Questions tagged [existence-theorems]
The existence-theorems tag has no usage guidance.
60 questions
22
votes
12
answers
2k
views
Instances where an existence result precedes the constructive version
The basic motivation here is to encourage and inspire - via examples - the pursuit of alternate proofs of existing results that might be more accessible and intuitive by cataloging success stories. ...
21
votes
3
answers
2k
views
Is this 1974 claim still valid?
In G. F. Simmons' Differential Equations book (p.141), the following claim is made:
“... As a matter of fact, there is no known type of second order linear differential equation- apart from those with ...
- 2,855
20
votes
3
answers
2k
views
Simplest diophantine equation with open solvability
What is the simplest diophantine equation for which we (collectively) don't know whether it has any solutions? I'm aware of many simple ones where we don't know (whether we know) all the solutions, ...
- 351
17
votes
2
answers
4k
views
Is there a relationship between a quotient group of the fundamental group of X and the fundamental group of a quotient topology of X?
Let ($X$, $x_0$) be a topological space with a base point, and denote the fundamental group of $X$ as $\pi_1(X)$. Let $N$ be a normal subgroup of $\pi_1(X)$.
Does there necessarily exist an ...
- 271
9
votes
3
answers
3k
views
Existence of Rational Orthogonal Matrices
Question:
Let $A\in\mathbb{R}^{n \times n}$ be an orthogonal matrix and let $\varepsilon>0$. Then does there exist a rational orthogonal matrix $B\in\mathbb{R}^{n \times n}$ such that $\|A-B\|<\...
9
votes
0
answers
244
views
Interesting geometric flow of space curves with non-vanishing torsion
Recently, while thinking about CMC surfaces, I came up with an interesting geometric flow for curves in $\mathbb{R}^3$ given by
\begin{equation}
\partial_t \gamma = \tau^{-\frac{1}{2}} n,
\end{...
- 151
6
votes
2
answers
250
views
Minimal assumptions for existence of solutions of First order PDE
I'm looking for a reference about existence of linear homogeneous first order PDE, in particular about the minimal assumption on the data. In literature I found that one require $C^1$-regularity on ...
- 81
5
votes
1
answer
285
views
Solvability of Yang-Mills equations
For which gauge groups is the initial-value problem for the classical Yang-Mills equations known to be uniquely solvable up to gauge?
Related: References for classical Yang-Mills theory
- 3,873
5
votes
1
answer
315
views
Finding a semi-sparse vertex in a grid
Let $H$ be a $r \times r$ grid. Suppose that at most $r/10^5$ vertices of this grid are colored red. For every vertex $v \in V(H)$, let $B_i(v)$ be the ball of radius $i$ centered at $v$. (Or for ...
- 51
5
votes
0
answers
244
views
When does a "stable" assignment of buyers into goods exist?
Consider a setting of $n$ buyers and $m$ goods.
We have a value matrix $V\in[0,1]^{n\times m}$ specifying how much each buyer values each good (everything is open information here and there is no ...
- 618
5
votes
0
answers
107
views
reference on existence result for nonlinear elliptic PDE
During my work, I came to the question of existence of weak solutions to the following elliptic equation
$$\triangle u + \partial_{1} u + \partial_2(f(x_1,x_2,u)) = 0 \hbox{ in } \Omega$$
with ...
- 51
4
votes
1
answer
1k
views
Existence of solutions to first-order PDE involving convolution
Let $f(x,\alpha)$ be a smooth function of compact support in $x$. Now, let its $\alpha$-dependence be determined by the following first-order equation,
\begin{align}
\frac{\partial}{\partial \alpha} ...
- 193
4
votes
2
answers
627
views
Source of equation - theorems about solving quadratic matrix equations
I have seen outlined in this comment os mathoverflow how to solve quadratic matrix equations of the form
$$
XCX + AX = I
$$
where $X \in \mathbb{R}^{n\times n}$, $C = C^T > 0 \in \mathbb{R}^{n\...
- 143
4
votes
2
answers
844
views
Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard)
What are modular forms or cusps forms, resp. ?
We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The ...
- 85
4
votes
1
answer
490
views
ODE in Banach space
Have I understood this correctly:
So originally we consider the following partial differential equation:
$$u_t= \frac{u_{xx}}{1+w}-\frac{1}{\epsilon}(1+w)u^3+\frac{wu}{\epsilon(1+w)} \text{ in } \...
- 41
4
votes
1
answer
317
views
Given convex l.s.c. function $f$, find decreasing convex function $\phi$ such that $f(x) \equiv \sup_y x\phi(y)-\phi(-y)$
Let $f: \mathbb R \rightarrow (-\infty,+\infty]$ be a lower-semicontinuous convex function.
Question
Under what futher conditions does there exists a convex decreasing function $\phi: \mathbb R \...
- 6,853
4
votes
0
answers
746
views
Maximum Principles in Parabolic PDE with Neumann Condition
I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...
- 1,759
4
votes
0
answers
103
views
Existence and uniqueness of an elliptic equation coupled with a parabolic equation (mean curvature flow)
Given a parabolic equation on a simply connected smooth domain $\Omega(t)$ with boundary $\Gamma(t)$
$$
\Delta u = 1 \quad on \quad \Omega(t)
\\
\nabla u \cdot n + u = g \quad on \quad \Gamma(t)
$$
(...
- 41
3
votes
1
answer
521
views
Is the set of real matrices with at least one real logarithm closed under multiplication?
Let $S$ be the set of real matrices with at least one real logarithm. For some couple of its elements, for example those with at least (one real logarithm each with submultiplicative norm smaller than ...
- 286
3
votes
1
answer
991
views
What is the most general Carathéodory-type global existence theorem?
I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$
\begin{equation}
\left\{ \begin{aligned}
x'(t) &= f(t, x(t)), \qquad t \in ...
- 192
3
votes
1
answer
241
views
Existence and uniqueness of solutions for continuous and directionally differentiable ODE
Given $f:\mathbb{R}^n \to \mathbb{R}^n$ continuous and directionally differentiable (i.e., such that the directional derivative of $f$ exists for any direction) at a neighborhood $N$ of $x_0\in\mathbb{...
- 63
3
votes
0
answers
113
views
On the "Peano phenomenon" in higher dimensions
The following result in one-dimensional differential equations is sometimes referred to as "Peano phenomenon" (see e.g. here).
If $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, the ...
- 63
3
votes
0
answers
116
views
Existence result for an operator obtained by integrating Laplace-Beltrami operator to normal direction in Fermi coordinate
I am going through some literature and encountered with some known facts about Fermi coordinate and Laplace-Beltrami operator. Let $u$ be a function on $\mathbb{R}^{n+1}$ and $\Gamma_0$ be a $0$ level ...
- 31
2
votes
1
answer
160
views
Well-posedness of PDE with $\partial_{tt}\Delta u$ - like term
I am looking for direct hints or references for the establishment of existence of suitable weak solutions admitted by a class of problems of the following type: We search $u$ satisfying
$$
\begin{...
- 35
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
2
votes
1
answer
400
views
Existence of linear stochastic differential equation given solution
Normally if you have a linear SDE given such as
$dx_t = (A(t)x_t + a(t))dt + \sigma(t) dW_t$, we want to find $x_t$, more precisely we want to find the mean and variance of $x_t$ at each timestep $t$. ...
- 29
2
votes
2
answers
220
views
$L^\infty_\mathrm{loc}$ assumption in global existence for Boltzmann equation
In short:
In P. Gérard's paper on the existence of global solutions to the Boltzmann equation from 1988 (or equivalently Cercignani's book), why are the stated assumptions (especially $A_n \in L^\...
- 191
2
votes
0
answers
82
views
Existence of SDE solution under integrability of Lipschitz coefficients
I am reading the paper Lan and Wu, Stoch. Process. Appl., 2014, on sufficient conditions weaker than Lipschitzianity for the existence of strong solutions of time-inhomoegneous $d$-dimensional SDEs. ...
- 135
2
votes
0
answers
152
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
2
votes
0
answers
62
views
Well-posedness or existence for a Poisson problem in Orlicz spaces
I know that the problem
\begin{equation}
\Delta_p u = f
\end{equation}
make sense if $f \in L^q$ with $n/p<q<n$ and that is there a existence theory for
$$
u_t -\Delta_p u = f
$$
For a given ...
- 191
2
votes
0
answers
125
views
Conditions replacing compactness
Reading this book, the authors used the following "classic" idea:
Let $X$ be a Banach space and $C$ a nonempty, weakly compact, convex subset of $X$. Let $T: C \rightarrow C$ be a ...
- 291
2
votes
0
answers
133
views
Why should we give special attention to at most polynomially growing solutions of PDEs?
The equation
\begin{gather}
\frac{\partial u}{\partial t} (t,x) = \frac{1}{2} \text{Trace}[\sigma(x) \sigma(x) (\text{Hessian}_x u)(x,t)] + \langle \mu (x) , (\nabla_x u) (t,x) \rangle, \\
u(0,x) = \...
- 153
2
votes
0
answers
79
views
Existence of a shift invariant selection map
Some time ago I asked this question, but now realise that it is harder than I anticipated. Therefore I am taking a step back to the following problem.
Let $X$ and $Y$ be two sets and $F$ a point to ...
- 479
2
votes
0
answers
117
views
Positive existential theory of $(\mathbb{Z}; +, |_n)$
I am reading a paper and there is the following theorem:
Let $n$ be a fixed integer, and $n >1$.
Denote divisibility in $\mathbb{Z}[\frac{1}{n}]$ by $|_n$, thus for
all $x, y \in \...
- 309
2
votes
1
answer
1k
views
Global Solutions of Ordinary Differential Equations
Background
Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,
$f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,
for every ...
- 123
2
votes
0
answers
297
views
Existence and uniqueness of heteroclinic orbits
I am looking for conditions on a nonlinear dynamical system $\frac{d\vec{x}}{dt} = \vec{F}(\vec{x})$ that guarantee the existence of a unique heteroclinic orbit between a stable attractor of this ...
- 229
1
vote
2
answers
164
views
Existence of directional heat equation without uniform ellipticity
I am asking for references, or for a proof idea on how to show that weak solutions of the following problem exist: search $u$ on a bounded domain $\Omega\times (0,T]$, where $\Omega\subset\mathbb{R}^d$...
- 35
1
vote
2
answers
150
views
Singularity of matrix pencil-like expression
I was working more on the topic on my previous question when I have to know whether the following statement is true to circumvent the "exception" caused by division by singular matrices; ...
- 286
1
vote
1
answer
175
views
The existence of a copy of a random variable with conditional expectation constraint
Let there be two random variables 𝑋 and 𝑌 with a certain joint copula. Is it always true that there is another random variable 𝑍 independent from 𝑌 such as the vectors $(X,Y)$ and $(X,Z)$ have the ...
- 375
1
vote
1
answer
175
views
Existence of Markov chain on nonnegative integers with specified rates
Let $\lambda_k,\mu_k\in\mathbb R_{\ge0}$ $(k\ge1)$ be nonnegative real numbers, let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, let $T=\mathbb R_{\ge0}$ be the nonnegative real numbers and ...
- 276
1
vote
1
answer
126
views
Existence of analysis regularization solution
I am interested in the optimization problem known as "analysis regularization":
$$ {\rm argmin}_{x \in \mathbb{R}^{p}}\frac{1}{2}\|y - Ax\|_2^2 + \lambda \|D^T x\|_1,$$
where $y \in \mathbb{R}^n$, $...
- 133
1
vote
0
answers
171
views
Proving that there are no solutions other than a few known ones
My question is mostly out of curiosity, with probably no other use, but here it is. I will need to provide a bit of background.
I heard from someone who works with elliptic curves that often proving ...
- 407
1
vote
0
answers
37
views
When does an optimal input sequence for a discrete-time system exist?
Suppose an LTI discrete-time system is given by the equations
$$
x_{k+1} = Ax_k + Bu_k,\\
y_{k} = Cx_k + Du_k
$$
with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\...
1
vote
0
answers
105
views
How using the standard Galerkin method
I am attempting to solve the following evolution problem using the standard Galerkin method
$$\begin{cases}
\dot y(x,t)=\Delta y(x,t) +b(t) \nabla y(x,t), \ (x,t)\in \Omega\times (0,T) \\
...
- 55
1
vote
0
answers
70
views
Pohozaev type obstruction for higher order elliptic operators
I was reading about Pohazaev type obstructions: precisely, I mean the following kind of results. Let $h\in C^1(\mathbb{R}^n)$ and consider the following Dirichlet problem
$$
\begin{cases}
\Delta u + ...
- 87
1
vote
0
answers
45
views
Existence of real solutions to nonlinear algebraic equation: conditions on coefficients
Good day. I am dealing with the following system of nonlinear algebraic equations:
$$
x_j = \prod_{k=1}^N (1 + x_k)^{A_{j,k}}\,,\quad j=1,\ldots,N\,,
$$
where $A_{j,k}\in\mathbb{Z}$.
I would like to ...
- 105
1
vote
0
answers
76
views
Help with understanding a proof of existence of solutions
In El Fatini and Boukanjime "Stochastic analysis of a two delayed epidemic model incorporating Lévy processes with a general non-linear transmission" paper, can someone give a detailed ...
- 121
1
vote
0
answers
111
views
Existence theory for geometric flow of space curves
Is there any existence theory applicable to general geometric flows of space curves in the following form?
$$
\partial_t \gamma = v_t t + v_n n + v_b b
$$
Here $\gamma$ is the evolving curve, $t$, $n$ ...
- 11
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
1
vote
0
answers
177
views
Do functions $f: \mathbb R \to \mathbb R$ with these properties exist?
Basically, I am trying to determine how exactly and in which ways everywhere discontinuous bijections $f: \mathbb R \to \mathbb R$ "behave when they are unusual".
More precisely, I am trying to ...
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