All Questions
Tagged with fa.functional-analysis ca.classical-analysis-and-odes
524 questions
2
votes
1
answer
265
views
A Volterra-type equation
Consider the following integral equation
$\phi(x) = f(x) + \frac{1}{x}\int_0^x N(x,y)\phi(y)\;dy$,
where $f$ and $N$ are continuous and bounded functions. Are solutions $\phi$ of the above equation ...
- 595
2
votes
2
answers
354
views
A bound on linear functionals over cotype 2 spaces
This is a modification of the somewhat naive question that I asked below.
Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\...
- 2,151
2
votes
1
answer
142
views
Estimating a solution to an Euler-type ODE
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer and $a$ be a real number.
Let $u(r)$ be a function on $[1,\infty)$ ...
- 969
2
votes
1
answer
295
views
Non-zero, bounded, continuous, differentiable at the origin, compactly supported functions with everywhere non-negative Fourier transforms
Do there exist functions $F(x) \! : \, \mathbb R \to \mathbb R$ which are non-zero and bounded:
$$ \mathrm {Range} (F) = [l, u] \, , \quad \mathrm {where} \quad l, u \in \mathbb R \land u > l \, ; \...
- 179
2
votes
2
answers
539
views
Graph with complex eigenvalues
The question I am wondering about is:
Can the discrete Laplacian have complex eigenvalues on a graph?
Clearly, there are two cases where it is obvious that this is impossible.
1.) The graph is ...
user144765
2
votes
1
answer
183
views
A minimal sequence in $L^2([-\pi, \pi])$
Let $\lambda_n = n + \delta_n $ for all $n \in \mathbb{Z}$ where $\delta_n$ are a sequence of real numbers in $\ell^2(\mathbb{Z})$. How can one show that the sequence $(x_n)_{n \in \mathbb{Z}} = (e^{i ...
- 309
2
votes
1
answer
234
views
ODE for functions with values in locally convex TVS
Given an ODE for a function $u \in C^1(I,V)$, where $V$ is some locally convex TVS (topological vector space) and $I \subset \mathbb{R}$, i.e.
$\frac{d}{dt} u = f(t,u)$
for some function $f: I \...
- 403
2
votes
4
answers
411
views
A Fractional Linear Transformation Class Property
Let $\mathcal{S}$ be the class of Fractional Linear Transformations (or FLT's) consisting of $f: [-1,0] \rightarrow [-1,0]$ such that $f(x) = \frac{ax+b}{cx+d}$ where
$a,b,c,d \in R$, and $f'(x)>0$...
- 81
2
votes
1
answer
1k
views
Green's function for wave equations in R² or R³
Hello,
For almost one year, I am searching for the Green's function for wave equation in R² or R³ with some boundary conditions. As far as I know, when the boundaries permit the method of images, we ...
- 1,649
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 ...
2
votes
1
answer
206
views
A simple 1-dimensional inequality, maybe Poincaré inequality or Hölder inequality?
I'm reading a paper on the classical Gagliardo-Nirenberg interpolation inequality arXiv link and there is a inequality used
$$
|v-\overline{v}|\le \left\Vert v' \right\Vert_{r,I} \ell^{1-\frac{1}{r}}, ...
- 65
2
votes
1
answer
230
views
Integration by parts with Hilbert transform
Is there a good integration by parts formula to compute
$$\int_{0}^\infty f \ H (f') dx,$$
where $H$ denotes the Hilbert transform and $f$ is a smooth function?
- 303
2
votes
1
answer
276
views
Weak-star approximation of smooth functions in weak $L^p$-space
It is well known that the weak space $L^{p,\infty}$ has less density property contrary to standard $L^p$ space. Related to this one, I'm struggling to prove the following statement which is given in ...
- 323
2
votes
1
answer
103
views
A density question
Suppose $\Omega= (0,1)\times(0,1)\subset \mathbb R^2$. Assume that $f, g \in C^{\infty}(\Omega)$ and that
$$ \int_\Omega \left(f(x_1,x_2)- \frac{m}{(n+1)}g(x_1,x_2)\right) x_1^n \,x_2^m \,dx_1\,dx_2 = ...
- 4,143
2
votes
1
answer
93
views
Lipschitz bound on semigroups
Let $T$ be a self-adjoint operator (possibly unbounded) and $S$ a bounded self-adjoint operator.
Then one can study the unitary groups $R_T(t):=e^{itT}$ and $R_S(t):=e^{itS}.$
Now if you think about ...
2
votes
1
answer
547
views
Equivalent references for Schwartz's book of the distribution theory
Hello,
It seems that there is no English translation of the Schwartz's book 1966. I may need to use the spaces like
$$
\dot{\mathcal{B}}(R),\quad \dot{\mathcal{B}}'(R),\quad \mathcal{B}(R),\quad \...
- 1,649
2
votes
1
answer
1k
views
Range of the Radon Transform
Let us consider the Radon transform in two dimensions:
$$\tag{1}Rf(r,\theta):=\int\limits_{-\infty}^{\infty} f(r\cos\theta-t\sin\theta,r\sin\theta+t\cos\theta) dt,$$
where $r\in\mathbb{R}$ and $0\...
- 931
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} =: \...
- 97
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
$$
\frac{\partial^2\gamma}{\partial x \, \partial y} = 0\:\:\...
- 134
2
votes
1
answer
145
views
Estimate for an oscillatory integral of the first kind
I am confused in finding the right bound for the following oscillatory integral
$$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$
Where $\psi(2^{-k} \xi)$ is a smooth ...
- 159
2
votes
1
answer
239
views
Injectivity of an integral transform
For a bounded function $F: \mathbb R_{\ge 0} \to \mathbb R$ (not necessarily non-negative), is it true that
$$\int_0^\infty \frac{x^ks}{(s^2+x^2)^{(k+3)/2}} F(x) dx = 0 \text{ for all $s >0$} \iff ...
- 303
2
votes
2
answers
317
views
Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative
What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative?
More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of
a function
$$...
- 839
2
votes
1
answer
997
views
Derivative and Jacobian determinant of solution of ODE [closed]
Let $\Phi$ be the unique solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
where we have assumed $f$ smooth.
...
user124345
2
votes
2
answers
261
views
Viscosity solutions for $u'(x) + \alpha u(x) - f(x) = 0$: supersolutions dominate subsolutions
Let $$u'(x) + \alpha u(x) - f(x) = 0,$$ with $x \in [0,\infty)$ and $\alpha \in \mathbb{R}$. Suppose $f \in C(\mathbb{R})$.
If
$u_1$ is a viscosity supersolution (or a viscosity solution, or a $C^...
user102027
2
votes
1
answer
158
views
Defect of Compactness for the Strichartz Estimates
I am trying to understand the motivation behind the main theorem of Keraani: See the top of page 356 and which is the motivation behind his Theorem 1.6
We consider
$$i\partial_t u + \Delta u =0, u(x,...
- 233
2
votes
1
answer
159
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 $V_0$.
...
- 23
2
votes
1
answer
267
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 ...
- 83
2
votes
1
answer
687
views
Solutions to Heat Equations with Obstacles!
Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = \...
2
votes
2
answers
181
views
convergence of the coefficients of lacunary series
I just want to find some standard reference to the following result: let $(a_k)_k$ be the sequence of coefficients of a lacunary Fourier series which converges to an $L_1(T)$ function in the sense of ...
2
votes
1
answer
152
views
Growth rate of elementary sequences
We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...
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 + ...
- 41
2
votes
1
answer
94
views
Decay rate for a small perturbation of a simple linear ODE
MOTIVATION.
Let $f:[0,+\infty)\to \mathbb{R}$.
Solutions to $\partial_tf(t) = -\lambda f(t)$, $f(0)\neq 0$ approach zero exactly as $e^{-\lambda t}$.
This property is preserved if we apply an ...
- 2,533
2
votes
1
answer
225
views
Prove convergence of whole sequence $f_n$ of solutions to a differential problem to a limit $f$ (without uniqueness assumptions)
Let $\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$ be a sequence of functions that solves a linear differential equation $F_n(f_n, \nabla f_n) = 0$. Suppose that there exists a subsequence $n_k$ ...
- 839
2
votes
1
answer
391
views
Entropy solution for linear transport equation
Consider the transport equations
$$ (1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$
and
$$ (2) \qquad \partial_t u + b \cdot \nabla u= 0$$
Can we define a notion of entropy solutions for (1) ...
- 839
2
votes
1
answer
300
views
Optimal control theory of PDEs
This is a somewhat openly phrased question because I am not quite sure what has been done in that direction.
Imagine one has two evolution equations
$$\partial_t u = p(x,\partial_x,f)u$$
$$\...
- 536
2
votes
2
answers
406
views
Solution to semilinear heat equation at $t=0$: $u_t(0,x) - \Delta u(0,x) + f(x,u,u_x)= 0 \ ?$
Consider the following Cauchy problem
$$u_t - \Delta u + f(x,u,u_x) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$
with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$
Suppose that $u \...
- 303
2
votes
1
answer
265
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 121
2
votes
1
answer
176
views
Two ODEs, why is one the solution of the other? (Caratheodory ODE)
This question is based on Zeidler II/B, Problem 30.2.
Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t.
$$u'(t) = F(t,u(t))$$
$$u(0) = u_0$$
given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ ...
- 251
2
votes
1
answer
535
views
about decomposition of a non-negative definite operators
Hello,
Many years before, I had the following problem.
We first give a definition. Given a non-negative definite real-valued definite matrix $n^2\times n^2$ matrix $M$, it is called separable if it ...
- 1,649
2
votes
1
answer
570
views
Is a polynomial positive on the sphere a sum of squares of spherical harmonic polynomials?
Let $p\in {\mathbb{R}}[x_1,\ldots, x_n]$ be a homogenous polynomial of even degree $2d$. If $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$, then does there exist some $m>0$ and ...
user2529
2
votes
1
answer
412
views
General Sobolev Inequalities
In Partial Differential Equation by Lawerence Evan p284 there is this theorem stated:
Let $U$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary. Suppose $u\in W^{k,p}$ then if $k>n/p$ ...
- 3,217
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
2
votes
0
answers
102
views
Existence of unique-up-to-shift solution of a Volterra equation
Let $\Delta=\{(t,s):\ 0<s\leq t\leq1\}$, and suppose $k:\Delta\to\mathbb R$ and $f:(0,1]\to\mathbb R$ are continuous. Further assume that for every $t\in(0,1]$, the function $k(t,\cdot):(0,t]\to\...
- 537
2
votes
0
answers
97
views
On the second order analog of the upper 1-Lipschitz envelope of a function
Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope
$$
\hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
- 355
2
votes
0
answers
81
views
Extension of a tangent vector field
Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
- 434
2
votes
0
answers
138
views
Sufficient initial conditions for "non-local" PDE
I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
- 71
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 $$ \norm{ \partial_{ij}u }_{L^p} \leq C \norm{\Delta u }_{L^...
- 457
2
votes
0
answers
136
views
Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?
Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$,
$$H(h')(t) = \lambda h(t)$$
...
- 20.2k
2
votes
0
answers
201
views
Green function of a 2D exterior domain
Consider solutions of the laplace equation
\begin{equation}
\begin{split}
-\Delta u=f, \ \ u|_{\partial D}=0,
\end{split}
\end{equation}
where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
- 379
2
votes
0
answers
64
views
Scaling limit of ODE with double-well potential
Let us consider the ODE
$$
\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