All Questions
Tagged with fa.functional-analysis ca.classical-analysis-and-odes
524 questions
5
votes
1
answer
566
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 ...
- 1,446
1
vote
0
answers
119
views
Continuity of a minimizing measure w.r.t a parameter
Let $V_t(x)=x^2+t\phi(x)$ where $t>0$ and $\phi\in C^\infty_c(\mathbb{R})$.
My question is what can be said about the continuity of the (unique) minimizer (among probability measures) of the ...
- 41
6
votes
1
answer
310
views
Surjectivity of a class of integrals in dimensions two
Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
- 1,075
1
vote
1
answer
179
views
On the compact embedding of Sobolev space
In dimension three, we know that the Sobolev space $W^{\frac{13}{11},11}(D)$ is compactly embedded into $W^{1,11}(D)$, where $D$ is a bounded domain in $R^3$ with smooth boundary. My question is: Does ...
- 63.9k
8
votes
3
answers
1k
views
Are all positive eigenfunctions principal eigenfunctions?
In a given domain $\Omega$, we have: $\Delta u=-\lambda u$ with $u>0$. Does this mean that $u$ is a principal eigenfunction for $\Delta$ in $\Omega$?
Also, more generally, does this also apply for $...
- 12.5k
5
votes
1
answer
1k
views
Analytic functions where all derivatives vanish at infinity and which are bounded
Let $C_0(\mathbb{R})$ denote the analytic functions $f : \mathbb{R} \rightarrow \mathbb{R}$.
I wonder whether there a functions $f \in C_0(\mathbb{R})$ with $f \neq 0$, such that there is a constant $...
- 232
0
votes
2
answers
238
views
Fractional Laplacian of $(a-x)_+^\alpha$ in $(0,1)$
How can I compute the spectral fractional Laplacian of $(a-x)_+^\alpha$ on $\Omega = (0,1)$?
Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
CommunityBot
- 1
4
votes
0
answers
127
views
What can be possible conditions for the solution of an autonomous ODE to be conservative with respect to the initial data?
Let $F : \mathbb{R}^n \to \mathbb{R}^n$ be a smooth mapping and consider the following autonomous ODE
\begin{equation}
y'(t)=F(y(t))
\end{equation}
with the initial data $y(0)=x \in \mathbb{R}^n$.
...
- 3,477
2
votes
1
answer
107
views
If $b\in C^1(E, \mathbb{R})$ and $b'$ is compact, then $b$ is weakly continuous — a reference request
While reading the well known book Minimax Methods in Critical Point Theory with Applications to Differential Equations by Paul Rabinowitz, in the proof of a generalisation of the Mountain Pass Theorem ...
- 128k
6
votes
2
answers
519
views
Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)
This question is concerning the paper, particularly the proof of Lemma 2.1 in Section 2.1:
Matas, A., Merker, J. Existence of weak solutions to doubly degenerate diffusion equations, Appl Math 57 (...
- 4,713
1
vote
1
answer
160
views
On an integral equation
Let $B: C^{\infty}([0,1]^3)$ satisfy
$$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$
Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation:
$$ \int_0^1 f(t,x)\,dx + \int_0^t\...
- 6,696
8
votes
3
answers
1k
views
Fourier dimension of the sum of sets
This question came up when my supervisors, my colleague, and I were considering arithmetic progressions in sets of fractional dimension. In particular, we were interested in "extracting" Salem sets ...
2
votes
1
answer
455
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
$$
\frac{\partial^2\gamma}{\partial x \, \partial y} = 0\:\:\...
CommunityBot
- 1
2
votes
2
answers
451
views
The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$
Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}...
- 2,494
0
votes
1
answer
245
views
Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e
The Riemann-Liouville integral is defined by
$$
I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t
$$
where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
- 141
6
votes
1
answer
134
views
Multi-parameter stationary phase asymptotic expansion
I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...
- 188k
5
votes
2
answers
707
views
Approximation of Hölder continuous functions "from below"
We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$.
I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\...
2
votes
0
answers
206
views
Failure of Calderón–Zygmund inequality at the endpoints
$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'd like to prove that the famous Calderón–Zygmund elliptic estimate $$ \norm{ \partial_{ij}u }_{L^p} \leq C \norm{\Delta u }_{L^...
- 457
0
votes
0
answers
127
views
Closure of BV paths in space of paths of finite $p$-variation
Let $p\ge1$ and $T>0$. Define $\mathscr D([0,T])$to be the space of partitions of $[0,T]$, where each partition is a finite collection of distinct points of $[0,T]$. Consider a continuous path $X:[...
- 177
5
votes
1
answer
430
views
Oscillation and Hölder continuity
Where can I find a proof of the following fact?
If
$$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$
for some function $u(x)$ satisfies
$$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda ...
- 4,219
1
vote
2
answers
213
views
How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$
Let $[a, b]$ be a nonempty interval, $o \in C^1([a, b])$ be such that $o>0$ and $o'<0$ and assume we found some $v \in L^\infty(\mathbb{R})$ such that
\begin{equation}\tag{1}\label{1}
\int_a^b \...
- 12.9k
1
vote
2
answers
106
views
Green function of symmetric stable process in dimension 1 and 2
Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
19
votes
4
answers
5k
views
Explicit extension of Lipschitz function (Kirszbraun theorem)
Kirszbraun theorem states that if $U$ is a subset of some Hilbert space $H_1$, and $H_2$ is another Hilbert space, and $f : U \to H_2$ is a Lipschitz-continuous map, then $f$ can be extended to a ...
- 60.5k
1
vote
1
answer
164
views
Complex interpolation of subspaces
Let $(X_0,X_1)$ be an interpolation couple of Banach spaces. Using complex interpolation we can form Banach spaces $X_\theta:=(X_0,X_1)_\theta$ where $0<\theta<1.$ Let $E_\theta\subseteq X_\...
- 1,879
3
votes
0
answers
111
views
Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces
The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is,
$[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$.
Now I ...
- 39k
0
votes
1
answer
171
views
Looking for English version of a paper of Jean Ginibre
I am in serious need of an English translation for the following paper:
Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires
périodiques en variables d’espace, d'après ...
- 26.3k
0
votes
1
answer
59
views
Given positive $\epsilon$ and $c$, find a density $\phi$ such that $t\phi(\epsilon/t) \ge c \|\phi'\|_\infty$ for all positive $t$
A nice density (on $\mathbb R$) is function $\phi:\mathbb R \to \mathbb R$ such that
(1) $\phi(x) \ge 0$ for all $x \in \mathbb R$,
(2) $\int_{-\infty}^\infty \phi(x) \mathrm{d}x = 1$,
(3) $\phi$ is ...
- 128k
0
votes
1
answer
161
views
Verifying the proof of a bilinear estimate in $L^2$
$\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert_{L^{#2}}}$
$\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert_{L^{...
- 159
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 $M$...
- 829
5
votes
1
answer
151
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
$$ f’’(x) \leq 0\quad \text{and} \...
- 10.6k
2
votes
1
answer
276
views
Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm
Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c,$ $|f(b)| = c$ and $\varepsilon > 0.$
It is easy to see that if $\|f\|_{\infty}< \frac{\varepsilon}{2} =: \...
- 42.8k
1
vote
1
answer
136
views
Construction of the Lipschitz function with a given Lipschitz constant and given two values
Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c$ and $|f(b)| = c$. Is there a Lipschitz function $g$ such that $|g| \geq c,$ $g(a)=f(a),$ $ g(b)=f(b)$ and Lipschitz ...
- 97
2
votes
0
answers
136
views
Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?
Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$,
$$H(h')(t) = \lambda h(t)$$
...
- 20.2k
1
vote
1
answer
177
views
Finding $W^{1,\infty}$ solutions to an integral equation by fixed point
Let $f \in L^\infty((0,1)\times(0,1))$ and consider the integral equation
\begin{align*}
u(x, y)= \int_0^y \int_0^{x-y} u(x-y+\tau, s+\tau) f(s+\tau, \tau) d s d \tau -\int_0^y f(x-y+s, s) d s.
\end{...
- 2,494
1
vote
1
answer
259
views
Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable?
Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively ...
- 12.9k
37
votes
4
answers
4k
views
Which differential equations allow for a variational formulation?
Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
\frac{d}{dt}\frac{\...
- 2,821
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 $\nabla f$...
- 128k
4
votes
2
answers
315
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:
$$Tf(x)=\int_{0}^{\infty}t^{\frac{n}{2 p^{\prime}}-1}e^{-t}
\int_{|x-...
- 852
2
votes
0
answers
201
views
Green function of a 2D exterior domain
Consider solutions of the laplace equation
\begin{equation}
\begin{split}
-\Delta u=f, \ \ u|_{\partial D}=0,
\end{split}
\end{equation}
where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
- 379
0
votes
0
answers
251
views
How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?
Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$.
How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{...
- 511
20
votes
6
answers
7k
views
Does the derivative of log have a Dirac delta term?
Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics":
$\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/1947pqm..book.....D ...
- 10.5k
1
vote
0
answers
100
views
N-wave solution of conservation law $u_t + (u - u^2)_x = 0$
How can we compute the "N-wave" source-solution of the conservation law
$$u_t + (u - u^2)_x = 0, $$
that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
- 839
4
votes
2
answers
610
views
Unit ball of the sum space
Let $V$ be a vector space and $\|\cdot \|_1$ and $\|\cdot\|_2$ two norms on $V$.
Let $\|\cdot\|_+$ be given by
$$ \|v\|_+ := \inf_{v = v_1 + v_2} \|v_1\|_1 + \|v_2\|_2 $$
It is well-known that $\|\...
- 128k
2
votes
1
answer
373
views
Bakry-Emery criterion
The most common use of the Bakry-Emery criterion is for the measure $\mu(x)=e^{-u(x)} /Z$ where $u \in \mathcal{C}^2$. I would like to ask for an application to a smaller class.
Consider $u(x)=|x|^2 + ...
- 1,081
1
vote
0
answers
47
views
Scaling limit of transport equation with double-well potential
Let us consider the transport PDE
$$
u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon)
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...
- 839
2
votes
0
answers
64
views
Scaling limit of ODE with double-well potential
Let us consider the ODE
$$
\frac{d}{dt}x_\epsilon(t) = -\frac{1}{\epsilon} W'(x_\epsilon(t))
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the ODE reads
$$...
- 839
0
votes
1
answer
166
views
Relationship between elliptic and parabolic problems and their discretizations
Let us consider the fully nonlinear problem
$$
\begin{cases}
F(x,u,Du,D^2 u) = 0 & \text{ in } \Omega \\
u=0 & \text{ in } \partial \Omega
\end{cases}
$$
Suppose that we know that the ...
CommunityBot
- 1
1
vote
0
answers
59
views
Identification of a limit point of a sequence of solution of ODE
Let $v^0$ and $v^1$ be the following vector fields over $\big(\mathbb{R}_+^*\big)^3$: for $x\in\big(\mathbb{R}_+^*\big)^3$ and $1\leq i\leq 3$,
\begin{align*}
& v^0_i(x)=x_i(x_{i-1}-x_{i+1}) \\
&...
- 6,045
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 $...
- 5,038
5
votes
1
answer
170
views
Regular Lagrangian flow for "square root example": $\frac{d}{dt} X(t,x) = \sqrt{X(t,x)}$
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \sqrt{X(t,x)}, &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
This is the prototype of non-uniqueness ...
- 2,504