Questions tagged [laplace-equation]
The laplace-equation tag has no usage guidance.
29
questions
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answers
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What's going on with the two-dimensional Helmholtz equation?
I've come to realize that its somehow harder to find results for this equation than for the three-dimensional one.
For example the wikipedia article on Green's functions has a list of green functions ...
- 241
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0
answers
58
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On the convergence of the spectral decomposition of a harmonic function
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
- 2,858
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votes
1
answer
297
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On eigenfunctions of the Laplace Beltrami operator [closed]
How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?
- 61
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votes
0
answers
44
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Harmonic function over a square with linear Neumann boundary conditions
For a rectangle with height 1 and length 2, here is the unique numerical solution
(showing contours of the equipotential from 0, defined by the bottom, to 0.54, the numerically-calculated maximum)
to ...
- 1,425
4
votes
1
answer
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Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$-smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation
\...
- 19k
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46
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Regularity of Laplace equation on non-convex polyhedral domain
This might be a known problem, but I could not find a precise answer.
I have the following Laplace equation
\begin{equation}
\begin{cases}
-\Delta u = f & x \in \Omega;\\
\quad\: u = g & x \in ...
- 203
2
votes
2
answers
196
views
Use stochastic process to express solution to Laplace equation in the whole space
Consider the Laplace equation in $\mathcal{R}^3$
\begin{equation}
\Delta u = f, ~~~\lim_{x\to \infty} u(x) = 0.
\end{equation}
Here we assume $f$ is a smooth, compactly supported function. Of course, $...
- 883
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votes
0
answers
53
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$W^{1,p}-$regularity on the boundary for solution of Laplace equation with Robin boundary condition
I came across with the attached paper and here is the part that I try to understand.
If the non-tangential maximal function of $\nabla u$, i.e $(\nabla u)^*$, belongs in $L^p(\partial \Omega)$, then ...
- 227
0
votes
1
answer
198
views
the curvature wave equation
I was referred here from this question I asked on stackexchange. And now that I'm here, I see that this other question about geometric wave equations is very closely related to mine. But I have a ...
- 103
3
votes
0
answers
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Elliptic equations in semi-infinite strips
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good ...
- 2,858
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votes
2
answers
536
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:
$$\Delta u = 0, ~~\frac{\partial u}{\partial n} = b(x) u(x)~~ \forall x \in \...
- 883
1
vote
0
answers
96
views
Can complex analysis be used to solve Laplace's equation in three dimensions? [closed]
Can complex analysis be used to solve Laplace's equation in three dimensions? Or is it restricted to cases which have 2D symmetry?
All of the examples I've seen are 2D, never 3D.
It would seem ...
- 111
2
votes
0
answers
80
views
Neumann problem on a convex domain
Let $\Omega$ be a convex open bounded subset of $\mathbb R^n$ and let $u$ be the solution of
$$
\begin{cases}
∆ u=1\quad\text{in $\Omega$,}
\\
\frac{\partial u}{\partial \nu}=\frac{\vert \Omega\vert_n}...
- 13.2k
1
vote
1
answer
308
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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 ...
- 47
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votes
0
answers
108
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 ...
- 563
1
vote
1
answer
270
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}\...
- 21
9
votes
1
answer
607
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 ...
- 307
1
vote
0
answers
168
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 ...
- 563
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vote
0
answers
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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 $X$ ...
- 203
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votes
1
answer
123
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 ... $\mathbb{P}_1$ ...
- 171
4
votes
1
answer
218
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 ...
- 2,221
2
votes
1
answer
204
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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 $u$ ...
- 2,221
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votes
3
answers
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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 ...
- 261
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vote
1
answer
505
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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
votes
0
answers
183
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]...
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1
vote
0
answers
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
0
answers
130
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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, ...
- 249
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votes
2
answers
597
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
$$\varphi(r \mathrm{e}^{\mathrm{i} \theta})\leq\frac{1}{2\pi} \...
- 11
3
votes
1
answer
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views
A series question related to solution of Laplace equation
Let $u(x,y)$ be the solution of the Laplace equation $\Delta u=0$ on the unit square $(0,1)\times (0,1)$ with boundary condition:
$$ u(x,1)=1, u(x,0)=0, u(0,y)=0, u(1,y)=0$$
The series solution is $$\...
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