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Results tagged with ap.analysis-of-pdes
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user 36688
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
23
votes
1
answer
1k
views
Eigenvalues of Laplace operator
Assume that $(M,g)$ is a Riemannian manifold.
Is there any relation between the sequence of eigenvalues of Laplace operator acting on the space of smooth functions and the sequence of eigenvalues of …
- 356
18
votes
3
answers
2k
views
Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order deriv...
EDIT: According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated t …
- 356
11
votes
2
answers
1k
views
Elliptic operators corresponds to non vanishing vector fields
Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting dy …
- 356
6
votes
0
answers
283
views
A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory
The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in thi …
- 356
6
votes
2
answers
613
views
On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemann...
Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field …
- 356
6
votes
1
answer
570
views
A vector space associated with a vector field on a symplectic manifold
$\DeclareMathOperator\Div{Div}$Edit: The correct formulation of the vector space $S(X)$ which is defined in this question is the following:$$S(X)=\{Y\in \chi^{\infty}(M)\mid X.\omega(X,Y)=(1 …
- 356
6
votes
0
answers
266
views
Elliptic foliations of the plane
A $1$ dimensional foliation of the plane $\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field $X$ with the following properties:
The differential operator …
- 356
6
votes
2
answers
387
views
A generalization of holomorphic functions
Assume that $U$ is an open set in the complex plane $\mathbb{C}$ and $A$ is a real $2\times 2$ matrix.
We define
$$\mathcal{S}_{A}=\{f:U\to \mathbb{C}\mid Df.A=A.Df \}$$
where $Df$ is the $2\times 2$ …
- 356
6
votes
2
answers
1k
views
The adjoint operators as elliptic operators
Edit:
It seems that the link "https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt" which contains a talk by Loic Teyssier about homological equations and vanishing cycles is temporally …
- 356
6
votes
1
answer
340
views
Fredholm theory of non elliptic operators
In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in th …
5
votes
0
answers
278
views
The Spectrum of certain differential operators
We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.
We consider the following polynomial vector field on …
- 356
5
votes
0
answers
217
views
A differential operator analogy of certain fact in real analysis of smooth functions
Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$.
Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$.
Ass …
- 356
5
votes
1
answer
450
views
An alternative representation of the principal symbol of the Laplace operator
Assume that $(M,g)$ is a $n$ dimensional Riemannian manifold. We denote by $\Delta$, the Laplacian associated to this Riemannian structure.
Are the following two conditions equivalent?
First condition …
- 356
5
votes
2
answers
459
views
A question on certain elliptic PDE
Consider the elliptic PDE
$$(CR)\;\;\;\;\;\;\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
And its consequence $$(LAP)\;\;\;\;\;\;U_{xxxx}+U_{yyyy}=0$$.
Somehow, these equations are si …
- 356
5
votes
1
answer
414
views
Fredholm index vs. Limit cycle theory
Let $A$ be the algebra of all smooth functions $f: \mathbb{R}^2 \to \mathbb{R}$ such that $f$ is flat at the origin and is real analytic on $\mathbb{R}^2 \setminus \{0\}$.
Let $B $ be …
- 356