-2 \quad \text{in} \quad W$$ $$u=0 \quad ...
zeyu hao's user avatar
  • 347
2 votes
1 answer
92 views

The relationship between the norm of the gradient solution of Poisson's equation in different regions

Question: Consider the following two special Poisson equations: $$\Delta u=-2 \quad \text{in} \quad L$$ $$u=0 \quad \text{on} \quad \partial L$$ and $$\Delta u=-2 \quad \text{in} \quad W$$ $$u=0 \quad ...
zeyu hao's user avatar
  • 347
1 vote
0 answers
81 views

Green's function for the screened Poisson equation in an n-dimensional ball

Consider the screened Poisson equation $\Delta u(r) - cu(r) = -g(r)$, in a ball (centered at the origin) of radius $R$, with boundary condition $u(R)=0$. In 3-dimensional space, the Green's function ...
ricm's user avatar
  • 11
0 votes
0 answers
73 views

How to solve 2D Poisson equation for coordinate independent source?

$$\frac{∂^2 u}{∂x^2}+\frac{∂^2 u}{∂y^2}=S(x)$$ Consider solution $u(x,y)$ in finite Cartesian domain $(0,L_x)\times(0,L_y)$. The domain is subjected to the source of $S(x)$, which is only dependent on ...
Alexander Newman's user avatar
0 votes
1 answer
44 views

Construct a manufactured solution of Poisson's equation with Chebyshev/Fourier expansions

I am solving a nonlinear Poisson's equation numerically using a mixed Chebyshev/Fourier spectral methods. Thus, assuming $x$ is periodic and $y$ is nonperiodic. I am trying to test my current ...
Jamie 's user avatar
  • 111
2 votes
2 answers
131 views

Solving Poisson equation with Fourier transform.

Is possible solving the poisson equation with Fourier transform? For example, with application of the Fourier transform, the problem \begin{align} (1-\partial_x^2)u=f \end{align} with $f\in L^2(\...
eraldcoil's user avatar
  • 3,586
0 votes
0 answers
46 views

variational formulation of a mixed boundary condition problem and prove the existence of solution

This is a mixed boundary value problem: \begin{align*} -\nabla \cdot(\nabla u) &= f, & x &\in \Omega \\ u &= g_D, & x &\in \Gamma_D \\ \frac{\partial u}{\partial \...
well's user avatar
  • 174
1 vote
1 answer
167 views

Evans: Method of Subsolutions and Supersolutions

Throughout this problem we assume $f:\mathbb{R}\to \mathbb{R}$ is a smooth function, with $|f'|\leq C$ for some fixed constant $C>0$. The constant $\lambda >0$ is defined to be a large enough ...
Ssay's user avatar
  • 488
1 vote
0 answers
41 views

Doubts in lemma 9.12 of Gilbarg Trudinger

Lemma $9.2$: Let $u\in W_0^{1,1}(\Omega^+),f\in L^p(\Omega^+),1<p<\infty$ satisfy $\Delta u=f$ weakly in $\Omega^+$ with $u=0$ near $(\partial \Omega)^+$. then $u\in W^{2,p}(\Omega^+)\cap W_0^{1,...
PNDas's user avatar
  • 937
3 votes
0 answers
77 views

Obtain $L^\infty$ estimate of Poisson Equation only using eigenfunction expansion

Let $\Omega\subseteq \mathbb{R}^d$ a bounded domain with sufficient smooth boundary. We consider the Poisson Equation $$\left\{\begin{array} --\Delta u = f, & \Omega\\ u=0, &\partial\Omega\end{...
MikeMichael_maths's user avatar
0 votes
1 answer
98 views

Prove that $u = |\nabla v|^2$ reaches its maximum in $\partial \Omega$, $v$ being a solution to the given Poisson pde

Let $\Omega \subset \mathbb{R}^2$ and $v: \mathbb{R}^2 \to \mathbb{R}$ be a solution to: $$\begin{cases} v_{xx}+v_{yy} = -2 &\text{in $\Omega$}\\ v = 0 &\text{in $\partial \Omega$} \end{cases}$...
nicoyanovsky's user avatar
0 votes
0 answers
74 views

How to properly apply 2D Green's function formula to nonhomogenous Poisson equation on unit disc

Brief: I am struggling with applying boundary condition to Poisson equation and being inspired by this work: https://dept.math.lsa.umich.edu/~sijue/greensfunction.pdf For eq. $\text{$\Delta $u}[\...
Rob's user avatar
  • 1
1 vote
0 answers
21 views

Solve Poisson problem $ \Delta u(\rho,\phi)=\ln \rho+2\cos^2\phi$ on annulus $1<\rho<3.$

Solve \begin{eqnarray} \left \{\begin {array}{lll} \Delta u(\rho,\phi)=\ln \rho+2\cos^2\phi&,~ 1<r<3,~0\leqslant \phi \leqslant \pi,~0\leqslant \phi < 2\pi\\ \displaystyle u_{\rho}(1,\phi)...
Nikolaos Skout's user avatar
0 votes
0 answers
62 views

Help with proof of Poisson's formula for ball

Here follows a theorem from L. Evans PDE and a short part of its proof Theorem Assume $g\in C(\partial B(0,r))$, and define $u$ by $$ u(x)=\frac{r^2-|x|^2}{n\alpha(n)}\int_{\partial B(0,r)}\frac{g(y)}...
JackpotWizard 180's user avatar
0 votes
0 answers
38 views

Fourier transform of Poisson's equation with periodic shifted boundary conditions

I am trying to solve following 2D Poisson's equation numerically: $$ \Delta \Phi = \rho $$ When $\rho$ and $\Phi$ are periodic in both directions, this can be solved straightforward thanks to Fourier ...
steven's user avatar
  • 1
3 votes
1 answer
88 views

Regularity of solutions to Poisson's equation on part of the boundary

Let $\Omega \subset \mathbb{R}^n$ be an open and bounded set and suppose $\partial \Omega$ is smooth on a relatively open subset $\Gamma \subset \partial \Omega$. Consider a weak solution of the ...
Cauchy's Sequence's user avatar
0 votes
1 answer
107 views

Solving Poisson equation with separate variable method

Here's a Poisson equation, $D$ is a unit square i.e. $[0,1]\times[0,1]$. \begin{equation*} \begin{aligned} \Delta u(x,y) &= \varphi(x,y) \quad (x,y)\in D \\ u(x,y) &= 0 \quad (x,y)\in\partial ...
Cunyi Nan's user avatar
  • 744
3 votes
1 answer
87 views

Showing uniqueness of solution to a non-linear Poisson problem

I'm trying to prove that a non-linear Poisson problem has a unique solution. The context is the following: Let $\Omega \subset \mathbb{R}^n$ be a bounded open subset of class $C^2$. Consider the ...
Matheus Andrade's user avatar
1 vote
1 answer
84 views

poisson's formula for half-spce, evans page 38

I have questions about the proof of Theorem 14 (poisson's formula for half-space) in Page 38. Let $K(x,y)$ be the Poisson's kernel for $\mathbb R^n_+$: $$K(x,y)=\frac{2x_n}{na(n)}\frac{1}{|x-y|^n}\...
topst's user avatar
  • 147
0 votes
0 answers
12 views

how to solve the reweighted Poisson equation efficiently

Consider the following reweighted Poisson equation: given $\operatorname{Q} $ and $g$, $$ (\nabla \cdot \operatorname{Q} \nabla ) f = g, $$ where $\operatorname{Q} $ is a diagonal matrix with ...
Xue Feng's user avatar
1 vote
1 answer
122 views

Poisson equation infinite strip Sobolev regularity

Consider the equation (with $l_-=0, l_+=l>0)$ \begin{align} \left\{\begin{array}{rclclcl} -\Delta u (s,t) & = & f(s,t) &&&& \text{for } (s,t)\in \mathbb{R}\times (0,l), \\...
snape1234's user avatar
1 vote
0 answers
85 views

Show that $\frac 1{4\pi |x|} e^{-c|x|}$ is a fundamental solution for $-\Delta+c^2$ on $\mathbb R^3$

[Introduction to Partial Differential Equations - Gerald B. Folland, chapter 2, section C, question 6] Show that $$\frac 1{4\pi |r|} e^{-c|r|}$$ is a fundamental solution for $$-\Delta+c^2\qquad (c\in ...
Sayan Dutta's user avatar
  • 9,474
1 vote
0 answers
77 views

Schauder estimates on boundary

I recently studied the Schauder estimates with the boundary and checked the wiki page. The following is the link (the boundary estimate is on the bottom part): https://en.wikipedia.org/wiki/...
jiaxinzerg's user avatar
1 vote
0 answers
72 views

Unique solvability of weak Poisson equation with Neumann boundary condition

I'm studying a Poisson BVP: Find $u \in W^{1,p}(\Omega)$ such that $$ \int_{\Omega} \nabla u \cdot \nabla v = F(v) \quad \forall v \in W^{1,p'}(\Omega), $$ where $p$ and $p'$ are Hölder conjugates and ...
bananab0y's user avatar
7 votes
1 answer
110 views

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 $\...
kam's user avatar
  • 312
1 vote
0 answers
79 views

Show that the following condition $\{p,\{p,\phi\}\}>0$ on $T_{\Sigma}^{*}\mathbb{R}^{n}$

I am reading an article of D. Tataru and I found the following situation, Suppose we have a differential operator $P(x,D)$ of order $m$ in $\mathbb{R}^n$. Let $\Sigma$ be an oriented hypersurface in $\...
Darek_'s user avatar
  • 41
4 votes
0 answers
211 views

Sharp constant in the $L^p$ regularity estimate?

Problem: Let us denote $\mathbb{W}^{2,p}(\mathbb{R}^2)$ the space of Sobolev functions in the plane. Let us denote with $\Delta$ the classic Laplacian operator. We know that there exists a constant $C&...
Filippo Giovagnini's user avatar
0 votes
1 answer
219 views

PDEs: Derivating the weak form for the nonlinear poisson equation

I was reading a PDE tutorial on solving the nonlinear poisson equation. The author of the tutorial defines the problem and asserts the weak form, but does not provide the derivation. I tried my own ...
krishnab's user avatar
  • 2,491
0 votes
1 answer
90 views

Constructing operator for mesh fairing, are these 2 approaches equivalent?

There is a 2004 paper in computer graphics called "An Intuitive Framework for Real-Time Freeform Modeling" which explains how to make fairing operators to smooth out meshes. In particular ...
Makogan's user avatar
  • 3,419
1 vote
1 answer
101 views

Going from discrete Poisson equation to (discrete) divergence calculation

Just to give some background: I am currently working on a fluid simulation and am trying to clear any divergence from my discretized velocity field (i.e. it's split up into grids). To eliminate such ...
Glace's user avatar
  • 61
0 votes
0 answers
60 views

Poisson equation spherical-radial only coordinates with second type boundary condition at the origin

I have an issue with a basic Poisson's equation: $$ \frac{1}{r^2} \partial_r r^2 \partial_r V = - f(r)$$ with $r > 0$, $f(r) > 0$ has the following property: $$ \int_0^{\infty} r^2 f(r) d r = 1 $...
Fefetltl's user avatar
  • 191
1 vote
1 answer
197 views

Why do mathematicians use BDM or Hdiv finite elements to solve the Mixed Poisson partial differential equation

I am still new to finite element methods, and I was looking at some tutorials on a specific formulation of the Poisson equation that introduces an additional variable. Some of the tutorials call this ...
krishnab's user avatar
  • 2,491
0 votes
2 answers
60 views

Find vector field whose divergence is a scalar field

Say we have a Poisson problem: $\nabla^2 \varphi = S$ As a boundary value problem, it requires the definition of boundary conditions on all surfaces of the domain. If we assume there is a vector field ...
Fernando Zigunov's user avatar
1 vote
0 answers
26 views

Error analysis of a numerical time-averaging via Poisson equation

I'm following the proof of the paper, Mattingly et al., Convergence of numerical time-averaging and stationary measures via Poisson equations (https://arxiv.org/pdf/0908.4450.pdf). Specifically, I'm ...
JHLee's user avatar
  • 11

1
2 3 4 5
10