# 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.

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### How does one make sense of singular solutions to constant mean curvature equation?

Background: Consider the following ODE: $$\left(\frac{r^2 \dot{f}}{\sqrt{1+r^2(\dot{f})^2}}\right)' = c r$$ where $c$ is some positive constant (Lagrange multiplier), $f:[0,\infty)\to [0,\pi]$ is the ...
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### What is the optimal asymptotic behavior of this integral over the sphere?

Let $k_{1},\dots, k_{d}>1$ be integers and consider the integral $$J_{\lambda }=\int_{\mathbb{S}^{d-1}}e^{-\lambda \left(x^{2k_{1}}_{1}+\dots+ x^{2k_{d}}_{d}\right)} d\sigma(x)$$ where $d\sigma$ ...
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### Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\prod_{k=1}^n f(k/n)<\infty$?

Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim\limits_{n\to\infty}\prod\limits_{k=1}^n f(\frac{k}{n})<\infty$ ? I do not see any reason why such a function could ...
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### The $L^\infty$ norm of Hardy-Littlewood function equal the $L^\infty$ norm of the original function

Since $$Mf(x) = \sup_{x\in I} \frac{1}{|I|} \int_I |f(y)| dy \leq \|f\|_{\infty},$$ we have $$\|Mf\|_{\infty} \leq \|f\|_{\infty}.$$ On the other hand, it holds $$|f(x)| \leq Mf(x) \quad a.e.,$$ ...
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### Quantitative analytic continuation estimate for functions small except on a small set

This question arises as a variation of this question, which was helpfully answered in the negative. It turns out that for my application, a substantially weaker conjecture suffices, which fails to be ...
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### General version of Weyl's lemma

The classical Weyl's lemma say, suppose $u \in L^1_{loc}(\Omega­)$ satisfies $$\int_{\Omega}u \Delta \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$$ then $u$ is harmonic in $\Omega.$ What I want ...
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### Construction of the Lipschitz function with a given Lipschitz constant and given two values

Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c$ and $|f(b)| = c$. Is there a Lipschitz function $g$ such that $|g| \geq c,$ $g(a)=f(a),$ $g(b)=f(b)$ and Lipschitz ...
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### Quantitative analytic continuation estimate for a function small on a set of positive measure

The following conjecture about analytic functions arose as a way to show the asymptotic growth for certain PDE solutions. As I am unfamiliar with any results of this type, I thought I'd ask here. In ...
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### Continuity of a reaching time of a submanifold

Let $\mathcal{O}$ be a bounded open subset of $\mathbb{R}^n$ $(n\geq 1)$ and $v\in\mathcal{C}^1(\mathcal{O},\mathbb{R}^n)$. For $x_0\in\mathcal{O}$, let $\big(x(t)\big)_{t\geq 0}$ be the solution of \...
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### One question about a specific first-order differential equation

Find the function-constant pairs $\langle f(x),c\rangle$ that satisfy the differential equation below: $$f'(x)=f(x+c),$$ where $c \in \mathbb{C}$ and $f(x) \in \mathbb{C}$. I found two families of ...
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### Could the range of $\sum_{k\geq 1}r^{n(k)}$ for $r\in \big(\frac{1}{2}, 1\big)$ be continuous?

Let $\mathcal{F}_N$ be the set of all strictly increasing sequences of positive integers. For every two $F_1, F_2\in\mathcal{F}_N$, if we use $\delta(F_1,F_2)$ to denote the first $n$-th coordinate ...
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### Elementary convexity example

I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for ...
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### Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?

Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$, $$H(h')(t) = \lambda h(t)$$ ...
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### A periodic integral inequality

(This problem comes in connection with a geometric problem exposed here.) Let $\gamma(x,y)$ be a (real) function on the unit disk such that \begin{align} \frac{\partial^2\gamma}{\partial x \partial y}=...
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### When does the sine transform result in a positive function?

For each $x>0$ is, $$\int_0^\infty \tanh(t)e^{-\cosh(t)} \sin(x t) dt > 0\ \ \ ?$$ In general, it is known that if the kernel is decreasing, then the sine transform is positive. Note that this ...
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### Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable?

Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively ...
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### Optimizing a smoothing function with the Prime Number Theorem in mind

Let $f:[0,\infty)\to \mathbb{R}$ be a function with $f(x)=1$ for $0\leq x\leq 1$. Write $Mf$ for the Mellin transform of $f$. Let $c>0$, $T>10^6$ be constants. We are interested in minimizing ...
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### Finding $W^{1,\infty}$ solutions to an integral equation by fixed point

Let $f \in L^\infty((0,1)\times(0,1))$ and consider the integral equation \begin{align*} u(x, y)= \int_0^y \int_0^{x-y} u(x-y+\tau, s+\tau) f(s+\tau, \tau) d s d \tau -\int_0^y f(x-y+s, s) d s. \end{...
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### Solution to the integral of Bessel $\int^1_0 x \sin(a x) J_1 (b x) dx$

I've been trying to work out the solution of this integral. I have seen in the Gradshteyn (6.669(9)) a similar integral: \int_0^1 x^\nu \sin(a x)J_\nu(a x)dx= \frac{1}{2\nu+1}\left[\...
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### Do two ways to differentiate Lipschitz functions coincide?

Let $f\colon \Omega \to \mathbb{R}$ be a Lipschitz function on an open subset $\Omega\subset \mathbb{R}^n$. By the Rademacher theorem $f$ has first derivative almost everywhere. We denote it $\nabla f$...
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### Boundedness of solutions to second order linear damped ODE

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a polynomial with $\inf_{x\in\mathbb{R}^n}f(x)>-\infty$. If the solution to $x'(t)+\nabla f(x(t))=0$ is bounded for any initial point $x(0)=x_0\in\mathbb{R}^n$, ...
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### Green function of a 2D exterior domain

Consider solutions of the laplace equation $$\begin{split} -\Delta u=f, \ \ u|_{\partial D}=0, \end{split}$$ where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
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### How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?
Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$. How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on \$\Bbb C^+_*=\{z\in\Bbb C:\text{...