All Questions
Tagged with ca.classical-analysis-and-odes fa.functional-analysis
524 questions
1
vote
1
answer
706
views
Plancherel-Polya Type Inequality for non-compactly Fourier-supported Functions??
Hi!
The Plancerel-Polya inequality can be stated as follows:
Let $0 < p\le \infty$ and $ \nu \in \mathbb{Z}$. Suppose that $g$ is a (smooth) function satisfying $\mbox{supp }\hat g \subset \...
- 979
1
vote
0
answers
205
views
Uniqueness for Volterra equation with initially (linearly) unbounded kernel
Letting $D:=\{(x,y):\ 0\leq x\leq y\leq1 \text{ and } y>0\}$, I have a continuous function $k:D\to[0,\infty)$ that satisfies some properties that I list below. I'm interested in continuous and ...
- 537
1
vote
1
answer
120
views
Characterization of an integral operator with a Bessel kernel
I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$
I am ...
- 141
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
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
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
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}) \\
&...
- 449
1
vote
0
answers
278
views
Vector convolution?
I am working on a research problem which leads to the following optimization problem:
\begin{equation}
\hat{M} = \operatorname*{arg\,max}_M \Bigl\lVert\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi ...
- 273
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
$$
\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, ...
- 839
1
vote
0
answers
72
views
Initial-boundary value problem for transport equation with $W^{1,p}$ velocity
Let us consider $v:\mathbb R_+ \times \mathbb R \to \mathbb R_+$ such that $v \in L^1(0,\infty, W^{1,p}(\mathbb R))$ and the transport equation
$$ \begin{cases}
u_t + v(t,x) u_x = 0 \qquad & (...
- 61
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 $u,v \in C^2(\bar \Omega)$?
$$
\int_\Omega (\Delta u(x,\nabla v) + \Delta v(x,\nabla v)) dx = \int_{\...
- 217
1
vote
0
answers
257
views
Cut-off function and fractional Laplacian
Is there a smooth function $u$ such that $u = 1$ in $B_r(0)$, $u=0$ in $\mathbb R^N \setminus B_{2r}(0)$, and
$$
|\nabla u| \le Cr^{-1}, \quad |\Delta u |\le Cr^{-2}, \quad |(-\Delta)^s| u \le Cr^{-2s}...
- 99
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
1
vote
0
answers
130
views
Fractional Sobolev embedding theorem
Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds
$$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{...
- 99
1
vote
0
answers
60
views
A determinantal mixture of probability densities
I came up with this operation after playing with determinantal point processes:
Given two probability densities $f,g$ defined on some measurable space with reference measure $\mu$, set
$$
f\star g(x)...
- 2,135
1
vote
0
answers
76
views
Existence of a `right' sequence
Suppose that $f_n: \mathbb{R} \times [0,\infty) \to \mathbb{R}$ is a sequence of bounded variation functions such that there exists $C>0$: $|f_n| < C$ for every $n$ and, as $n \to \infty$, for $...
- 41
1
vote
0
answers
177
views
A consequence of De Giorgi oscillation lemma
The following lemma is true (see DeGiorgi oscillation lemma)
Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$
where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\...
- 839
1
vote
0
answers
107
views
Level sets of a BV function and its derivative
Given $u \in BV(\Omega; \mathbb{R}^M)$, where $\Omega \subset \mathbb{R}^N$, what is the relationship between its level sets and its distributional derivative $Db$?
More specifically, does Alberti ...
- 839
1
vote
0
answers
97
views
Fredholm integral equation of third kind
Let us consider the following integral equation
$$a(x)u(x) + \int\limits_0^2 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p \in [1,\infty]$ and let $K \in L^q((0,2) \times (0,1))$. Assume ...
- 617
1
vote
0
answers
110
views
Using semigroup theory for nonautonomous semilinear equations
We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the ...
- 11
1
vote
0
answers
324
views
Conditions for Poisson summation (for discontinuous functions)
Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
- 3,799
1
vote
0
answers
237
views
On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
1
vote
0
answers
146
views
Functional equation with Fourier transform
What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$:
$$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$
Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant....
- 1,199
1
vote
0
answers
183
views
Trace theorem for boundary value problem
Consider the inhomogeneous boundary value problem on the infinite strip $(x,y)\in \mathbb{R}\times [0,1]$ defined by
$$\begin{cases}\partial_{x}u + \partial_{y}v=f & {(x,y)\in \mathbb{R}\times (0,...
- 873
1
vote
0
answers
66
views
How to define spectral multiplier for −Δ?
Put $e_{n}(t)=e^{int}=\prod_{j=1}^{d}e^{in_{j}t_{j}}$, $t\in \mathbb T^d, n\in \mathbb Z^d.$ ($t=(t_{1},..., t_d), n=(n_1,..., n_d)$)
We note that $\{e_{n}\}_{n\in \mathbb Z^d}$ forms an orthonormal ...
- 233
1
vote
0
answers
105
views
compactness of sequence of harmonic functions
Let $ \Omega$ denote a smooth bounded domain in $ R^N$ and let $u_m \in C^\infty( \overline{\Omega})$ harmonic functions. We also suppose $ u_m$ is bounded in $L^2(\Omega)$ (uniformly in $m$).
...
- 1,385
1
vote
0
answers
86
views
Least restrictive condition such that $-y''(x)+q(x)y(x)=\lambda y(x)$ has two solutions
Let $\lambda \in \mathbb{R}$ and consider some function $q$ on the interval $[0,1].$
What is known to be the least restrictive condition on $q$ such that there are two linearly independent $H^2$ ...
- 11
1
vote
0
answers
148
views
References for the Sturm oscillation theorem
What is the most general form of the Sturm oscillation theorem?
So far I have only seen cases that treat either unbounded intervals or weighted $L^2$ spaces. I would be especially interested in ...
- 263
1
vote
0
answers
93
views
inverse problem to resolution of the identity
Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define $$E_\mathfrak{m}:L^2\...
- 587
1
vote
0
answers
86
views
Asymptotics of a elliptic pde when exponent gets large
I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball $...
- 1,385
1
vote
0
answers
129
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)" (...
- 1,051
1
vote
0
answers
133
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 $j, ...
- 93
1
vote
1
answer
416
views
Limit-circle and limit-point at endpoints
I was wondering if the following holds:
If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the ...
- 13
1
vote
0
answers
136
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) ...
- 11
1
vote
0
answers
182
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< \infty$...
- 539
1
vote
0
answers
1k
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 f(...
- 133
1
vote
0
answers
135
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
$$
A = b(y) \partial + \frac{1}{2} \sigma^2(y) \partial^2
+ \int_\...
- 11
1
vote
0
answers
100
views
Conditions on a measure to satisfy certain relation on moments.
Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>-1$ $t^s\in L^1(\mathrm d\mu(t))$.
I'd like to impose some conditions on $\mu$ so the function
$$f:p\to \frac{\int_0^\infty t^...
- 173
1
vote
0
answers
121
views
showing convergence of a function recursion relation
I have obtained (formally) a perturbative solution
$$
H(y) = \sum_{n=0}^\infty \delta^n H_n(y)
$$
to the following integro-differential equation ($\delta$ is a small constant, $\nu$ is a L\'evy ...
- 351
1
vote
0
answers
180
views
iterated traces for sobolev functions
It is well known that if $M$ is a smooth $(n-1)$-dimensional surface in $\mathbb R^n$ (e.g. a subspace) then there is a continuous trace operator $W^{s,p}(\mathbb R^n)\to W^{s-1/p,p}(M)$. Now suppose ...
- 2,041
1
vote
0
answers
693
views
A question about an equivalent definition of the Schwartz distribution
Hello,
Does anyone know a reference or proof of the "if" part of the following statement?
$$
\mu\in \mathcal{S}'(R)\quad\text{if and only if}\quad \mu*\alpha\in\mathcal{S}(R),\forall \alpha\in C_c^\...
- 1,649
1
vote
0
answers
477
views
A norm ratio inequality
Let $y,z\in(0,1)^n$ satisfy $||y||_1 = ||z||_1=1$.
Then
$$
\frac{||z||_3}{||z||_2} \le
K_n
||z/y||_\infty
\frac{||y||_3}{||y||_2}
$$
where $z/y\in\mathbb R^n$ is the coordinate-wise quotient of $z$ ...
- 6,541
1
vote
2
answers
515
views
continuity of extension of maps along curves
Let $a\le b$ and $k\ge 0$ be given and fixed. Let furthermore $x$ and $y$ denote two different elements of a Hilbert space $H$. Suppose $u:\mathbb{R}\rightarrow H$ is a $C^k$-embedding connecting $x$ ...
- 2,935
1
vote
0
answers
283
views
Density of Dolean exponentials in L2 and Wiener Measure
Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...
- 19
1
vote
1
answer
268
views
Best constant for Hölder inequality in Lorentz spaces
It's well known (and proved by R. O'neil) that there is a version of Hölder's inequality for Lorentz spaces, namely
$$\|fg\|_{L^{p, q}} \lesssim_{p_1, p_2, q_1, q_2} \|f\|_{L^{p_1, q_1}}\|g\|_{L^{...
0
votes
2
answers
162
views
Superquadratic boundedness from $L^2$ convergence
Assume $f_n,f\in L^2(\mathbb{R}^3)$ and $f_n\to f$ strongly in $L^2$. It seems there is a common fact that there exists a superquadratic maps $\beta\in C([0,\infty);[0,\infty))$ such that $\beta(0)=0$,...
- 71
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 ...
- 6,853
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
0
votes
2
answers
387
views
Derivative of fractional Laplacian is the fractional Laplacian of the derivative
Is it true that $$\partial_x ((-\Delta)^s u(x)) = ((-\Delta)^s \partial_x
u(x))?$$
user104647
0
votes
2
answers
425
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.
- 153