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Results tagged with ap.analysis-of-pdes
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user 13972
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
5
votes
Accepted
Asymptotic parametrization for negatively curved surfaces
As asked, the answer to the question is 'no'. The simply-connected cover $f:\mathbb{R}^2\to S$ of Sherck's first surface $S$ (which is defined in $\mathbb{R}^3$ by the equation $\mathrm{e}^{z} \cos x …
- 108k
10
votes
Accepted
Understanding exterior differential systems
Here's an expansion of my comment that the natural formulation of this problem as an EDS is on the coframe bundle $\pi: P\to M$, which, I hope, will be helpful. Also, because it will match my usual n …
- 108k
3
votes
Integrability of modified diagonalizable Jacobian
It has taken me a while to find time to write a more comprehensive answer to the above question. It turns out that for general dimension $N$, the overdetermined PDE system involved is not involutive, …
- 108k
6
votes
Accepted
Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Yes, there is a standard procedure to analyze such systems, essentially, it is Cartan's method of prolongation combined with his theory of involutive systems. There are other approaches as well, but …
- 108k
1
vote
Accepted
Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
In their comments to my first answer, the OP has clarified that they did not mean to regard the metric $h$ as a given, but, rather, an output of the problem of prescribing coframings by specifying the …
- 108k
3
votes
Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Let me phrase the problem as I understand the given data and then describe how the 'theory of exterior differential systems' would be applied.
One starts with a compact Riemannian $3$-manifold $(M,h)$ …
- 108k
3
votes
Heat kernel of left-invariant metric on 3-sphere
I don't know a formula for $d(e^x,e^y)^2$, and I suspect that there is no simple formula, but the answer to Q2' is 'no'. The right hand side of (3) is linear in $Q^{-1}$, but it is not hard to see th …
- 108k
5
votes
Accepted
On a result of Cartan for homogeneous manifolds arising from a quotient of discrete subgroups
The result that you are looking for is not in Élie Cartan's 1936 book La topologie des groupes de Lie because it was not known to be true at the time the book was written. Indeed, as Cartan remarks i …
- 108k
8
votes
Accepted
Building a geodesic conjugate parameterization on catenoid
I thought about your problem and realized that there is no coordinate parametrization of the catenoid with the properties that you want.
Here is my argument: First, note that, in the given $uv$-param …
- 108k
3
votes
Accepted
Method of characteristics with 2 dependent variables in 3 dimensions
The method of characteristics is a bit strange here because the equation is underdetermined, so one can't expect to be able to specify a solution by fixing initial data for $u$ and $v$ along a surface …
- 108k
5
votes
Accepted
Approximate isometric embeddings of surfaces
I think that the answer is 'yes' if $U$ is simply-connected, because there is a way to construct a candidate 'approximate surface' from 'approximate solutions' of Gauss and Codazzi, but a more useful …
- 108k
3
votes
Accepted
Hyperbolic system of PDEs with elliptic-like boundary contions
Here is an example for which there is no solution: Let $\Omega_1$ be defined by $x^2+y^2\le 1$ and $\Omega_1$ be defined by $X^2+Y^2\le R^2$, where $R>0$ is large. Take $Z(X,Y) = 0$. Then one is as …
- 108k
9
votes
Accepted
Existence of solution to linear inhomogeneous first order PDEs systems
You are correct that Cauchy-Kovalevskaya does not apply directly to this problem, but there are other theorems that give sufficient conditions, provided that you make certain basic regularity assumpti …
- 108k
13
votes
Accepted
Does the first Laplacian eigenfunction on a homogeneous space have a unique maximum?
The flat torus $\mathbb{T} = \mathbb{R}^2/\Lambda$ gives a counterexample: The first nontrivial eigenvalue is of the form $\lambda_1 = \xi_1^2+\xi_2^2$, where $\xi = (\xi_1,\xi_2)$ is a nonzero eleme …
- 108k
9
votes
Accepted
Solution existence in a pde system
Yes. Here is a general approach to this problem: Suppose that one has two functions a>0 and b on some interval $I\subset\mathbb{R}$ and one wants to know whether there is a solution $f$ to the syste …
- 108k