Questions tagged [uniqueness-theorems]
The uniqueness-theorems tag has no usage guidance.
15 questions with no upvoted or accepted answers
9
votes
0
answers
203
views
Reduction of the $0$-handle data in Lurie's classification of TFT
A vital part of Jacob Lurie's classification of fully extended topological
field theories [1], very roughly, says that any representation of the
n-Cobordism category $Z: {\rm Cob}_{{n}} \to C$ depends ...
- 5,230
5
votes
0
answers
234
views
Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$
My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
- 1,075
4
votes
0
answers
122
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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
3
votes
0
answers
102
views
Uniqueness continuation property for parabolic equation
Consider the following parabolic equation:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial \rho }{\partial t}-\Div\left( a\left( x\right) \nabla
\rho \right) +p(x)\rho = 0 & \...
2
votes
0
answers
102
views
Existence of unique-up-to-shift solution of a Volterra equation
Let $\Delta=\{(t,s):\ 0<s\leq t\leq1\}$, and suppose $k:\Delta\to\mathbb R$ and $f:(0,1]\to\mathbb R$ are continuous. Further assume that for every $t\in(0,1]$, the function $k(t,\cdot):(0,t]\to\...
- 537
2
votes
0
answers
154
views
Uniqueness of the solution to systems of first-order linear PDEs
Context:
Let $\Omega \subset \mathbb{R}^p$ be an domain.
For functions $A_{jk}^i : \Omega \to \mathbb{R}$ and $B_k^i : \Omega \to \mathbb{R}$ with some regularity, I am interested in the following ...
- 51
1
vote
0
answers
93
views
+100
Uniqueness of bubbling points in Struwe's global compactness theorem
I am reading the following paper of Struwe in which he proves the following result:
Proposition 2.1:
Let $n\geq 3$, $\lambda \in \mathbb{R}$ and $\Omega$ be a smoothly bounded domain in $\mathbb{R}^{n}...
- 547
1
vote
0
answers
205
views
Uniqueness for Volterra equation with initially (linearly) unbounded kernel
Letting $D:=\{(x,y):\ 0\leq x\leq y\leq1 \text{ and } y>0\}$, I have a continuous function $k:D\to[0,\infty)$ that satisfies some properties that I list below. I'm interested in continuous and ...
- 537
1
vote
0
answers
133
views
Infinite dimensional matrix solvability
In order to solve a boundary problem for a conducting hemisphere, the following matrix equation arises (derived from the boundary condition on the curved part of the surface) where we must solve for ...
- 573
1
vote
0
answers
82
views
Uniqueness of global solution
I am reading Section 3.3 of this paper, and trying to understand the proof of uniqueness of a global solution to the following equation defined on the Torus $\mathbb{T}^3$
\begin{align*}
\mathrm{d} \...
- 101
1
vote
0
answers
94
views
Reference request: existence/uniqueness of solutions to convection diffusion equations
I am looking for a reference wherein existence and uniqueness results are proven for a system of PDEs of the form
$$
\frac{\partial Q}{\partial t} + A \frac{\partial Q}{\partial x} = f(Q,x,t) + \frac{...
- 111
0
votes
0
answers
46
views
Uniqueness results for linear first order systems of PDEs
Context: I have the following system of PDEs, for an unknown function $u:\mathbb{R}^{n+1}\to \mathbb{C}^m$ (it is a system in the components of $u$):
$$u_{x_0}=\sum_{i=1}^n A_iu_{x_i} + B(x)u\qquad u(...
- 1,205
0
votes
0
answers
62
views
Proving the uniqueness of the solution to a functional equation involving integral
Consider the functional equation
$$
g\left(a\right) = \int_0^1 \frac{e^{c(a,h)+f(h)}}{1+e^{c(a,h)+f(h)}}dh
$$
and this holds for all $a$. $g(a)$ and $c(a,h)$ are known functions on a continuous ...
0
votes
0
answers
133
views
Implicit function theorem when $dF/dy = 0$ but under monotonicity constraint of the implicit function $y(x)$
I am looking for an extended version of the implicit/inverse function theorem that would show uniqueness of a strictly increasing implicit function, even when the derivative condition is violated (e.g....
- 151
0
votes
0
answers
41
views
Existence and Uniqueness of lifting Hele-Shaw problem
I am researching for the existence and uniqueness of solutions for the equation in figure below
enter image description here
$$\nabla\cdot u = \frac{\dot b(t)}{b(t)} \text{ in }\Omega(t) \tag{1}$$
The ...
