Questions tagged [ca.classical-analysis-and-odes]
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
2,175 questions
0
votes
0answers
33 views
Ellipticity-type condition
An elliptic operator $L=\mathrm{div}(A(x)\nabla u)$, is called uniformly elliptic if
$$C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$$
If $A$ depends also on $u$, what is the condition
$$C^{-1} + C^...
4
votes
1answer
87 views
Oscillation and Holder continuity
Where can I find a proof of the follwing fact?
If
$$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$
for some function $u(x)$ satisfies
$$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \...
-1
votes
1answer
29 views
Integral representation of the Digamma function vs. Integral representation of the Polygamma function
Yesterday I asked for the derivation of the Integral representation of the Digamma-Function:
https://math.stackexchange.com/questions/3113119/integral-representation-of-digamma-function
Thanks again @...
2
votes
1answer
110 views
Must $q$ be analytic?
I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that
$$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$
which also takes $\mathbb{R}^+ \to \...
-1
votes
0answers
56 views
Spectrum of non compact operator is 0 [migrated]
Can you please help me to give an example of non compact operator which spectrum is {0}
1
vote
1answer
122 views
Why is this series summable?
Let $\delta, \epsilon \in \mathbb{R}$, $\delta >0$, $\epsilon >0$.
Let $\{ a_k\}^\infty$,$\{ b_k\}^\infty$ be sequences of positive integers such that
$\lim \sup_{k \rightarrow \infty} \frac{...
3
votes
1answer
123 views
ODE with a measurable vector field
Suppose we have a bounded Borel measurable vector field $F:\mathbb{R}^n\to\mathbb{R}^n$. To make the question non-trivial, assume that $F\neq 0$ eveywhere.
Question. Does there exist at least one ...
1
vote
0answers
85 views
3D Homogenous Laplace equation with integral boundary conditions
I have the 3D Laplace equation:
$$\nabla^{3} T_w = 0$$
where $\nabla^{(3)}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ defined on $x \in [...
3
votes
0answers
48 views
system of Euler like ode's
I am interested in solving some linear elliptic system like
$$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$
$$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
0
votes
1answer
49 views
Empty interior lack of minima
Suppose that $U \subseteq \mathbb{R}^d$, and satsifies
$U$ is dense in $\mathbb{R}^d$,
U has empty interior,
Then is it possible that
$$
\inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x),
$$
...
0
votes
0answers
101 views
Laplace equation with integral source terms
I am not specifically asking for a solution, but any reference on any method i could read about would be a big help. This I clarify as I am aware of the fact that MathOverflow only deals with research ...
0
votes
0answers
32 views
Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
5
votes
1answer
141 views
“One half of a theta-function” - is there something in the literature about it?
In an answer to my question Enumeration of lattice paths of a specific type (which was in turn caused by an older one, "Special" meanders) I encountered the series
$$
F(t,q):=\sum_{n=1}^\...
1
vote
1answer
121 views
Time ordered integral involving beta function:
Any help on unpacking integrals of the following type, would be
helpful: $$ \int_0^1 \int_0^r r^a (1-r)^b t^n (1-t)^m dr dt $$ where $a, b, n, m \in \mathbb{N}$ and $0 \le t \le 1$.
Edit/...
23
votes
1answer
472 views
+50
Rademacher theorem
If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...
2
votes
2answers
232 views
ODE of the form $y'=\exp(-(\cos(2\pi y))$
Consider the function $h:[0,1]\to \mathbb{R}$
$$h(\theta):=\sum_{k\geq 1}\frac{a_{k}}{\sqrt{k}}\cos(2\pi k \theta)+\frac{b_{k}}{\sqrt{k}}\sin(2\pi k \theta),$$
where $a_{k},b_{k}\in\mathbb{R}$. For ...
4
votes
0answers
122 views
+100
Local “boundary comparison principle” for harmonic functions
Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = - ...
2
votes
0answers
32 views
Connection Problem for the Confluent Heun Equation
Consider the Confluent Heun Equation (CHE) written in its non-symmetrical canonical form, i.e,
$$y''(z)+\left(4p+\frac{\gamma}{z}+\frac{\delta}{z-1}\right)y'(z)+\left(\frac{4p\alpha z-\sigma}{z(z-1)}\...
0
votes
0answers
51 views
Find an integral form for a target polynomial
For $x\in \mathbb{R}^d$ and multi-index $J=(j_1,\dots,j_d)$ denote $x^J=x_1^{j_1}\cdots x_d^{j_d}$.
We are given some polynomial: $p(x) = \sum_{J\leq k}\alpha_Jx^J$ where $J\leq k$ means that $j_1+\...
1
vote
1answer
127 views
Closed form expression for this infinite series?
Is there a closed-form expression for this series?
$\displaystyle\sum_{k\geq 1}^\infty \frac{\lambda^k e^{-\lambda}}{k!}\cdot[1-(1-x)^k]\cdot \frac{1}{k}$
Any answers, ideas or references would be ...
0
votes
1answer
72 views
Averaged Parseval Relation for Sampling a Function on Integers
This was asked a long time ago on math.stackexchange with no answers.
Let $f:\mathbb{Z}\rightarrow \mathbb{C}.$ Assume that the support of $f$ is finite, say it is $[1,N],$ and that $\mid f\mid$ is ...
0
votes
0answers
96 views
Asymptotic expansion in epsilon gets worse with more terms?
I'm trying to get an order bound on the following integral as $\epsilon \to 0$:
$$g(\epsilon) = \int_a^{a+\epsilon} f(\epsilon,v) dv$$ where $f$ is an ugly, but smooth, function when $0< a \leq v &...
0
votes
1answer
69 views
Alternate forms of the Bessel equation
I have a question regarding an alternate form of the Bessel equation and how that alternate form translates to the modified Bessel equation and its solution. The modified form is from:
http://...
0
votes
0answers
79 views
Search trajectory point close to line
There is a 2d mechanism:
Link AB can be rigidly tied to a point at a distance less than the radius of the circle R0 with center B to show the trajectories:
If 4 random points are rigidly tied to the ...
4
votes
3answers
151 views
Uniqueness of minimizers in a problem in the Calculus of Variations - Part II
Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f_A$ be the associated Minkowski functional
$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$
which ...
1
vote
1answer
67 views
Product of independent random variables and tail deconvolution
Suppose $X, Y$ are two independent non-negative random variables. The conditions
(i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$
(ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...
3
votes
2answers
126 views
Example of convex functions fulfilling a (strange) lower bound
I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...
2
votes
1answer
72 views
Uniqueness of minimizers in the Calculus of Variations
Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by
$$
f(x,y):= (x^+)^2 + (y^+)^2
$$
where $a^+ = \max\{a,0\}$ for any real number $a$.
Given a Lipschitz regular domain $\Omega \...
2
votes
1answer
87 views
The regularity of ODE with Zygmund coefficients
A zygmund function $f\in\mathscr C^1$ is a continuous function satisfies $|f(x+h)+f(x-h)-2f(x)|\le C|h|$ for all $x,h\in\mathbb R^n$ in the domain.
According to Markus' paper A uniqueness theorem for ...
1
vote
0answers
59 views
How to prove a set of delay differential equations never converge (the delay is not constant)
Two functions $x(t)$ and $y(t)$ are coupled via:
$$\dot x(t) = a y(t)-b,y(t+x(t))=x(t)$$
where $a<0$, $b\neq 0$ is some constant.
I am mostly confused with the second equation. What is the ...
8
votes
2answers
839 views
Differentiating an integral that grows like log asymptotically
Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...
1
vote
1answer
74 views
Interpolation of a trilinear functional
Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If ...
3
votes
1answer
201 views
A certain generalisation of the golden ratio
Consider a real number $a \ge 1$, and let $g(a)$ be the unique positive solution $x>0$ of $x^a - x^{a-1} - 1 = 0.$
We have $g(1) = 2$, $g(2) = {1+\sqrt{5}\over2}$ (the golden ratio), and $g$ is ...
1
vote
0answers
114 views
On the infimium of a functional
Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$
where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...
9
votes
1answer
2k views
Is There an Induction-Free Proof of the 'Be The Leader' Lemma?
This lemma is used in the context of online convex optimisation. It is sometimes called the 'Be the Leader Lemma'.
Lemma:
Suppose $f_1,f_2,\ldots,f_N$ are real valued functions with the same ...
0
votes
1answer
64 views
Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
0
votes
0answers
60 views
Critical growth and geodesic connectedness in Lorentz manifold
What is the deep ("heuristic") reason why the quadratic growth of $\beta$ is critical for the study of geodesic connectedness in standard static Lorentz spacetime $\mathcal M = \mathcal M_0 \times \...
0
votes
0answers
34 views
Anisotropic elliptic problems
Where can I find a reference on the following anisotropic problem?
$$(\bullet) \qquad \begin{align*} \nabla_x (A(x) \nabla_x u(x,y)) + \Delta_y(K(y) \ast u(x,y))\end{align*}=0, $$
where $(x,y) \in \...
3
votes
1answer
122 views
Lower estimate of the minimal eigenvalue of a Hamiltonian
Consider a linear operator $H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ given by
$$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$
where $V\colon \mathbb{R}^3\to \mathbb{R}$ is a continuous (or ...
2
votes
1answer
67 views
Elliptic interface problem without conditions on the interface
Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...
6
votes
1answer
469 views
Famous but unavailable paper of Jan Boman
The following paper is well known, but hard to find:
J. Boman, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982.
In this paper ...
0
votes
0answers
30 views
Geodesic connectedness in static Lorentz manifold vs connectedness by trajectories with potential in Riemann manifold
What is the relationship between the study of geodesic connectedness in a standard static Lorentz manifold and the connectedness of two points by trajectories with potential (i.e. solutions to $x''(t) ...
0
votes
0answers
56 views
Prove that an iterative estimate implies Holder continuity
Let $u$, $w$ be nonnegative continuous functions such that $\frac{u}{w}$ is bounded on $B_{2^{-1}}$. Why the inequality
$$a_k \le \frac{u}{w} \le b_k \quad \text{ on $B_{2^{-k}}$} , \qquad b_k - a_k ...
12
votes
3answers
334 views
(Sharp) inequality for Beta function
I am trying to prove the following inequality concerning the Beta Function:
$$
\alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0,
$$
where as usual $B(a,b) = \...
2
votes
1answer
86 views
Approximate sequence of numbers
Let $n \in \mathbb N$ and $k_n \in \left\{0,..,n \right\}$ then we define the numbers
$$x_{n,k_n} = \frac{k_n+n^2}{n^3+n^2}.$$
It is easy to see that these numbers satisfy
$$x_{n,0} = \frac{1}{n+1} ...
0
votes
1answer
65 views
Exterior cone condition for $\mathrm{supp}\, u$ and Lebesgue points of $u$
Let $u:\mathbb{R}^n \to \mathbb{R}$ be an $L^1$ function with compact support. Let $\bar x \in \partial \mathrm{supp}\, u$ and assume that $\mathrm{supp} \, u$ satisfies the exterior cone condition at ...
1
vote
1answer
126 views
How to recognize if a continuous vector field in the Euclidean space is a gradient
How to recognize, by "analytic" methods, if a $C^0$ vector field $v:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is the gradient of a function $h:\mathbb{R}^n \rightarrow \mathbb{R}$, given that the ...
1
vote
0answers
25 views
Fredholm integral equation of third kind
Let us consider the following integral equation
$$a(x)u(x) + \int\limits_0^2 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p \in [1,\infty]$ and let $K \in L^q((0,2) \times (0,1))$. Assume ...
1
vote
0answers
57 views
On the convergence of an integral of Hardy's maximal function
Let $f:\mathbb{R}\times \mathbb{R}^N \to \mathbb{R}^N$ be an $L^1$ function.
Assume that
$$\mathcal M f(x,y) = \sup_{r< \bar r}\frac{1}{B_r(y)} \int_{B_r(y)} f(x,z)dz \to 0 $$ as $\bar r \to 0$ ...
1
vote
0answers
56 views
Time-varying perturbations of continuous-time hyperbolic orbits
My question is the following: Assume that the flow of an autonomous ODE $\dot{x} = f(x)$ ($f$ is $C^1$) has a periodic hyperbolic orbit $\varphi^t(x_0)$, $\varphi^{t+T}(x_0) = \varphi^t(x_0)$. Then ...
