All Questions
8 questions with no upvoted or accepted answers
5
votes
0
answers
374
views
A question about Carleman linearization
Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻²
Let $f$ be ...
- 1,590
3
votes
0
answers
124
views
Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different ODE.
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
- 969
2
votes
0
answers
445
views
Lax Milgram for non coercive problem?
I obtained the variational form of my problem. and the bilinear form is below.
Bilinear Form Let $\Omega\subset\mathbb{R}$ be an open set. For $u,v\in H^1_0(\Omega)$. I have
$$a(u,v)=\int_\Omega u(t)...
- 55
2
votes
0
answers
87
views
1D inhomogeneous linear Schrodinger equation
I have the following problem:
$iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(...
2
votes
0
answers
79
views
Point Spectrum of a Second Order System of Differential Equations
Consider the following operator acting on $H^1(\mathbb{R})$
$$
\mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) \...
- 516
1
vote
0
answers
93
views
inverse problem to resolution of the identity
Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define $$E_\mathfrak{m}:L^2\...
- 587
1
vote
0
answers
135
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growth bound for solution of an ordinary integro-differential equation
I am considering the following ordinary integro-differential equation
$$
A g = \sigma^2(y) + \int_\mathbb{R} \nu(y,dz) z^2
$$
where
$$
A = b(y) \partial + \frac{1}{2} \sigma^2(y) \partial^2
+ \int_\...
- 11
0
votes
0
answers
71
views
Sufficient condition to be increasing, following a vector field
Let $f\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ $(n\geq 1)$ be an observable, and let $v^1,v^2\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ be two vector fields such that for any $(x^1_t)_{t\geq 0}$ ...
- 449