**3**

votes

**0**answers

63 views

### behaviour of first eigenfunction near the boundary

Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $ \phi_1(x)>0$ is the first eigenfunction of $ -\Delta$ in $H_0^1(\Omega)$ normalized however one chooses.
My interest is in how $ ...

**-1**

votes

**0**answers

146 views

### Disprove this Piece of Jensen's Inquality “Black Magic”

Jensen's inequality states that if a real valued function $f(x)$ is concave, like $f(x)=\ln |x|,$ then $E(f(X))\le f(E(X)).$ A classic application of this is $E(X) \le \ln |E(e^{X})|.$
Now consider ...

**4**

votes

**2**answers

145 views

### Existence of a measure-preserving bijection

Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and
g is radially symmetric,
the function $ (0, \infty )\ni t \mapsto g ...

**2**

votes

**0**answers

72 views

### Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form
...

**1**

vote

**1**answer

155 views

### Is there an analytic solution for this partial differential equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$:
\begin{align}
\frac{\partial P(\theta,t)}{\partial ...

**0**

votes

**1**answer

76 views

### methods for situations where well-posedness criteria hold but global solutions do not exist

I have been learning PDEs (more specifically, nonlinear dispersive equations (Schrödinger/wave/ Klein-Gordan equations etc...)) through the harmonic analysis methods. And I have read a couple of ...

**0**

votes

**0**answers

33 views

### Orthogonality relation for associated Legendre functions

The associated Legendre Polynomials $P_l^m(x)$ obey orthogonality relations for fixed $m$ and fixed $l$:
$$
\begin{align}
\int_0^\pi P_k^m(\cos\theta)P_l^m(\cos\theta)\sin\theta d\theta ...

**1**

vote

**0**answers

40 views

### A fundamental lemma involving a certain exponential kernel

Let $h \in L^1(\mathbb R^n, \mathbb R)$ be a scalar field and let $\Psi_t: \mathbb R^n \to \mathbb R$ be smooth mappings, parameterized by $t \in \mathbb R$.
Suppose that we are given data $$D(v,t) = ...

**8**

votes

**1**answer

107 views

### Nonconventional ergodic averages for commuting transformations

Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about ...

**1**

vote

**0**answers

43 views

### persistence of regularity for nonlinear Klein-Gordon equation

I have been reading the paper on nonlinear Klein-Gordon equation(NLKG) for initial data in modulation space: For detail please see the paper "Klein-Gordon Equations on Modulation Spaces (2014)" ...

**2**

votes

**0**answers

30 views

### Global error estimates for numerical solutions of ODEs in Matlab or Mathematica [closed]

I need to find the first zero (smallest positive root) of the solution of the initial value problem
$ry''+y'+f(r)y=0, \ \ y(0)=y'(0)=1$
for certain $f \in C^{\infty}(R)$. One can easily use ...

**2**

votes

**1**answer

108 views

### Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given ...

**2**

votes

**2**answers

161 views

### “C^0 estimate for solutions to $\Delta(u)+e^{-u} \geq 0$”

Let $u: \mathbb{R}^2 \to \mathbb{R}.$ Suppose I have a solution to the equation
$$\Delta(u)+e^{-u} \geq 0$$ on $\mathbb{R}^2$. Let r be the radial coordinate on $\mathbb{R}^2$. Suppose that $$lim_{r ...

**6**

votes

**2**answers

357 views

### Texts about Dwork's work

I want to ask about references to papers, that probably exist, which explain the articles of Bernard Dwork starting from "The rationality of the zeta function of an algebraic variety" to "On the ...

**2**

votes

**0**answers

69 views

### Oscillation aspects of two-way infinite alternating series (a followup from the MO-question “functions that eventually oscillate”)

In the recent question on "eventually oscillating function" I had a heuristic for the function $d(x)$ that its amplitude is constant, but could not further describe that function. I just found a ...

**0**

votes

**0**answers

48 views

### What can we say about variational energies here?

Let $V_{ij}^{lk}$ be any $nm \times nm$ real symmetric matrix, $\forall i,j,k,l$
\begin{equation}
V_{ij}^{kl}=V_{ji}^{lk}
\end{equation}
(So for the indices we have $1 \leq k,l \leq m$ and $1 \leq i,j ...

**4**

votes

**0**answers

111 views

### Interesting triple integral

Some time ago I stumbled on an alleged identity
$$\int\limits_0^\infty \frac{dx}{x} \int\limits_0^x \frac{dy}{y}
\int\limits_0^y \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]=
...

**0**

votes

**0**answers

102 views

### Solving the transcendental equation $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$

I need to solve the following equation:
$Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$
for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $Li_{3}$ and $Li_{2}$ are the third and ...

**5**

votes

**0**answers

93 views

### Hyperbolic toral automorphisms, and maximizing over orbits the minimum along an orbit of a function

Setup:
Let $\phi\colon T^2 \to T^2$ be a hyperbolic toral automorphism. Let $f\colon T^2 \to \mathbb{R}$ be a continuous function.
For $x \in T^2$, let $\underline{f}(x) = \inf_{n \in \mathbb{Z}} ...

**4**

votes

**1**answer

79 views

### Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'

Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?:
Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...

**1**

vote

**1**answer

41 views

### Generating function for products of laguerre polynomials?

In a quantum physics context, I would like to evaluate $S=\sum_{n=0}^\infty z^n\cos(\pi L_n(x))$ for $z<1$. I found generating functions for squares of Laguerre polynomials but not for any higher ...

**3**

votes

**2**answers

300 views

### Summing cosines

Suppose I want to compute $$\sum_{n=1}^\infty \cos(n x)/n^{2k}.$$ Now, For any fixed $k$ this is, at least in principle, doable (see the excellent answer to my math.SE question a while back), but the ...

**0**

votes

**1**answer

113 views

### Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero [closed]

By playing around with assoc. Legendre polynomials, I arrived at
$$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$
Now, I want to show that we don't have equality ...

**1**

vote

**2**answers

339 views

### More recently published comprehensive reference on inequalities in the spirit of Hardy-Littlewood-Pólya

Is there a comprehensive reference book on inequalities in the
spirit of the one written by G.H. Hardy, J.E. Littlewood, and G. Pólya(*), but more up-to-date (i.e., published in more recent years and ...

**4**

votes

**1**answer

119 views

### Symmetric inequality on generalized means

Do there exist two functions $f$ and $g$ continuous and strictly increasing $[0,1] \to \mathbf{R}$ such that
$$ f^{-1}\left(\frac{1}{3} f(x) + \frac{2}{3} f(y)\right)<g^{-1}\left(\frac{1}{3} g(x) + ...

**2**

votes

**0**answers

61 views

### Global existence for infinite dimensional ODE

Let us consider the ODE $\hskip3pt \dot x=F(t,x)\hskip3pt $ in an infinite-dimensional Banach space $E$, where the flux $F$ is defined and continous from the whole $\mathbb R\times E$ into $E$.
...

**2**

votes

**0**answers

37 views

### Trigonometric multiple integral identity

How this alleged multiple integral identity can be proved?
$$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {
...

**2**

votes

**0**answers

86 views

### Uniqueness of solution of elliptic equation with exponential nonlinearity

Consider the following equation
$$\Delta v + p(r)e^v = 0$$ on $\mathbb{R}^n$
where $p(r)$ is a polynomial in $r = |(x_1,..., x_n)|$. I want to understand when equations like these have unique ...

**4**

votes

**1**answer

110 views

### Infinite dimensional Cauchy-Lipschitz theorem [duplicate]

From the answers to Mathoverflow Question "Continuity in Banach space for non-linear maps", it is possible to infer that the assumption of the Cauchy-Lipschitz theorem for the autonomous equation
$$
...

**1**

vote

**0**answers

19 views

### Renormalization for Transport Equations with SBD velocity field

In the paper Traces and fine properties of a $BD$ class of vector fields and applications by Ambrosio, Crippa and Maniglia (to be found here)the authors prove a chain rule for vector fields $B\in ...

**34**

votes

**6**answers

2k views

### On an example of an eventually oscillating function

For $x\in(0,1)$, put
$$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$
This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate ...

**2**

votes

**1**answer

142 views

### Multivariate Maximal Hilbert Transform

One way to define the maximal Hilbert transform of a function, $f$, is by
$$\mathcal{H}[f](x):=\underset{\varepsilon>0}{\sup}\;\left|\int_{|x-t|\geq\varepsilon}\dfrac{f(t)}{x-t}dt\right|,\quad ...

**0**

votes

**0**answers

41 views

### Searching for conditions?

I have this operator $$Au(t)=\int_0^1 G(t,s) f(s,u(s)) ds$$defined from $H^1_{0}$ to $H_0^1$ and satisfy the problem: $$\begin{cases} -(Au)''(t)=f(t,u(t)), t\in[0,1]\\Au(0)=Au(1)=0\end{cases}$$
Where ...

**5**

votes

**1**answer

191 views

### Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$

I'm interested in an asymptotic expansion of the following Riemann zeta-type function
$$
\begin{align}
\displaystyle \zeta(s \mid a,b) := \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)},
\quad \Re a ...

**15**

votes

**1**answer

522 views

### Interesting integral

Numerical evidence shows the validity of the following identity
$$\int\limits_0^z\frac{xdx}{\sin{x}\sqrt{\sin^2{z}-\sin^2{x}}}=\frac{\pi}{4\sin{z}}\ln{\frac{1+\sin{z}}{1-\sin{z}}},\tag{1}$$
if $0< ...

**1**

vote

**0**answers

66 views

### Uniqueness of analytic center manifold

In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...

**1**

vote

**0**answers

100 views

### A Characterization of Closed Ideals in $C^{\infty}(\mathbb{R}^n)$

The space $C^{\infty}(\mathbb{R}^n)$ can be turned into a topological ring using the Whitney topology. Whitney's Spectral Theorem says that the closure of an ideal in this ring is the ideal of all ...

**0**

votes

**1**answer

84 views

### Number of critical points

Let $f:[0,2\pi]\rightarrow R^2$ be a smooth function such that $f([0,2\pi])$ is a smooth closed simple curve $C$. Suppose $(0,0)$ lies inside the the bounded open region enclosed by $C$ and ...

**0**

votes

**0**answers

70 views

### Condition for boundedness in stationary phase theorem

I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1.
Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and ...

**0**

votes

**0**answers

48 views

### Maximum modulus principle and restriction

Suppose I have a complex multivariate polynomial $f\in \mathbb{C}[z_1,\ldots,z_n]$ and a connected, bounded open set $U$. The maximum modulus principle says $f$ achieves its maximum value on the ...

**51**

votes

**2**answers

1k views

### History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$

Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:
1º that
$$
\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2},
\tag1
$$
2º that, via the fundamental theorem of calculus, this is ...

**3**

votes

**0**answers

101 views

### Homogeneous polynomial

Can a homogeneous harmonic polynomial mapping $p(x)=(p_1(x),p_2(x),p_3(x))$, $p(0)=0$, of odd degree $m\ge 3$, of the unit ball $B^3$ into $R^3$ be injective. This means that $p_i$ are harmonic ...

**2**

votes

**0**answers

94 views

### Linear forms with best approximation vectors lying in a subspace

Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in ...

**0**

votes

**0**answers

82 views

### Positive curvature of the boundary away from a point implies regularity?

In a paper I'm refereeing, the authors make use of the following geometric fact:
Let $U$ be an open subset of $\mathbb{R}^2$. If there is a point $p\in \partial U$ so that $\partial U \backslash p$ ...

**2**

votes

**2**answers

123 views

### Limiting Ratio of Solutions to Ordinary Differential Equations

I'm trying to find the limit of the ratio of two functions
$ \lim_{t \rightarrow \infty} \frac{f(t)}{g(t)} $ but only have the initial conditions and the differential equations they solve, but the ...

**0**

votes

**0**answers

35 views

### Comparing Calculation Error in Divergent Numerical Methods

I'm not an expert in numerical methods, but I'm doing a simulation based on non-linear differential equations (General Relativity), there solutions has singularities, thus at some points numerical ...

**0**

votes

**0**answers

291 views

### What is the Fourier transform of this function?

Consider the function
$$
f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du.
$$
It is known that $f(x_1,x_2)\in ...

**12**

votes

**2**answers

482 views

### Surjectivity of curl

Let: $\mathbb R^3\ni x\mapsto v(x)\in\mathbb R^3$ be a vector field with null divergence belonging to the Schwartz class such that
$$
\int_{\mathbb R^3} v(x) dx=0.
$$
Is it true that there exists a ...

**0**

votes

**0**answers

68 views

### variation norm of a Fourier transform

Motivated by certain uniform estimate in oscillatory integrals, I am now trying to calculate the Fourier transform of the function ${\large e^{i|t|^{\epsilon}}/t}$ on $\mathbb{R}$, where $\epsilon\in ...

**4**

votes

**1**answer

445 views

### Elementary Proof of the Uniqueness of Smooth Structures on R

Is there any 'elementary' proof of the uniqueness of smooth structures on $\mathbb{R}$? By elementary, I mean that the proof does not use any sophisticated topological machinery. In particular, I'm ...