All Questions
Tagged with fa.functional-analysis ca.classical-analysis-and-odes
524 questions
0
votes
1
answer
102
views
Gronwall-type inequality with nonlinearity
Let $u: (0,\infty) \times \mathbb R \to \mathbb R$. Suppose that $\int_{\mathbb R} u(t,x) dx \ge 0$ (but not necessarily $u >0$). Let $A:(0,\infty) \to \mathbb R$ with $A \ge 0$. Let $\alpha \ge 0$...
user139844
1
vote
0
answers
99
views
Estimate on integral with logarithmic weight
Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have
$$
\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, ...
- 839
2
votes
0
answers
73
views
Question about Gidas-Ni-Nirenberg result
Background:
So I know that the Euler Lagrange equation associated with the Sobolev inequality takes the following form,
$$-\Delta u = u^p$$
where $p=2^*-1$ and here we assume that $u>0$ on $\mathbb{...
- 537
2
votes
0
answers
55
views
An integral average condition and its relationship with BMO, VMO, and Sobolev spaces
Let $V: \mathbb R^n \to \mathbb R^n$ be a vector field which satisfies
$$
\lim_{l \to \infty} \sup_{x \in \mathbb R^n} \left|\frac{1}{l^n} \int_{[0,l]^n}V(x+y) dy \right| = 0
$$
What is the ...
- 839
3
votes
2
answers
129
views
Link between controllability of ODEs and controllability of transport equations
What is the relationship between the controllability of the ODE
$$\dot x(t) = v(x) + u(t)$$
using a control $u$ and the controllabilty of the transport equation
$$\rho_t(t,x) + \mathrm{div}(v(x) \rho(...
- 839
2
votes
2
answers
176
views
Direct limit of the sequence $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category of Banach spaces
Recently I have been reading the paper The categorical origins of Lebesgue integration by Tom Leinster (https://arxiv.org/pdf/2011.00412.pdf). In this paper, he said that：
For $n \geq 0$, let $E_{n}$ ...
- 123
0
votes
1
answer
139
views
Build an explicit "small perturbation" of the identity satisfying some properties
How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant ...
user139844
0
votes
1
answer
239
views
A proof for the existence of smooth solution of PDE in form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III
This part is from page 107 in Michael E. Taylor's book Partial Differential Equations III.
In this part, we want a proof for the existence of smooth solution of the PDE
$\Delta u=f(x, u)$ on $U$ with ...
- 809
7
votes
1
answer
413
views
Criteria for operators to have infinitely many eigenvalues
Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem.
For non-normal operators this no longer has to be true.
There ...
- 536
1
vote
1
answer
113
views
Prove $\int_\Omega \left(\rho_{1} \ln \frac{\rho_{1}}{\rho_{2}}\right)dx dy \leq C\int_\Omega |\rho_1-\rho_2|dxdy$ for $0 \le \rho_1, \rho_2 \in L^1$
Let $\rho_1, \rho_2 \in L^1(\Omega;\mathbb R_+)$ such that $\int \rho_i|\ln \rho_i| < \infty$. Is it true that there exists a constant $C>0$ such that
\begin{align*}
\int_\Omega \left(\rho_{1} \...
user140746
3
votes
1
answer
171
views
Discrete singular integrals
Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:
$\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
- 1,879
-1
votes
1
answer
78
views
Fundamental of a signal
Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$
or any reasonable Euclidean norm such that bounded ...
- 1,902
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. ...
user452741
1
vote
2
answers
155
views
Construct suitable cutoff function
Let $\bar x \in \mathbb R$. Is there a cut-off function such that $\phi_\epsilon \in C^\infty(\mathbb R)$, $0 \le \phi \le 1$, and
$$\phi_\epsilon(x) = \begin{cases}
1 &\text{ if } |x - \bar x| \...
- 839
1
vote
1
answer
338
views
Does the Implicit Function Theorem in Banach spaces holds if the differential is only one-to-one (not onto!)?
Is the Implicit Function Theorem in the following form correct:
Let $V_1,V_2,W$ be Banach spaces, and $Ω⊂V_1×V_2$ an open subset containing $(x_0,y_0)$. Let consider a continuously differentiable map $...
- 13
0
votes
1
answer
162
views
Iterated integrations by parts using the fractional Laplacian
Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...
- 839
3
votes
1
answer
621
views
Strong convexity inequality w.r.t. infinity norm $\lVert\cdot\rVert_{\infty}$
Consider a twice differentiable 1-strongly convex function $f:\mathbb{R}^n \to \mathbb{R}$.
Is it true that there exists $\alpha>0$ independent of $n$ such that, for all $x \in \mathbb{R}^n$:
\...
- 33
1
vote
1
answer
134
views
If $f \circ u \in BV$ and $f$ is strictly monotone, then is $u \in BV$?
Let $f: \mathbb R \to \mathbb R$ be a Lipschitz strictly monotone (so, in particular, invertible) function. Let $u: \mathbb R \to \mathbb R$. If $f \circ u \in BV$ can we conclude that $u \in BV$?
- 161
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,...
- 537
1
vote
1
answer
182
views
Proving an estimate for the Neumann problem on $\mathbb{R}^3 \setminus B_1$ in Weighted Sobolev spaces
Let $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
It is known that for every $g \in H^{\frac{1}{2}} (\partial M)$, and for an appropriate $\delta$, there exists a unique solution $u$ ...
- 969
2
votes
0
answers
84
views
Method of characteristics and explicit formula for an IBVP for the transport equation
Let us assume $c:(0,1) \to (\epsilon,+\infty)$ and consider the PDE
$$
\begin{cases}
u_t+c(x)u_x = 0, \\
u(0,x) = g(x) \\
u(t,0) = f(t)
\end{cases} \qquad (t,x) \in (0,T)\times(0,1) \label{1}\tag{$\...
- 839
2
votes
0
answers
225
views
Sobolev (Triebel-Lizorkin) norm estimate for $F \circ u - F \circ v$
Let $F \in C^1(\mathbb R^d;\mathbb R)$ be such that $F(0) = 0$ and
$$|F'(\tau v + (1 - \tau)w)| \leq \mu(\tau)(G(v) + G(w))$$
for some $\mu \in L^1([0,1])$ and some non-negative $G \in C^0(\mathbb R^d;...
- 233
0
votes
1
answer
417
views
Application of Green function for non linear PDE [closed]
In the case of linear PDE, say $$Lu=0$$ if we have its green function say $G(x,y)$ then using that one can give solution of non homogenous PDE i.e. $Lu_f=f$ where $u_f=G*f$.
Is the same thing hold for ...
- 309
14
votes
1
answer
830
views
Spectrum of matrix involving quantum harmonic oscillator
The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator
$$a ^* = x- \partial_x \text{ and }a = x+\partial_x.$$
Fix two numbers $\alpha,\beta \...
- 192
7
votes
1
answer
545
views
Explicit isomorphism between $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$?
As Hilbert spaces, $L^2(\mathbb{R}^2)$ and $L^2(\mathbb{R})$ are isomorphic. Of course the isomoprhism is vastly not unique. I wonder if there are any particularly nice explicit isomorphisms. E.g. I ...
- 679
12
votes
1
answer
727
views
A generalization of Rubio de Francia's inequality
Suppose that $\{I_m\}$ is a sequence of pairwise disjoint intervals in $\mathbb{Z}$. The well known Rubio de Francia's inequality says that for any function $f\in L^p(\mathbb{T})$, $2\le p<\infty$, ...
3
votes
1
answer
216
views
Linear transport equation with Lipschitz conditions
Given the equation here, I would like to ask the following relaxed question:
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = ...
- 124
5
votes
2
answers
358
views
Linear transport equation with unbounded coefficients
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its ...
- 124
3
votes
1
answer
264
views
How to prove that this one-parameter family of distributions converges to the Dirac measure?
While trying to understand a proof in a paper, I came upon the following a calculation needing the following identity:
$$\lim_{t\to 0} \int_{-\infty}^\infty \left(e^{-\log(4\pi i t)/2} e^{ik^2/4t} -\...
- 840
2
votes
0
answers
162
views
$\int_{\mathbb{R}^{N}\setminus\Omega}\vert x-z\vert^{-N-\alpha} dz = c \ \forall x\in\partial U$ implies $dist(x,\partial\Omega)=c, x \in \partial U$?
Let $\alpha \in \mathbb R_+$, $\Omega \subset \mathbb R^N$ and $U \subset \Omega$. Is it true that if
$$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-\alpha} dz = \text{constant} \quad \text{for all ...
- 61
1
vote
0
answers
72
views
Initial-boundary value problem for transport equation with $W^{1,p}$ velocity
Let us consider $v:\mathbb R_+ \times \mathbb R \to \mathbb R_+$ such that $v \in L^1(0,\infty, W^{1,p}(\mathbb R))$ and the transport equation
$$ \begin{cases}
u_t + v(t,x) u_x = 0 \qquad & (...
- 61
2
votes
0
answers
65
views
On a question relating integral equation:
I don't know if the following question qualifies as research level. If it isn't, sorry.
Set the following terminology:
$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$
$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
- 41
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
$$
\int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \...
- 303
6
votes
1
answer
376
views
Lavrentiev phenomenon between $C^1$ and $C^2$
Does there exist a (onedimensional) functional that exhibits the Lavrentiev phenomenon between $C^1$ and $C^2$ that is
$$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \quad\text{or possibly}\quad F(y)=\int_a^b f(...
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 $d \ge 3$.
- 99
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 ...
- 192
7
votes
2
answers
1k
views
Energy levels of double well potential
Consider the (quantum) Hamiltonian on the real line
$$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$
Let us assume that the potential $V$ is an even smooth functions with exactly two non-degenerate ...
- 21.8k
6
votes
1
answer
365
views
Is the Besov space $B_{\infty,1}^0(\mathbb{R}^d)$ a multiplication algebra?
Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Consider the Besov scale of spaces $B_{p,q}^s(\mathbb{R}^d)$ defined by the norm
$$\|f\|_{B_{p,q}^s} := (\sum_{j=0}^\infty \|P_{j} f\|_{L^p}^q)^{1/q},$$
...
- 873
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 ...
- 303
1
vote
1
answer
122
views
Existence and uniqueness for the equation $u_t + \nabla |u| = 0$
How does one prove the existence, uniqueness, and regularity for the equation
$$u_t + \nabla_x |u| = 0 $$
with initial data $u(0,x) = u_0(x)$ and where the unknown function is $u:\mathbb (0,\infty)\...
- 41
2
votes
0
answers
162
views
Explicit computation of a norm in context of operator-semigroups and differential equations
I am interested in the explicit calculation of the following norm $\vert \cdot \vert$.
Let $X$ a Banach space with norm $\Vert \cdot \Vert$ and $(T(t))_{t \geq 0}$ a strongly continuous one-parameter ...
- 185
1
vote
0
answers
94
views
Obtaining identity from Pokhozhaev formula
From the classical Pokhozhaev formula, how can I obtain that the following identity holds for $u,v \in C^2(\bar \Omega)$?
$$
\int_\Omega (\Delta u(x,\nabla v) + \Delta v(x,\nabla v)) dx = \int_{\...
- 217
0
votes
1
answer
116
views
Fractional Laplacian and support
Let $u:\mathbb [-1,1] \to \mathbb R$ such that $\mathrm{supp}(u) \subset B_{1/2}(0)$. Under what assumptions on $u$ does it hold $$\mathrm{supp}\Big((-\Delta)^s u\Big) \subset B_{1/2}(0),$$
where $(-\...
- 217
0
votes
0
answers
239
views
Fractional Laplacian for the product of two functions
Considering the following definition for the fractional Laplacian
\begin{eqnarray}
\label{pointwisedef} (-\Delta)^{s}u(x) & : = & \mathrm{ \mbox{p. v}} \quad a_{d,s} \int_{\mathbb{R}^d}\frac{...
1
vote
0
answers
257
views
Cut-off function and fractional Laplacian
Is there a smooth function $u$ such that $u = 1$ in $B_r(0)$, $u=0$ in $\mathbb R^N \setminus B_{2r}(0)$, and
$$
|\nabla u| \le Cr^{-1}, \quad |\Delta u |\le Cr^{-2}, \quad |(-\Delta)^s| u \le Cr^{-2s}...
- 99
2
votes
0
answers
175
views
Boundary terms in integration by parts for the fractional Laplacian
Let $u,v \in C^\infty(\Omega)$ and assume that $v$ is compactly supported inside a domain $\Omega$.
Is it true that
$$
\int_\Omega v (-\Delta)^su \, d x = \int_\Omega (-\Delta)^{s/2}v(-\Delta)^{s/2}u \...
- 99
0
votes
0
answers
53
views
Explicit computation related to the fractional Laplacian
Suppose $$c_{n,s}\int_{\mathbb R^n}\int_{\mathbb R^n} \frac{(u(x+y+z)-u(x+y)-u(x+z)-u(x))^2}{|y|^{n+2s}|z|^{n+2s}} dydz = C$$
for some constants $c_{n,s}$, $C$, and $s \in (0,1)$.
Is it true that $$u =...
- 161
2
votes
0
answers
162
views
Bochner's formula for fractional Laplacian
Is there an analogue of the classical Bochner formula
$\frac{1}{2} \Delta |\nabla u|^2 = |\nabla^2 u|^2$ for harmonic functions that holds for $s$-harmonic functions?
- 161
1
vote
1
answer
133
views
Integrability of fractional heat kernel
In Estimates of fractional heat kernel, it was stated that $$ \partial_{x_j} p_t^{(n)}(x) = -\frac{x_j}{2 \pi} \, p_t^{(n+2)}(\tilde x) $$
where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (...
- 109
1
vote
0
answers
144
views
Liouville theorem for elliptic equation with advection term
How can one prove that any $L^2$ solution of
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N $$
is zero if $a(x)$ is a divergence-free vector field such that
$\int |\...
- 99