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Approximate constant function

Let $f:[0,1]^2 \rightarrow \mathbb C$ be an $H^1$ function with the property that $f(x,x)=0$ and $\Vert f \Vert_{L^2[0,1]}=1.$ Does there exist a constant $c>0$ such that any such function ...
Kung Yao's user avatar
  • 192
4 votes
2 answers
2k views

Inclusions of $C^{k,\alpha}$ spaces

When is $C^{k,\alpha}(\bar{\Omega})$ a subset of $C^{k',\alpha'}(\bar{\Omega})$? Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k + \...
Spencer's user avatar
  • 1,771
4 votes
2 answers
261 views

A convexity question

Let $Q=[0,1]\times[0,1]$ and let $a$ be a positive smooth function on $Q$. Does there exist a smooth positive function $u$ On $Q$ such that there holds $$ \frac{\partial^2}{\partial x_1^2}u <0 $$ ...
Ali's user avatar
  • 4,145
4 votes
1 answer
390 views

Existence of periodic solution to ODE

We shall consider the matrix-valued differential operator $$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$ This is ...
Kung Yao's user avatar
  • 192
4 votes
1 answer
387 views

Asymptotic formula for fractional Laplacian

For the solution of $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan ...
Jun's user avatar
  • 303
4 votes
2 answers
784 views

Gradient flows: convex potential vs. contractive flow?

Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$). Consider the autonomous gradient-flow $$ \dot ...
leo monsaingeon's user avatar
4 votes
1 answer
1k views

Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization

The following result is well-known: Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have $$H \left( (1 - \lambda)(x_1,y_1) + \...
Sascha's user avatar
  • 536
4 votes
1 answer
2k views

On the nascent delta 'function'

I have two questions regarding the sinc function in the weak limit, where it can be used as a nascent delta function. The definition: $\lim_{\varepsilon \rightarrow 0}\frac{1}{\pi }\int_{-\infty}^{\...
mohammad-83's user avatar
4 votes
2 answers
372 views

How close to uniform are Perron-Frobenius eigenvectors?

Let $A=(a_{i,j})$ be a square matrix with non-negative entries. (Assume $A$ is symmetric, if it helps.) Let $v$ be a Perron-Frobenius eigenvector. What do we need to assume about $A$ in order to have ...
H A Helfgott's user avatar
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4 votes
1 answer
555 views

Construct smooth functions with prescribed derivatives

To be specific, suppose we are given a sequence of smooth functions $\{f_k\}_{k\geq 0}$ on flat torus $\mathbf{T}^2$(you may think of it as doubly periodic functions on $\mathbf{R}^2$ and smooth). ...
Jingrui Cheng's user avatar
4 votes
1 answer
145 views

Power series in functions other than monomials

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example. Consider the interval $[-\pi,\pi]$ let's say. ...
Sascha's user avatar
  • 536
4 votes
1 answer
909 views

Riemann-Lebesgue lemma for measures

Riemann Lebesgue Lemma states that Fourier transform of an $L^1$ function, $\hat{f}(\lambda)$ is continuous and goes to zero as $|\lambda|\to \infty$. If $\mu$ is a finite nonatomic measure then is it ...
spr's user avatar
  • 415
4 votes
2 answers
364 views

Equivalence between two Sobolev norms on manifolds

On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following. Use pseudo-differential operators on $M$...
kuuga's user avatar
  • 71
4 votes
1 answer
166 views

Compute $ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions where $u$ solves a PDE

Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator. How can I compute the ...
Jay's user avatar
  • 109
4 votes
1 answer
1k views

Sobolev-Slobodeckij spaces for p=infinity

For $1\leq p<\infty$ an approach to define fractional Sobolev spaces is by Sobolev-Slobodeckij spaces a generalisation of Hölder continuity. For example letting $U\subset\mathbb{R}^n$ then, $ \...
alext87's user avatar
  • 3,217
4 votes
1 answer
195 views

Asymptotic spectrum of a complex Sturm-Liouville differential operator

Let $\varepsilon > 0$ and consider the (complex) Sturm–Liouville differential operator on $[0,1]$ given by $$ \mathcal{L}_\varepsilon f(x) = \varepsilon^2 f''(x) + i V(x) f(x), $$ with Neumann ...
Matheus Manzatto's user avatar
4 votes
1 answer
210 views

On some convergence theorems by Felix E. Browder (1967)

I have been reading Felix E. Browder's Convergence Theorems for Sequence of Nonlinear Operators in Banach Space and I was hoping I could find answers to a couple of questions I have about the paper. ...
user avatar
4 votes
2 answers
928 views

Rate of convergence of mollifiers // Sobolev norms

Following up to the question raised here, I am searching for a reference (or a simple argument) to establish (in the whole space) the following (suggested) equivalence : Given $N_1$ and $N_2$ two (...
Ayman Moussa's user avatar
  • 3,425
4 votes
0 answers
127 views

What can be possible conditions for the solution of an autonomous ODE to be conservative with respect to the initial data?

Let $F : \mathbb{R}^n \to \mathbb{R}^n$ be a smooth mapping and consider the following autonomous ODE \begin{equation} y'(t)=F(y(t)) \end{equation} with the initial data $y(0)=x \in \mathbb{R}^n$. ...
Isaac's user avatar
  • 3,477
4 votes
1 answer
339 views

Mikusiński's approach to Bochner integrals; replace absolute by unconditional?

In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions: Defn. Let $X$ be a Banach space. ...
Willie Wong's user avatar
4 votes
0 answers
174 views

Techniques for showing non-degeneracy results (PDE)

Motivation: Consider the equation, $$-\Delta u = u^p$$ in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
Student's user avatar
  • 537
4 votes
0 answers
126 views

Relationship between three different definitions of solutions for ODE with irregular coefficient

What is the difference between the notions of Regular Lagrangian flow Filippov solution Caratheodory solution of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...
Riku's user avatar
  • 839
4 votes
0 answers
170 views

Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$

Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^...
user avatar
4 votes
0 answers
100 views

Commuting flows problem for non-Lipschitz vector fields

Let $X$ be a continuous vector field on a (say compact) manifold $M$, if $X$ has ODE uniqueness then we can define its associated flow $\mathcal F_X:\mathbb R\times M\to M$ uniquely given by $\mathcal ...
Liding Yao's user avatar
4 votes
0 answers
125 views

Properties of solution to Schrödinger equation

Given a Schrödinger equation with, let's say continuous, periodic potential $$-y''(x)+V(x)y(x)=\lambda y(x)$$ where $V(x+1)=V(x)$ and $V$ is even, i.e. for $x \in (0,\frac{1}{2})$ we have $V(x+\frac{...
Zinkin's user avatar
  • 501
4 votes
0 answers
188 views

Evaluate a multiple integral

I want to compute this integral and I would appreciate any help: $N\geq 1$ is fixed. $$I_N=\int_{0\le r_n\le r_{n-1}\le\cdots\le r_1} e^{-(r_1^2+\cdots+r_n^2)} \prod_{i<j} \sinh(r_i-r_j) dr_1\...
Hatem's user avatar
  • 41
4 votes
0 answers
820 views

Calderón's complex interpolation: what is the corresponding classical theorem?

This question is closely related to my answer to Dan's question, which contains the definitions of some terms I use here. In addition, the notion of exact interpolation functor of exponent $\theta$ is ...
Mark Kim-Mulgrew's user avatar
4 votes
0 answers
487 views

Convolutions and Toeplitz Operators

Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$. Let be $...
Leandro's user avatar
  • 2,044
3 votes
3 answers
584 views

Polynomials and L^p(R)

As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are no non-trivial ...
Jacques Carette's user avatar
3 votes
6 answers
1k views

Reference for complex analysis jargon

I am not a (complex) analyst but it seems that some of the questions I am working on are related to the following concepts: logarithmic capacity transfinite diameter Green's function of a compact ...
Hadi's user avatar
  • 741
3 votes
2 answers
809 views

Growth of $L^p$ norms as $p \to \infty$

Let $f$ be a non-negative function defined on the unit interval. It is well known that $N(p) := \left(\int_0^1 f^p(t) dt\right)^{\frac{1}{p}} $ converges to $\operatorname{esssup}_{[0,1]} f$ when $p \...
Hammerhead's user avatar
  • 1,211
3 votes
2 answers
279 views

Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...
Lewandowski's user avatar
3 votes
2 answers
716 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 $\nabla f$...
asv's user avatar
  • 21.8k
3 votes
1 answer
480 views

Is there a uniform upper bound for this oscillatory integral?

I am wondering whether the following uniform upper bound holds: $|\int_a^{2a}\frac1t\sin(N b^2t)\exp(iNbt^2)dt|\le Cab^2,$ where $0<a<b<1$, $N>N_0(a,b)\gg1$, and $C$ is a constant ...
Right's user avatar
  • 187
3 votes
2 answers
322 views

Hausdorff dimension of the graph of the sum of two continuous functions

How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions: Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
user avatar
3 votes
2 answers
1k 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, ...
Adrien Hardy's user avatar
  • 2,135
3 votes
2 answers
1k views

Limit of the logarithm of the $L^p$ norm over the logarithm of $p$ as $p$ goes to infinity

Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit $$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$ exists in $[0,\infty]$ for ...
J. E. Pascoe's user avatar
  • 1,429
3 votes
1 answer
233 views

A special approximation of the Heaviside function

Is there a $C^m$ approximation $f_\epsilon$ of the Heaviside function such that $$f_\epsilon(x) = f_1(x/\epsilon) = \begin{cases} 0 & \text{ if } x < 0 \\ 1 & \text{ if } x/\epsilon \ge 1 \...
Hiro's user avatar
  • 131
3 votes
3 answers
580 views

Approximate identities and pointwise convergence

I'm studying Fourier analysis and have a question about approximate identities. Let $k_{\epsilon}$ be an approximate identity on $L^{1}(\mathbf{T})$. We know that $k_{\epsilon}*f\to f$ in $L^{1}$ as $...
yun's user avatar
  • 41
3 votes
1 answer
916 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 ...
username's user avatar
  • 135
3 votes
1 answer
432 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 ...
Ali Taghavi's user avatar
3 votes
2 answers
354 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 ...
W.J.'s user avatar
  • 379
3 votes
1 answer
264 views

How to prove that this one-parameter family of distributions converges to the Dirac measure?

While trying to understand a proof in a paper, I came upon the following a calculation needing the following identity: $$\lim_{t\to 0} \int_{-\infty}^\infty \left(e^{-\log(4\pi i t)/2} e^{ik^2/4t} -\...
Dispersion's user avatar
3 votes
2 answers
324 views

An integral transform and the Stone-Weierstrass theorem

For a bounded function $\operatorname{F}: \mathbb{R}_{\,\ge\ 0} \to \mathbb{R}$ (not necessarily non-negative), if $$ \int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \...
Jun's user avatar
  • 303
3 votes
2 answers
717 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 ...
TheBook's user avatar
  • 155
3 votes
1 answer
2k views

fourier transform of radon measure

hi, assume that I have a function $q$ which is a Fourier Multiplier of order zero, i.e. $$ \left|\left( \frac{d}{dx}\right)^nq(x)\right|\lesssim \left(\frac{1}{1+|x|}\right)^n\quad \mbox{for all ...
Philipp's user avatar
  • 979
3 votes
1 answer
425 views

Regularity of boundary of a level set of a $C^{1,\alpha}$ function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^{1,\alpha}$ function. Denote $S_C=\{x\in\mathbb{R}^2\mid f(x)=C \}$ the level set of $f$ with value $C$. What i want to ask is, if $S_C$ is nonempty for some $...
W.J.'s user avatar
  • 379
3 votes
1 answer
453 views

Duality argument

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$Throughout my studying for some papers, in particular, the proof of localized Strichartz estimates, I encountered a use of the ...
Mr. Proof's user avatar
  • 159
3 votes
1 answer
216 views

Linear transport equation with Lipschitz conditions

Given the equation here, I would like to ask the following relaxed question: Consider the PDE $$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$ with Schwartz initial data $f(0,x) = ...
Pritam Bemis's user avatar
3 votes
1 answer
213 views

Unique solution of a 1-D ODE with a bounded positive right-hand-side

Consider the initial value problem $$\dot x(t) = F(t,x), \quad t \in (0,T)$$ with given initial datum $$x(0) = x_0 \in \mathbb R.$$ More precisely we consider the integral equation $$x(t)=x(0)+\int_0^...
Riku's user avatar
  • 839

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