All Questions
Tagged with uniqueness-theorems differential-equations
7 questions
3
votes
1
answer
243
views
Existence and uniqueness of solutions for continuous and directionally differentiable ODE
Given $f:\mathbb{R}^n \to \mathbb{R}^n$ continuous and directionally differentiable (i.e., such that the directional derivative of $f$ exists for any direction) at a neighborhood $N$ of $x_0\in\mathbb{...
- 63
2
votes
2
answers
135
views
Uniqueness of a second order linear ode
I am currently confronted with the following equation $$
0=w''(t)(t^2-t)+w'(t)((2n-1)t^2-n)+w(t)(n-1)^2t
$$ for $t\in(-1,1)$. So $w:(-1,1)\rightarrow\mathbb{R}$. The following assumption is also in ...
- 21
0
votes
0
answers
133
views
Implicit function theorem when $dF/dy = 0$ but under monotonicity constraint of the implicit function $y(x)$
I am looking for an extended version of the implicit/inverse function theorem that would show uniqueness of a strictly increasing implicit function, even when the derivative condition is violated (e.g....
- 151
0
votes
0
answers
41
views
Existence and Uniqueness of lifting Hele-Shaw problem
I am researching for the existence and uniqueness of solutions for the equation in figure below
enter image description here
$$\nabla\cdot u = \frac{\dot b(t)}{b(t)} \text{ in }\Omega(t) \tag{1}$$
The ...
3
votes
2
answers
376
views
Is the converse of Osgood criterion for ODEs also true?
Namely, Assuming that $f$ is a continuous real function and $f(0)=0$ , $f(x)>0 $ when $x\neq 0$,
Consider the differential equation $x'= f(x)$ with the initial value $x(0)=0$ , is it true that if ...
- 141
4
votes
0
answers
122
views
Ricci flow on locally symmetric noncompact manifold
As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ...
- 123
2
votes
1
answer
155
views
Lotka Volterra existence of Caratheodory solution
I strive to prove that the following system of differential equations:
$$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$
has a unique Caratheodory solution ...
- 1,759