Questions tagged [existence-theorems]

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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 + ...
Sarthak's user avatar
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2 votes
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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 ...
Paruru's user avatar
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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 ...
Stefano's user avatar
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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 ...
Leo's user avatar
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3 votes
1 answer
155 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{...
Todd Chavez's user avatar
2 votes
0 answers
101 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 ...
Todd Chavez's user avatar
2 votes
0 answers
58 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 ...
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Asymmetric strictly balanced graphs

I am interested in the existence of strictly balanced, asymmetric graph with given number of vertices and edges. A known result of Rucinski and Vince shows that for every $(v,e)$ with $1\leq v-1 \leq ...
35T41's user avatar
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1 vote
1 answer
257 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 $t$. ...
mathemagier's user avatar
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Follow-up question regarding real singular matrices with additional details

After my question whose answer turned out to be false, I re-examined the course of my proof, which is actually seperate from the one in my question, and found out that there's another condition, at ...
Kanghun Kim's user avatar
1 vote
2 answers
144 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; ...
Kanghun Kim's user avatar
3 votes
1 answer
514 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 ...
Kanghun Kim's user avatar
3 votes
0 answers
91 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 ...
user494715's user avatar
4 votes
1 answer
336 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 } \...
PeterSo's user avatar
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2 votes
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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 ...
Motaka's user avatar
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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 ...
fayez ahmed's user avatar
2 votes
1 answer
656 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 ...
A. Pesare's user avatar
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1 vote
0 answers
102 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, $t$, $n$ ...
Iruka's user avatar
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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 ...
StaTik's user avatar
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1 vote
0 answers
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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{...
Eddy's user avatar
  • 111
1 vote
1 answer
168 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 ...
xFioraMstr18's user avatar
1 vote
1 answer
131 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 ...
Averroes's user avatar
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19 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, ...
Veky's user avatar
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-1 votes
1 answer
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(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 $...
user153033's user avatar
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 ...
user avatar
4 votes
0 answers
592 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 ...
Bogdan's user avatar
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4 votes
1 answer
299 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 \...
dohmatob's user avatar
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2 votes
0 answers
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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) = \...
Joker123's user avatar
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9 votes
0 answers
238 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{...
Jiří Minarčík's user avatar
2 votes
1 answer
146 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 ...
Bogdan's user avatar
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1 vote
0 answers
128 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 ...
Darkwizie's user avatar
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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} ...
user143410's user avatar
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 ...
Malachi Holden's user avatar
1 vote
0 answers
195 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 $\Omega$ $\...
Nick's user avatar
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5 votes
1 answer
314 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 ...
user113298's user avatar
4 votes
0 answers
96 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 $\Gamma(t)$ $$ \Delta u = 1 \quad on \quad \Omega(t) \\ \nabla u \cdot n + u = g \quad on \quad \Gamma(t) $$ (...
Josiki's user avatar
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0 answers
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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 ...
visitor's user avatar
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2 votes
0 answers
77 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 ...
Marc's user avatar
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1 vote
1 answer
124 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 $y \in \mathbb{R}^n$, $...
user23658's user avatar
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4 votes
2 answers
606 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\...
marc's user avatar
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2 votes
0 answers
111 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 \...
Mary Star's user avatar
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1 vote
0 answers
150 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 ...
stefanbschneider's user avatar
5 votes
0 answers
239 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 ...
R B's user avatar
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0 votes
0 answers
145 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 ...
Lucy's user avatar
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2 votes
2 answers
214 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^\...
thomas's user avatar
  • 191
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 ...
orlandoweber's user avatar
5 votes
0 answers
101 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 ...
Marin's user avatar
  • 51
2 votes
0 answers
281 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 ...
Danny W.'s 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
833 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 ...
Tom's user avatar
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