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Results tagged with ap.analysis-of-pdes
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user 1540
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
7
votes
fixed point arguments in PDE
The graphical minimal surface equation is a great example of a PDE where Leray-Schauder fixed point theory is applied:
$$
\left(\delta_{ij} - \frac{D_i u D_j u}{1+|Du|^2}\right)D_{ij}u = 0
$$
This r …
- 7,197
4
votes
Accepted
Derivation of yamabe flow
The Yamabe flow is (up to a constant) the gradient flow of the Yamabe functional on the unit volume conformal class, as you expected. The comment by @Mark Peletier hints at your error: you aren't usin …
- 7,197
4
votes
Accepted
Analytic dependence on the metric
I find it strange that Besse does not discuss this! Here's my understanding of the issue:
Sun and Wang care about analyticity because they want to apply the Łojasiewicz--Simon inequality. This inequa …
- 7,197
6
votes
metric has morse index 2
I assume you mean these notes: http://www.math.jhu.edu/~js/Math646/schoen.totalscalar.pdf?
In general it does not make sense to say that a "metric has morse index 2" without any context. Before proce …
- 7,197
22
votes
5
answers
16k
views
Physical interpretation of Robin boundary conditions
In a (bounded) domain $\Omega \subset \mathbb{R}^n$, if we're studying the Laplace equation or heat equation or such PDE's we can impose the Dirichlet
$u|_{\partial\Omega} \equiv 0$,
Neumann
$D_{\nu} …
- 7,197
4
votes
Progress on isospectral plane domains
The Dirichlet/Neumann spectrum determines the domain among the class of analytic domains with some discrete symmetries by work of Zelditch.
The unit disk is determined by its Dirichlet spectrum (amon …
- 7,197
2
votes
Replacing large-dimensional ODE systems with one PDE
This idea is studied in the field "kinetic PDE." See these lecture notes by Clément Mouhot:
http://cmouhot.wordpress.com/1900/10/25/mathematical-topics-in-kinetic-theory-part-iii-course/
Ch 2 in par …
- 7,197
4
votes
Accepted
On-diagonal to off-diagonal heat kernel lower bounds, Davies' argument
This is a classic trick for heat kernel proofs:
The idea is that if $f(t,x)$ is a solution to the heat equation with initial data $f_0(x)$, then $g(t,x) := f(t+s,x)$ is a solution to the heat equatio …
- 7,197
5
votes
Accepted
What is the infinite Morse index solution?
NEW ANSWER:
Yes there exist infinite Morse index solutions in all dimensions $N\geq 3$. For example you can take the solution in the Li Chen paper
$$
\phi(x,y) = \frac{\ln(32)}{(4+|(x,y)|^2)^2}
$$
and …
- 7,197
15
votes
Accepted
A long-lasting conjecture of Pólya & Szegő
I'm pretty sure that this is still open for $n$-gons (with $n\geq 5$). As far as I know, basically no progress has been made since the original proofs for triangles/quadrilaterals.
There have been som …
- 7,197
10
votes
Accepted
Morse index in PDEs
In finite dimensional Morse theory, you study a function $f:M\to\mathbb{R}$ and look for it's critical points, i.e., where $(df)_p = 0$. Then, Morse theory says that a count of these critical points i …
- 7,197
5
votes
Accepted
Finding vector fields on $S^2$ with equal divergence
I think that this is not possible:
Per my comment on Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics you want to solve
$$
\textrm{div} (Y) = -2a\cdot x
$$
for $Y$ or …
- 7,197
5
votes
3
answers
2k
views
Characterizing the harmonic oscillator creation and annihilation operators in a rotationally...
I am interested in a characterization of the creation and annihilation operators that is in some sense invariant under $O(n)$ rotations of $\mathbb{R}^n$:
Background
The Harmonic Oscillator on $\mat …
- 7,197