# Questions tagged [laplace-equation]

The laplace-equation tag has no usage guidance.

38
questions

7
votes

2
answers

497
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### Intuition for Agmon-Douglis-Nirenberg ellipticity

First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently.
I am trying to understand the definition of ellipticity of systems due to ...

1
vote

0
answers

44
views

### Gradient estimate of the eigenfunction of Laplacian on hyperbolic space

I am trying to understand the asymptotic behaviors of the gradient of the eigenfunction function of the Laplace-Beltrami operator on the hyperbolic plane $\mathbb{H}^2$. Specifically, my focus lies on ...

2
votes

0
answers

84
views

### A question on the maximum principle of second order elliptic equations

Let $Lu=a^{ij}u_{ij} + b^i u_i$ be an elliptic operator of second order in a bounded domain $\Omega$. Assume that $a^{ij}$ is uniformly elliptic. Then it's well known that the following maximum ...

0
votes

0
answers

68
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### How to calculate the weights for Discrete Laplacian Operator?

I am following this paper step by step and want to build an isotropic Laplacian kernel. As shown in the following figure, I can understand until using Taylor to expand the 2D discrete Laplacian ...

4
votes

1
answer

217
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### Laplace beltrami eigenspaces of compact Lie groups

For a Riemannian manifold $\mathbb M$, let $0=\lambda_0<\lambda_1<\cdots$ be the eigenvalues of (negative of) its Laplace-Beltrami $-\Delta_{\mathbb M}$, with corresponding eigenspaces $\mathcal ...

1
vote

0
answers

74
views

### Are eigenfunctions of the Dirichlet problem for the Laplace equation uniformly bounded?

Let $Q\subset \mathbb R^n$ be a bounded domain with boundary $\partial Q\in C^\infty$ and $\varphi_1,\varphi_2,\ldots$ are eigenfunctions of the Dirichlet problem for the Laplace equation in $Q$ ...

0
votes

0
answers

101
views

### Estimate value of harmonic function in the annulus

Let $D = B_{2r}(0)\backslash \overline{B}_r(0)$. Assume $Lu = 0$ in $D$ where $L$ is a uniform elliptic operator with constant coefficients
$$
Lu = \sum_{i,j} a_{ij}u_{x_i}u_{x_j}, \qquad \lambda |\xi|...

8
votes

1
answer

477
views

### Dirichlet-to-Neumann map on Lipschitz domains

Let $\Omega$ be a bounded domain with a Lipschitz boundary. Consider the Dirichlet-to-Neumann map $\Lambda:H^{\frac{1}{2}}(\partial \Omega)\to H^{-\frac{1}{2}}(\partial \Omega)$ defined via
$$ \langle ...

6
votes

1
answer

364
views

### Convex solutions of the Poisson equation

Let $D$ be a planar, bounded, convex open domain. Given a positive function $f:D\to(0,+\infty)$, let us consider the Poisson equation
$$\Delta u=f\quad\hbox{in }D.$$
Not specifying any boundary ...

3
votes

1
answer

1k
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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 ...

4
votes

0
answers

81
views

### 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 $(M,g)$...

7
votes

1
answer

975
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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)?

2
votes

0
answers

124
views

### 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 ...

4
votes

1
answer

407
views

### 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
\...

1
vote

0
answers

125
views

### 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 ...

2
votes

2
answers

435
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, $...

2
votes

0
answers

107
views

### $W^{1,p}-$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 ...

1
vote

1
answer

493
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 ...

3
votes

0
answers

64
views

### 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
votes

2
answers

761
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
vote

0
answers

124
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
votes

0
answers

89
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

1
answer

459
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

0
answers

118
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
vote

1
answer

391
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}\...

10
votes

1
answer

794
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
vote

0
answers

181
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 ...

1
vote

0
answers

134
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

1
answer

176
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 ... $\mathbb{P}_1$ ...

4
votes

1
answer

339
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
votes

1
answer

225
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 $u$ ...

10
votes

4
answers

2k
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
vote

1
answer

755
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
votes

0
answers

205
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
vote

0
answers

40
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

132
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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, ...

0
votes

2
answers

752
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

1
answer

388
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 $$\...