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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.
25
votes
Accepted
When is a given matrix of two forms a curvature form?
The answer is generally 'no'; for most $F$ that satisfy your condition, there will not exist an $A$ that satisfies $F = dA + A\wedge A$.
The easiest counterexample I know of is when $n=4$ and the m …
- 108k
22
votes
Accepted
Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order deriv...
As Raziel wrote, the local question is whether one can find a local basis of orthonormal vector fields that are divergence-free.
It's true that, in dimension $2$, this can only be done if the metri …
- 108k
19
votes
Vanishing eigenvalues of Jacobian
Actually, I made a sign mistake in my original calculation of $\det(Df)$, so my original argument was not right. Sorry. Here is (I hope and believe) a correct one.
In fact, when $n=2$, such an $f$ …
- 108k
18
votes
Accepted
partial differential equation for ruled surfaces
Here is a test for when a surface of the form $z = f(x,y)$, where $f$ is a sufficiently smooth function of two variables, is ruled.
To begin, set $I\!I = f_{xx} dx^2 + 2f_{xy}dxdy + f_{yy}dy^2$. If …
- 108k
16
votes
Accepted
Representing immersions from a surface into 3-space
I may have to enter this as a sketch and fill in details later, but I thought that I'd go ahead and get the main ideas out there.
The first thing to notice is that the given problem is equivalent to …
- 108k
13
votes
Special Second-Order PDE
This is not really an answer, just a sequence of comments that are all related, but are too long to put into a comment field.
First, some good news:
When $n=1$, there's always a (unique) solution f …
- 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
12
votes
Accepted
Theorems that tell if an explicit analytical solution is possible for nonlinear PDEs
As far as (local) power series solutions go (i.e., in the analytic category) the main existence theorem is the Cauchy-Kowalewski Theorem (in the determined, non characteristic case) and its generaliza …
- 108k
12
votes
Accepted
Vector field with Harmonic flow
Well, right away, you can see that the answer is 'no', in general. Consider the round $n$-sphere $S^n$ with its standard metric. When $n>1$, it has no nonzero harmonic $1$-forms, but it has nontrivi …
- 108k
12
votes
Accepted
Symbols of elliptic operators
Maybe I'm misunderstanding something, but it seems that the answer is probably 'no', at least if $d = \dim_\mathbb{C} V$ is large enough.
What really matters is the $n$-dimensional real subspace $\ …
- 108k
12
votes
Accepted
Pde system problem
I assume that, in the surface case, the OP wants to interpret $S$ as a surface endowed with a Riemannian metric and wants to understand the solutions to the equations $\Delta f - hf^2 = 0$ and $|\nabl …
- 108k
12
votes
Accepted
Global orthogonal coordinates on the open unit ball
This problem is just the classical problem of finding global Tchebychev coordinates on hyperbolic $n$-space. By Hilbert's Theorem, this is impossible when $n=2$. The problem remains open in higher d …
- 108k
11
votes
Accepted
Vanishing eigenvalues of Jacobian
For some reason, the 'edit' button didn't appear for my earlier answer, maybe because it was already accepted. Thus, I'm adding the general $n$ argument as a separate answer.
In fact, there is a str …
- 108k
11
votes
Accepted
Solutions to the eikonal equation
Note: I have realized that, using the Stable Manifold Theorem, one can prove the smoothness of the solution $\phi$ that I describe below. Thus, I am modifying my answer to incorporate that.
Local …
- 108k
10
votes
Accepted
A question on certain elliptic PDE
1.If $U$ satisfies LAP then there exists a $V$ such that $(U,V)$ satisfies CR. In fact, $V$ is unique up to the addition of a term of the form $a + bx + cy + d xy$, where $a$, $b$, $c$, and $d$ are c …
- 108k