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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5 votes
1 answer
115 views

On existence of a concave function

Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that $$ f’’(x) \leq 0\quad \text{and} \...
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3 votes
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What is the supremum of 1-dim Hausdorff measure of the nodal set of Neumann eigenfunction $u$ for planar convex domain

All descriptions of this question are limited to 2-dimension for simplicity. Recently, I read some papers on the nodal set of Laplacian eigenfunctions. Denote $\Sigma=\{u(x)=0\}$ be the nodal set of a ...
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0 answers
91 views

How can I check whether this function is periodic?

I have a function $$F(t)=\sum_{k=1}^n{\left[\eta_ke^{-i{\omega_k}\ t}+\eta_k^{*}e^{i{\omega_k}\ t}\right]},$$ where $\omega_k\in \mathbb{R}$ and $\eta_k\in\mathbb{C}$. In fact, I don't care about the ...
3 votes
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A new arranging of discrete sine transform

Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix $$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$ Let us denote by $s_{-,l}$ the $l^{\text{...
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2 votes
0 answers
48 views

Coupled 1-dimensional Allen-Cahn system

Suppose we solve Allen-Cahn on the interval $[-1,1]$ $$\epsilon^2 u_{xx} = u(u^2 - 1)$$ $$u(-1) = 0, \qquad u(1) = 0$$ For small $\epsilon$, such a solution is unique and can be chosen to be positive ...
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Stability of a special singular perturbation problem

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a lower bounded smooth function, i.e., $\inf_{x\in\mathbb{R}^n} f(x)>-\infty$. Consider the following singular perturbation problem: $$\begin{cases}\dot{...
3 votes
1 answer
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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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23 votes
3 answers
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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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1 vote
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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 $$|f(x)| \leq Mf(x) \quad a.e.,$$ ...
4 votes
1 answer
99 views

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 ...
3 votes
1 answer
141 views

Does gravity constant affect boundedness of solution?

Consider a second order gradient-like system with linear damping $$\ddot{x}+\dot{x}+\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$ Suppose $f\in C^2(\mathbb{R}^n)$ and $\inf_{x\in\mathbb{R}^n}f(x)&...
3 votes
1 answer
127 views

Entire function with almost periodic boundary condition?

Let $v_1 =\lambda_1 \zeta_1$ and $v_2 = \lambda_2 \zeta_2$ with $\zeta_1 = \frac{4\pi i\omega}{3}$ and $\zeta_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and ...
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2 votes
1 answer
148 views

Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm

Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c,$ $|f(b)| = c$ and $\varepsilon > 0.$ It is easy to see that if $\|f\|_{\infty}< \frac{\varepsilon}{2} =: \...
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3 votes
1 answer
177 views

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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1 vote
1 answer
102 views

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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3 votes
3 answers
366 views

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 ...
2 votes
1 answer
134 views

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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3 votes
1 answer
234 views

Bounds on zeros of rational function

Let $(x_n)_{n=1}^N$ be a sequence taking values in $[1,2]$ with the property that $x_1<x_2<...<x_N$ and $$\frac1N \gtrsim \vert x_j-x_{j-1} \vert \gtrsim \frac1N.$$ We then define a function $...
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"More" cyclical monotonicity

Let $X$ and $Y$ be some finite sets. For a given function $f:X\times Y\rightarrow \mathbb{R}$, we say a set $S\subset X\times Y$ is $f$-cyclically monotone if for any sequence $(x_1,y_1),...,(x_n,y_n)\...
2 votes
2 answers
218 views

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 ...
3 votes
1 answer
126 views

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 ...
2 votes
2 answers
160 views

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 ...
2 votes
0 answers
91 views

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 $\lambda\in \mathbb{R}$, $$H(h')(t) = \lambda h(t)$$ ...
2 votes
0 answers
237 views

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}=...
1 vote
1 answer
82 views

Existence for a singular Sturm-Liouville "eigenvalue" problem with non-homogeneous boundary condition

Consider the following singular Sturm-Liouville problem: $$ -(r^{N - 1}h')' - r^{N - 1}c(r)h = \lambda r^{N - 3} h \text{ in } (0, 1), \qquad h(1) = \alpha $$ where $N \in \mathbb N$, $N \geq 3$; $c(...
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0 answers
77 views

Newton and affine curvature

This is a reference request for the following modern formulation of one of the central results of mathematical physics—Newton’s deduction of the inverse square law from Kepler’s description of the ...
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10 votes
1 answer
411 views

A basic estimate of exponential sums

Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate: \begin{equation} \sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\...
1 vote
1 answer
116 views

Does linearity of cofactor imply linearity of determinant for 3×3 symmetric matrices?

$\DeclareMathOperator\cof{cof}$If we suppose that a finite family (greater than 3) of $3\times3$ symmetric matrices $A_i$ and positive reals $a_i$ such that $\sum_ia_i=1$ satisfies $\cof(\sum_i a_i ...
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4 votes
1 answer
268 views

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 ...
1 vote
1 answer
196 views

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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5 votes
2 answers
461 views

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 ...
1 vote
1 answer
168 views

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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0 votes
0 answers
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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: \begin{equation} \int_0^1 x^\nu \sin(a x)J_\nu(a x)dx= \frac{1}{2\nu+1}\left[\...
3 votes
2 answers
426 views

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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2 votes
0 answers
74 views

For $\Phi$ a majorant of $1_{[-1/2,1/2]}$, how small can the total variation of $\widehat\Phi$ be?

Let $\Phi:\mathbb{R}\to \mathbb{R}$ be a real-valued, symmetric, non-negative function such that $\Phi(t)\geq 1$ for $|t|\leq 1/2$. Assume furthermore that $\Phi$ and $\widehat\Phi$ are both in $L^1\...
3 votes
1 answer
112 views

Is there a theory of "elementary closed form solution" at the operator level for differential equations?

We begin by considering the usual general first order linear equation of the form $$ a_0 y' + a_1 y + a_2 = 0 $$ Where $a_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone'...
2 votes
0 answers
52 views

Questions about article "Ordinary differential equations, transport theory and Sobolev spaces" by DiPerna-Lions

I am reading the article, and I am more or less halfway through it. I have some questions though on some parts I am not understanding, so I wanted to ask about these here. I apologize for listing the ...
8 votes
1 answer
485 views

Trivial (?) product/series expansions for sine and cosine

In an old paper of Glaisher, I find the following formulas: $$\dfrac{\sin(\pi x)}{\pi x}=1-\dfrac{x^2}{1^2}-\dfrac{x^2(1^2-x^2)}{(1.2)^2}-\dfrac{x^2(1^2-x^2)(2^2-x^2)}{(1.2.3)^2}-\cdots$$ $$\cos(\pi x/...
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3 votes
1 answer
296 views

Showing convergence of an infinite ODE system

Suppose $\{a_n(t)\}_{n \geq 0}$ is a collection of differentiable (or simply smooth) functions such that i) $0 \leq a_n(0) \leq 1$ for all $n\in \mathbb N$ (ii) $a_n(t) \approx 1 - \mu2^{-n}$ ...
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3 votes
1 answer
99 views

Restriction to dense subset of functions whose graph is dense

Let $f: [0, 1] \to \mathbb R$ be a measurable function. A function $g: [0, 1] \to \mathbb R$ is said to be a condensation limit of $f$ if $g$ is continuous and agrees with $f$ on a dense subset of $[0,...
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0 votes
0 answers
42 views

A certain property of positive-semidefinite infinite matrices

In this answer I concluded with this: For which arrays $\big(\sigma_{ij}\big)_{(i,j)\in\mathbb N^2}$ [of real numbers] whose every upper-left corner is positive-semidefinite does line $(1)$ above ...
4 votes
1 answer
128 views

$L^2$ norm for solutions of evolution equations driven by different elliptic operators

Let $u$ be a solution of the heat equation $$u_t - \Delta u = 0, \qquad t >0, \ x \in \mathbb T^d$$ and $v$ be a solution of the bi-harmonic heat equation $$v_t +\Delta^2 v = 0, \qquad t >0, \ x ...
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4 votes
2 answers
216 views

Is this an $L^p-L^{\infty}$ operator?

Let $1\leq p <\infty$ and let $p^{\prime}$ denote its conjugate exponent. Consider the following operator on Schwartz functions: $$Tf(x)=\int_{0}^{\infty}t^{\frac{n}{2 p^{\prime}}-1}e^{-t} \int_{|x-...
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0 votes
0 answers
25 views

Functional inequality for fractional Laplacian

Let $f$ be a nonnegative function on the $d$-dimensional torus $\mathbb{T}^d$, which you can take to be smooth. Let $\bar{f}:=\int_{\mathbb{T}^d}fdx$. I am interested in whether the following ...
5 votes
2 answers
726 views

Do non-zero derivatives imply tangent lines (and vice versa)?

Let $\gamma : \mathbb{R} \rightarrow \mathbb{R}^2$ be any continuous function, with image given by $C_\gamma$. We can say that $\gamma$ has an image tangent at $t \in \mathbb{R}$ if there exists $\...
5 votes
0 answers
103 views

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 $x(0)=x_0\in\mathbb{R}^n$, ...
2 votes
0 answers
36 views

Green function of a 2D exterior domain

Consider solutions of the laplace equation \begin{equation} \begin{split} -\Delta u=f, \ \ u|_{\partial D}=0, \end{split} \end{equation} where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
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3 votes
0 answers
94 views

A special type of differential equations

Working in optimal control of PDEs, I came across a type of evolution problem that has instead of an initial condition a link between the initial state and the final state. Here is a simplified ...
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0 votes
0 answers
240 views

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{...

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