Questions tagged [existence-theorems]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4 votes
1 answer
177 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 } \...
user avatar
  • 41
2 votes
0 answers
113 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 ...
user avatar
  • 311
0 votes
0 answers
33 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 ...
user avatar
1 vote
1 answer
187 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 ...
user avatar
  • 172
1 vote
0 answers
86 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, $t$, $n$ ...
user avatar
  • 11
0 votes
0 answers
38 views

Global existence for one Cauchy problem based on global existence of other two auxiliary Cauchy problems

I have a Cauchy problem for the differential equation \begin{equation} y' = f(t, y), \end{equation} with initial condition $y(0) = y^0$; here, $y$ and $f$ are two-dimensional vector-functions. The ...
user avatar
  • 19
1 vote
0 answers
49 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{...
user avatar
  • 111
1 vote
1 answer
136 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 ...
user avatar
1 vote
1 answer
75 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 ...
user avatar
  • 355
18 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, ...
user avatar
  • 323
-1 votes
1 answer
91 views

(maximal) antichains with respect to two different partial orders on the same set

In my recent work I stumbled across a problem of this type: G with two partial oders $\leq$ and $\preceq$ on every set, i.e. for every $n \in \mathbb{N}$ $A_n \subset A_{n+1}$ and $(A_n, \leq) $ and $...
user avatar
1 vote
0 answers
171 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 ...
user avatar
4 votes
0 answers
346 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 ...
user avatar
  • 892
4 votes
1 answer
265 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 \...
user avatar
  • 5,840
2 votes
0 answers
106 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) = \...
user avatar
  • 153
9 votes
0 answers
211 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{...
user avatar
1 vote
1 answer
119 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 ...
user avatar
  • 892
1 vote
0 answers
109 views

Reference for Existence and uniqueness of an Integro-Differential Equation

I have an Integro-Differential Equation (IDE) of the following form: $$ x'(t) = f(t,x(t)) + \int_0^t K(t-s, x(s), x(t)) ds, $$ I have found this classical reference, but the IDEs considered therein ...
user avatar
  • 121
4 votes
1 answer
904 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} ...
user avatar
17 votes
2 answers
3k 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 ...
user avatar
1 vote
0 answers
184 views

Theory on interior Helmholtz Equation with mixed Neumann and Robin BC

Let $\Omega \subset \mathbb{R}^n$ bounded with smooth boundary $\partial \Omega$. Let k>0 and $\partial \Omega = \Gamma_1 \cup \Gamma_2$. Consider the problem $\Delta u + k^2 u = 0$ in $\Omega$ $\...
user avatar
  • 11
5 votes
1 answer
304 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 ...
user avatar
4 votes
0 answers
87 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 $\Gamma(t)$ $$ \Delta u = 1 \quad on \quad \Omega(t) \\ \nabla u \cdot n + u = g \quad on \quad \Gamma(t) $$ (...
user avatar
  • 41
0 votes
0 answers
274 views

Existence and uniqueness of solution for nonlinear system

Under what conditions will a solution $y \in \Re^n$ exist for this system of nonlinear equations? If it exists, will it be unique? $$ \mu_k = \int_0^\infty g_k(x) \, f(x, y) \, dx \qquad \forall \, k ...
user avatar
  • 101
2 votes
0 answers
73 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 ...
user avatar
  • 479
1 vote
1 answer
116 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 $y \in \mathbb{R}^n$, $...
user avatar
  • 133
3 votes
2 answers
491 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\...
user avatar
  • 133
2 votes
0 answers
109 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 \...
user avatar
  • 299
1 vote
0 answers
133 views

Probabilistic proof for expander existence [closed]

I am new to probabilistic proofs and trying to understand them better. Apparently, a common probabilistic proof focuses on the existence of expanders (eg. vertex expanders). I've been using the search ...
user avatar
  • 111
5 votes
0 answers
230 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 ...
user avatar
  • 608
0 votes
0 answers
144 views

Does there exist this special kind of homeomorphism?

Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
user avatar
  • 173
2 votes
2 answers
197 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^\...
user avatar
  • 191
2 votes
1 answer
837 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 ...
user avatar
5 votes
0 answers
95 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 ...
user avatar
  • 51
2 votes
0 answers
266 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 ...
user avatar
  • 229
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. ...
4 votes
2 answers
812 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 ...
user avatar
  • 85
9 votes
3 answers
2k 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\|<\...
user avatar
0 votes
1 answer
561 views

Continuous variation from solution of easy problem to solution of hard problem

I asked this question a week ago over on math.stackexchange and got no reply, so I am asking here with slightly different wording. I am trying to prove that there exists a solution to a problem. I ...
user avatar
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 ...
user avatar
  • 2,657