Questions tagged [laplace-equation]

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1answer
195 views

Laplace equation on the disk with Robin boundary condition

Consider the following two dimensional Laplace equation on the unit disk $D$ with homogeneous Robin boundary condition: $$\Delta u = 0, ~~\frac{\partial u}{\partial n} = b(x) u(x)~~ \forall x \in \...
1
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0answers
93 views

Can complex analysis be used to solve Laplace's equation in three dimensions? [closed]

Can complex analysis be used to solve Laplace's equation in three dimensions? Or is it restricted to cases which have 2D symmetry? All of the examples I've seen are 2D, never 3D. It would seem ...
2
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0answers
76 views

Neumann problem on a convex domain

Let $\Omega$ be a convex open bounded subset of $\mathbb R^n$ and let $u$ be the solution of $$ \begin{cases} ∆ u=1\quad\text{in $\Omega$,} \\ \frac{\partial u}{\partial \nu}=\frac{\vert \Omega\vert_n}...
1
vote
1answer
177 views

Laplace equation with integral source terms

I am not specifically asking for a solution, but any reference on any method i could read about would be a big help. This I clarify as I am aware of the fact that MathOverflow only deals with research ...
0
votes
0answers
99 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 ...
1
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1answer
183 views

Laplace equation, medium discontinuity and finite difference method

The main question is: How to deal with the Poisson equation in the presence of the medium interface. Let's say we have 1D Laplace equation: \begin{equation} -\frac{d}{dx}\left(\epsilon(x)\frac{d}{dx}\...
7
votes
1answer
335 views

Variation of the Green function with respect to the metric

Consider a (closed) Riemann surface and let $G(x,y)$ be the Green function of the Laplace-Beltrami operator. We can informally identify $G$ with the two-point correlation function for the Gaussian ...
1
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0answers
149 views

Expansion of prolate spheroidal harmonics

For two coordinate frames $O'$ and $O''$ both offset along the $z$-axis by $\pm R$ respectively, with corresponding offset spherical coordinates $r'$, $\theta'$, $r''$ and $\theta''$, and with prolate ...
2
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0answers
101 views

heat kernel for powers of some degenerate elliptic operators

Let $\Omega$ be a bounded open domain in $R^{n}$ with smooth boundary and $X=(X_{1},X_{2},\cdots,X_{m})$ be a system of real smooth vector fields defined on $\Omega\subset \mathbb{R}^{n}$. If $X$ ...
2
votes
1answer
107 views

Finite Element Method on a single triangular element

Consider the Laplace equation on a single triangular domain with a Dirichlet condition on two of the sides and a Neumann condition on the remaining side. I am using a linear element ... $\mathbb{P}_1$ ...
3
votes
1answer
164 views

$H^1$-continuity of Laplace's equation with respect to boundary data

Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's ...
3
votes
1answer
190 views

Generalization of maximum principle to other norms

Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ ...
8
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2answers
1k views

Separable coordinate systems for the Laplace and Helmholtz equations?

According to Mathworld, in three dimensions there are 13 coordinate systems in which Laplace's equation is separable, and 11 for the Helmholtz equation. I've read the relevant chapters of the book by ...
1
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1answer
334 views

W^{2,∞} regularity of solutions of Poisson's equation if the right hand side is in L^{∞}

Let $u$ be solution of $-\Delta u = f$ in $\Omega$ and $\frac{\partial u}{\partial n} = 0$ on $\partial \Omega$. Is it true that if $f \in L^{\infty}(\Omega)$ then $u \in W^{2,\infty}(\Omega)$? (...
4
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0answers
172 views

explicit solution for Laplace equation on punctured cylinder

I need an explicit formula for the rotationally invariant solution of $\Delta u=0$ in cylindrical coordinate $(r,\theta,z)$ for a domain like $D=[0,2]\times [0,2\pi]\times [-2,2] - [0,1]\times [0,2\pi]...
1
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0answers
38 views

2-d laplace equation with corrugated isothermal boundary [closed]

Consider a 2-d laplace equation $\Delta\Theta(x,z)=0$ with a corrugated boundary $ \Theta(x,f(x))=\Theta_0$. You can assume $f(x)$ to be a sinusoidal function. 1.My idea is to set $p=z-f(x)$. But ...
1
vote
0answers
128 views

Basic doubt in a free boundary problem for the Laplacian

I am studying the following article : http://hal.archives-ouvertes.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf In this article the authors considers $K \subset \{ x \in R^n ; x_1 = 0 \}$ a smooth, ...
0
votes
2answers
504 views

Poisson inequality for subharmonic functions

This is probably a very basic matter, but I am looking for a proof of the Poisson inequality for subharmonic functions, which reads $$\varphi(r \mathrm{e}^{\mathrm{i} \theta})\leq\frac{1}{2\pi} \...
3
votes
1answer
271 views

A series question related to solution of Laplace equation

Let $u(x,y)$ be the solution of the Laplace equation $\Delta u=0$ on the unit square $(0,1)\times (0,1)$ with boundary condition: $$ u(x,1)=1, u(x,0)=0, u(0,y)=0, u(1,y)=0$$ The series solution is $$\...