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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.
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
2
votes
1
answer
131
views
Global first integral for certain $3$ dimensional system
A physicist colleague asks me the following question. I have no idea to answer him. Your answer is very appreciated.
Is there a global first integral on $\mathbb{R}^3$ for the following vector field? …
- 356
0
votes
0
answers
125
views
A heat equation approach to the perturbation of vector field with center
Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it.
We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)= …
- 356
1
vote
0
answers
171
views
The Lie algebra of Harmonic functions
Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ with corresponding volume form $\omega= \sqrt{det(g_{ij})} dx \wedge dy$ and the corresponding Laplace operator $\Delta$ such that the space …
- 356
0
votes
1
answer
307
views
Vector field with Harmonic flow
Assume that $(M,g)$ is a Riemannian manifold. A vector field $X$ on $M$ is called a harmonic vector field if the corresponding $1$-form $\alpha$ with $\alpha(Y)= \langle X,Y \rangle_g …
- 356
2
votes
0
answers
608
views
Is Laplacian a surjective operator?
For a closed manifold the laplacian is almost surjective operator since the index of $\Delta$ is zero and there is no a non constant harmonic function. So the codimension of the image …
- 356
0
votes
1
answer
147
views
Is it a sufficient condition for linearity?
Edit: According to the comment by LSpice we come back to the initial version of this question
Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth function such that for every $x\in \mathbb{R}^n$ the der …
- 356
3
votes
1
answer
155
views
Is the space of harmonic functions invariant under the derivational operator associated with...
Assume that $V$ is a vector field on a
Riemannian manifold $(M,g)$ with natural volume form $\Omega$ arising from $g$.
Assume that the solution curves of $V$ are parametrized geodesics of the Riema …
- 356
0
votes
0
answers
62
views
A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact paralleliza...
Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2} …
- 356
3
votes
0
answers
89
views
Cycloid on manifolds
Inspired by differential equation
$$y(1+y'^2)=c$$
which generates the cycloid we consider the following differential equation on a Riemannian manifold:
$$f(1+|\nabla f|^2)=c$$
On the other hand th …
- 356
4
votes
0
answers
243
views
Dynamical obstruction for a vector field to have a Harmonic divergence
Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic …
- 356
2
votes
1
answer
497
views
The flow of Harmonic vector fields
A map or a vector field $g: \mathbb{R}^n \to \mathbb{R}^n $ is called a harmonic map if all its components are harmonic functions.
Motivated by conversations on this questions we ask: …
- 356
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
1
vote
1
answer
123
views
Invariance of the space of harmonic functions under derivation associated to a non-vanishing...
Let $X$ be a non-vanishing real analytic vector field on an open manifold $M$. What kind of obstructions would appear when we search for a Riemannian metric on $M$ such that the space of …
- 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