11
votes
Accepted
On eigenfunctions of the Laplace Beltrami operator
For $\mathrm{SU}(2)$, with the scale for the biïnvariant metric so that it becomes isometric to the unit $3$-sphere $S^3$ in Euclidean $4$-space, it is well-known what the eigenvalues of the Laplace-...
- 108k
9
votes
Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation
The first derivatives of $u$ are harmonic too. Therefore the Maximum Principle tells us
$$\|\nabla u\|_{C^0(\bar\Omega)}=\left\|\left.\nabla u\right|_{\partial\Omega}\right\|_{C^0(\partial\Omega)}.$$
...
- 52.3k
8
votes
Convex solutions of the Poisson equation
I apologize for having posted this question too early. I realize that the answer to the first question is negative.
Actually suppose that $D=D(0;R)$ is a disk and $f=f(r)$ is a radial function. If a ...
- 52.3k
6
votes
Accepted
Use stochastic process to express solution to Laplace equation in the whole space
If $f(x) / (1 + |x|)$ is integrable, then the solution $u$ is equal to the Newtonian potential of $f$:
$$ -u(x) = \frac{1}{4\pi} \int_{\mathbb R^3} \frac{f(y)}{|x - y|} \, dy . $$
And the Newtonian ...
- 17.2k
6
votes
Accepted
Intuition for Agmon-Douglis-Nirenberg ellipticity
The idea of the weights used in the ADN definition of the principal symbol is quite natural in the context of graded vector spaces or more generally graded modules.
In the equation $u = Dv$, $u=(u_1,\...
- 21.5k
6
votes
Laplace beltrami eigenspaces of compact Lie groups
Too long for a comment, but this is not a full answer.
I'm not sure about the sums of LB eigenvalues, but the analogue with highest weights instead of the corresponding eigenvalues should hold.
Recall ...
- 3,186
5
votes
Accepted
Dirichlet-to-Neumann map on Lipschitz domains
Let $u$ be the solution of the Dirichlet problem for Laplacian in a Lipschitz domain with boundary data $g$. Then, for every $s\in [1/2,3/2]$,
$$
\| u \|_{H^{s}\,(U)} \leq C \| g \|_{H^{s-1/2}\,\,\,\,(...
- 2,155
4
votes
Intuition for Agmon-Douglis-Nirenberg ellipticity
The question whether appropriate weights exist is discussed in the following paper:
L.R. Volevich, A problem of linear programming arising in differential equations,
Uspekhi Mat. Nauk 18 (1963), No. 3,...
- 13k
3
votes
Accepted
Variation of the Green function with respect to the metric
It seems that the naive derivation from the path integral only picks up the term coming from quasiconformal variations. Combining the known result for the quasiconformal variation with the much more ...
- 661
3
votes
Separable coordinate systems for the Laplace and Helmholtz equations?
There is a modern geometric theory of (orthogonal) separation of variables on spaces of constant curvature (of arbitrary dimension), which includes an exhaustive classification. It follows from the ...
3
votes
Accepted
the curvature wave equation
A similar equation that's been used is the Penrose wave equation
\begin{equation}
\square R_{a b c d} = 2 R_{a e d f} R{_b}{^e}{_c}{^f} - 2 R_{a e c f} R{_b}{^e}{_d}{^f} - R_{a b e f} R{_{c d}}{^{e f}}...
- 712
2
votes
Neumann problem for the Laplacian with Dirac delta functions
It occurs to me that perhaps one can split the problem in two
$$\begin{cases}
\Delta f^+= 0 & B \\
\partial_n f^+=\delta(x-x_0) & \partial B \\
\end{cases}$$
and
$$\begin{...
- 21
2
votes
Accepted
Laplace equation on the disk with Robin boundary condition
The normal trick is to set it up as an eigenvalue problem, namely to look instead at
$$\Delta u = 0, ~~\frac{\partial u}{\partial n} = \lambda b(x) u(x)~~ \forall x \in \partial D. (1)$$
You first ...
- 2,494
1
vote
What's going on with the two-dimensional Helmholtz equation?
You seek the solution of
$$(\nabla^2+\kappa^2+i\epsilon)G(\mathbf{r})=\delta(\mathbf{r}),$$
in the limit $\epsilon\rightarrow 0^+$, which is given by a Hankel function of the first kind,
$$G(\mathbf{r}...
- 188k
1
vote
Laplace equation with integral source terms
More a long comment than an answer, but perhaps it could be useful to you: one reference I know that deals extensively with non-local equation (read "integrodifferential" and "with ...
- 6,357
1
vote
Laplace equation, medium discontinuity and finite difference method
A surface charge will accumulate on the interface where the dielectric constant has a discontinuity. You need to calculate this surface charge and include it into the discretised Poisson equation. ...
- 188k
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