2
votes
1answer
84 views

ODE system has zero as the only solution?

Let $V \subset H$ be a continuous, compact and dense embedding with $V$ and $H$ Hilbert spaces. Let $\beta_j:[0,T] \to \mathbb{R}$ be functions for each $j$, and let $v_j$ be a basis of $V_0$. ...
3
votes
1answer
114 views

Hardy-type inequality for point boundary

Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...
1
vote
0answers
87 views

A linear operator equation (PDE) with non-monotone term

I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple. There is a linear operator $L:{D}(L) ...
3
votes
0answers
77 views

Operator theory of initial-value ODE problems

The theory of elliptic boundary value problems is usually treated from the perspective of functional analysis, and the theory of operators between Hilbert spaces. In contrast to that, the theory of ...
3
votes
2answers
221 views

Uniqueness of solutions to an ODE system

For each $i$ (up to infinity), let $u_i \in C^1(0,T)$ satisfy $$\frac{d}{dt}u_i(t) + \sum_{j=1}^\infty b(t;w_j,w_i)u_j(t) = 0$$ $$u_i(0) = u_i(T)$$ where $b(t;\cdot,\cdot)$ is an inner product on some ...
11
votes
3answers
729 views

Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for ...
9
votes
3answers
309 views

$L^p$ norm means

Consider the unit sphere $S_p^{n-1}$ of an $L^p$ normin $\mathbb{R}^n.$ The question is: what is the expected value of the $L^q$ norm on $S_p^{n-1}?$ Since (I assume) this is intractable in closed ...
0
votes
2answers
176 views

A book about almost periodic functions [closed]

Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot.
0
votes
0answers
60 views

The trivility of Besov space for large parameter

For all $s>0$, $1\le p<+\infty$ and $u\in L^p(\mathrm{d}x)$, we define $$D_{s,p}(u)=\sup_{r>0}\frac{1}{r^{sp+n}}\int_{\mathbb{R}^n}\int_{B(x,r)}|u(y)-u(x)|^p\mathrm{d}y\mathrm{d}x$$ and ...
4
votes
0answers
171 views

Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
1
vote
0answers
82 views

Laplacian mapping on various function spaces

I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$. If $ 1 <p< ...
1
vote
0answers
156 views

Is there an asymptotic bound for this oscillatory integral?

I have an oscillatory integral: $$ \int u(x,y) e^{i\lambda f(x,y)} dx $$ with $f(x,y)\in \mathbb{C}^{\infty}$ a complex-valued function in a neighborhood of $(0,0)$ satisfying: $$ \text{Im} f \geq ...
6
votes
1answer
217 views

Spectrum of this ODE

I noticed something interesting studying this Sturm-Liouville Problem: $$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + ...
3
votes
0answers
78 views

Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...
0
votes
0answers
87 views

elliptic regularity when right hand side in weak $L^p$

I am interested in the following question (whose answer i assume is well known) but just not by me. Suppose $u,f$ are smooth functions defined on $B_1$ and $ \Delta u = f$ in $B_1$ with $u=0$ on $ ...
2
votes
0answers
57 views

Holder continuity of Poisson equation with divergence free drift

I am interested in the following PDE. Suppose $u_m$ is a smooth solution of a elliptic equation of the form $$ -\Delta u_m(x) + a_m(x) \cdot \nabla u_m(x) = f_m(x) \qquad B_1 $$ with $ u_m=0 $ on ...
3
votes
2answers
210 views

Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

Let $m \geq 2$ and let $m'$ be its conjugate. Let $w_j$ for $j=1, ..., k$ be a basis of $H_1 \cap L^{m'}$. The task is to show that there is a $u(t) \in \text{span}(w_1, ..., w_k)=:A$ such that ...
1
vote
2answers
355 views

Spectrum of Mathieu equation

I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = ...
3
votes
1answer
197 views

Can I approximate Schwartz functions which integrate to zero by $C_0^\infty$ functions which integrate to zero?

Let $X$ be the closed subspace of Schwartz space $\mathcal{S}(\mathbb{R}^N)$ defined by \begin{equation*} X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}. \end{equation*} My ...
10
votes
1answer
251 views

smooth Luzin theorem

For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...
4
votes
2answers
298 views

Abstract ODE; PDE; uniqueness of solution

I have a somewhat vague question regarding an abstract ODE in a Banach space. Suppose $A:D(A) \subset X \rightarrow X$ is some linear operator (let's assume it's closed) and maybe add some other ...
17
votes
2answers
639 views

What can be said about the Fourier transforms of characteristic functions?

What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular, What properties are common to ...
0
votes
1answer
205 views

Theorem with an example [closed]

i have this theorem in the paper they gives an example: but here $H_1$ is not satisfied ! How to correct it please?
3
votes
2answers
162 views

Existence for ODE in Banach space (accretive operators and Crandall-Liggett)

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some ...
3
votes
1answer
162 views

A differentiable version of the Michael selection theorem

Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map. Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id_{Y}$?
3
votes
1answer
238 views

A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable

Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet ...
1
vote
0answers
273 views

Inverse Transpose of Jacobian Matrix

Let $f:\mathbb{R}^n\mapsto\mathbb{R}^n$ be a bijective function. Fixed $a\in\mathbb{R}^n$. For any $x$ closes to $a$, using Taylor's series we can approximate $f(x)$ by \begin{equation} f(x)\approx ...
1
vote
1answer
134 views

Is the speed of a curve in $ \ell^\infty $ zero a.e. if the derivative of each component is zero a.e.?

Let $ A $ be an $ \mathcal{H}^1$-measurable subset of $ \mathbb{R} $ and $ \gamma \colon A \subseteq \mathbb{R} \to \ell^\infty $ be a Lipschitz mapping with the Lipschitz constant $ L $. Also, assume ...
2
votes
0answers
96 views

A two dimensional integral equation

I have the following integral equation: $\phi(x, y) = \frac{a}{x-y} \int_y^x \phi(s, y) ds + \frac{b}{x-y} \int_y^x \phi(x, s) ds$ where $a > 1$ and $b> 1$ are constants, and $x \geq y$. The ...
1
vote
0answers
153 views

matrix Khintchine inequality

The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then \begin{equation*} \left( ...
5
votes
2answers
424 views

What is the translation in Fourier transform for a function to have exp. decay at $x\to -\infty$

It is known that smooth functions with exponential decay at $\pm\infty$ are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if ...
27
votes
2answers
2k views

Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function. Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e. $\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...
1
vote
3answers
257 views

What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$, for numerical purpose, what are the possible basis function for $X$? In finite element method, the basis functions are tooth functions, or polynomial functions. Is ...
1
vote
2answers
204 views

specific improper integral involving erf

I have encountered an integral, and kindly ask for help with a solution. It is beyond my own capabilities, and neither Maple nor Mathematica were of any help: $$ \int_{1}^{\infty} ...
1
vote
0answers
91 views

growth bound for solution of an ordinary integro-differential equation

I am considering the following ordinary integro-differential equation $$ A g = \sigma^2(y) + \int_\mathbb{R} \nu(y,dz) z^2 $$ where $$ A = b(y) \partial + \frac{1}{2} \sigma^2(y) \partial^2 + ...
5
votes
1answer
293 views

Pseudo-differential operators with compactly supported symbols

If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support. Is the reverse true? Namely that if some PDO ...
2
votes
0answers
156 views

A contradiction to do with continuity? (involves chain rule)

Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE $$\frac{d}{dt}D^0_t(\cdot) = ...
4
votes
0answers
210 views

decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
4
votes
1answer
294 views

On the convergence of the the function series $\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$

Let $f$ be a smooth real function defined around origin. If we informally differentiate from the series $\hat{f}(x):=\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$ term by term we get ...
1
vote
0answers
83 views

Conditions on a measure to satisfy certain relation on moments.

Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>-1$ $t^s\in L^1(\mathrm d\mu(t))$. I'd like to impose some conditions on $\mu$ so the function $$f:p\to \frac{\int_0^\infty ...
4
votes
2answers
269 views

Bruhat-Schwartz functions and derivatives in p-adic numbers

First of all, I am not an expert in neither classical, nor $p$-adic functional analysis, but anyway, I stumbled over the following lately: Let $\varphi:\mathbb{Q}_p\rightarrow\mathbb{C}$. ...
3
votes
0answers
211 views

Constructing a Sobolev space containing the differential k-forms of a Riemannian manifold

I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let $M$ be a compact, orientable n-dimensional ...
1
vote
0answers
98 views

Fundamental solutions for degenerate elliptic equations

Hello, I am looking for a paper or a book that says about the existence and some estimates (like these in the non-degenerate case) of the fundamental solutions for degenerate elliptic equations $L = ...
0
votes
1answer
192 views

Variation on Fatou's lemma for Sobolev norms

Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions $$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$ If I am not ...
6
votes
1answer
304 views

Could we interpolate the compactness of compact operators?

Classical theorems of Marcinkiewicz and Riesz and their extensions to general Banach spaces by Calderón, Lions, Peetre, et al. allow us to interpolate the continuity of two operators, viz., the ...
2
votes
2answers
278 views

A sufficient condition for a probability measure to have compact support

Consider a probability measure $\mu$ on, let's say, $\mathbb R$. Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ? I agree this question is too vague, ...
6
votes
2answers
438 views

For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?

(This is essentially a continuation of my previous question, here.) Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
4
votes
3answers
1k views

Does the derivative of log have a Dirac delta term?

Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics": $\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/1947pqm..book.....D ...
1
vote
1answer
151 views

weak*closure of {f:||f||=1} in dual.

What is the weak* closure of {f:||f||=1}? I am sure this set is not closed in weak* topology. So what is the weak* closure of this set. Thanks.