All Questions
Tagged with fa.functional-analysis differential-equations
67 questions with no upvoted or accepted answers
10
votes
0
answers
845
views
Witt's proof of Gelfand-Mazur / Ostrowski's Theorem
Previously asked on Math Stackexchange without answers.
Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
- 2,454
6
votes
0
answers
219
views
Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
- 370
6
votes
0
answers
201
views
Dependence of Neumann eigenvalues on the domain
I have the following problem in hands, in the context of a broader investigation:
Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following:
For any $\...
- 101
6
votes
0
answers
113
views
A continuity argument for a dispersive $gKdV$ estimate
I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at
$$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$
where $F(u) = u^5$ (for example). The ...
- 163
5
votes
0
answers
419
views
Nonlinear variation of constants formula
Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), \; x(0)=x_0\in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X \rightarrow X $ are two nonlinear functions. Furthermore, ...
- 51
5
votes
0
answers
262
views
Weighted reverse Poincare inequality over a function class of neural networks
We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
- 325
5
votes
0
answers
374
views
A question about Carleman linearization
Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻²
Let $f$ be ...
- 1,590
5
votes
0
answers
179
views
Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?
Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...
5
votes
0
answers
240
views
Linear ODEs in a locally convex vector space
Let $X$ be a complete, locally convex, Hausdorff topological vector space over $\mathbb{C}$. Let $J \subset \mathbb{R}$ be an open interval. Consider the space $M = C^\infty(J,X)$ of smooth ...
- 235
4
votes
0
answers
126
views
Darboux integral for non-polynomial ODEs
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$
\dot{x}_j=f_j(x)=\sum_{i_1,\dots,i_n=1}^d a_{i_1,\dots,i_n}^j x_1^{i_1}\dots x_n^{i_n} \quad \forall j=1,\ldots,n
$$
we define ...
- 247
4
votes
0
answers
258
views
Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel
Let $\Omega$ be an open subset of $\mathbb{R}^d$. Under regularity conditions, we know that the $s$-th order Sobolev space $H^s(\Omega)$ with $s \geq d/2$ is a reproducing kernel Hilbert space. In ...
- 1,127
4
votes
0
answers
410
views
Spectral Gap of Elliptic Operator
Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled?
The boundary condition is that the ...
- 325
4
votes
0
answers
145
views
An embedding question: Morrey spaces
Question. If $u\in L^1$ and $Du$ is in the dual of the Holder space $C^\alpha$, then is it possible to say $u$ belongs to some Morrey space $L^{1, \delta}$?
- 43.2k
4
votes
0
answers
176
views
Distributional PDE solutions as topological linear duals of PDE solutions
Let
$$
P \;\colon\; \Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)
$$
be a formally self-adjoint hyperbolic linear differential operator ($\tilde E^\ast$ denotes the densitized dual of a ...
- 19.8k
4
votes
0
answers
433
views
Scattering for rapidly decaying solutions of NLS
Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...
- 403
3
votes
0
answers
108
views
A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$
Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
- 1,171
3
votes
0
answers
124
views
Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different 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, $R&...
- 969
3
votes
0
answers
99
views
Definition clarification: "regular directed distributions"
(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of ...
- 1,171
3
votes
0
answers
182
views
Parabolic regularization for the Navier-Stokes equations
I'm looking for some references about a result on the Navier-Stokes equations which seems to be folklore but for which I didn't manage to find a proof. The setting is the following :
Let $Q=\mathbb{R}^...
- 161
3
votes
0
answers
104
views
Regular Lagrangian flows on a domain of $\mathbb{R}^d$ with a boundary
I'm looking for some references about the theory of regular Lagrangian flows on a smooth domain $\Omega$ of $\mathbb{R}^d$ (say a smooth bounded open set of $\mathbb{R}^d$ or a half space).
Here, ...
- 161
3
votes
0
answers
245
views
Regularity of the dependence of the flow on the vector field definining it
Let $M$ be a smooth compact manifold and $k \geqslant 1$.
Define $\mathfrak{X}^k(M,TM)$ to be the set of vector fields $M \rightarrow TM$ of class $C^k$. As $M$ is compact, endowing $\mathfrak{X}^k(M,...
3
votes
0
answers
498
views
PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field
Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let $<\cdot,...
- 800
2
votes
0
answers
143
views
How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...
- 21
2
votes
0
answers
126
views
Differential equations: trying to connect a nonlinear equation to a linear one
The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
- 383
2
votes
0
answers
153
views
Riesz’s representation theorem in a weak form
Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$, $\phi\in H_0^1(\Omega)$ is a solution of $$
\begin{cases}\Delta \phi+ \phi=h & \text { in } \Omega, \\ \phi=0 &...
- 471
2
votes
0
answers
266
views
Compactness of a nonlinear operator
Let $H^{1}_{0}(0;\pi)=\{f\in L^{2}(0; \pi): f^{\prime}\in L^{2}(0; \pi)\ \text{and}\ f(0)=f(\pi)=0 \} .$ equipped with the following norm $$\|f\|=\Big(\int_{0}^{\pi}|f'(x)|^2dx \Big)^{\frac{1}{2}}$$
...
- 39
2
votes
0
answers
445
views
Lax Milgram for non coercive problem?
I obtained the variational form of my problem. and the bilinear form is below.
Bilinear Form Let $\Omega\subset\mathbb{R}$ be an open set. For $u,v\in H^1_0(\Omega)$. I have
$$a(u,v)=\int_\Omega u(t)...
- 55
2
votes
0
answers
551
views
Euler-Lagrange equations on a differentiable manifold
I am following the conventions of https://arxiv.org/abs/math-ph/9902027
Let $M$ be a differentiable manifold, $E \to M$ a vector bundle over $M$ with fibre $F$, $J^1(E)$ the rank-one jet bundle over $...
- 651
2
votes
0
answers
77
views
How we can do the derivative for this equation w.r.t.to time t>0
Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
- 199
2
votes
0
answers
150
views
Limit circle/point of an ODE with finite eigenvalues
Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
- 155
2
votes
0
answers
169
views
Stochastic Approximation in Reproducing Kernel Hilbert Space
Consider an iterative algorithm with incremental updates
\begin{align}
x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}],
\end{align}
where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert ...
- 1,127
2
votes
0
answers
63
views
How do spectrums interact with bi-Lipschitz maps?
If it makes things simple, we can just stick to bi-Lipschitz maps from $S^k \rightarrow \mathbb{R}^d$ (w.r.t geodesic distance on the sphere with the standard round metric and the $2-$norm on the ...
- 617
2
votes
0
answers
87
views
1D inhomogeneous linear Schrodinger equation
I have the following problem:
$iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(...
2
votes
0
answers
112
views
Is the Wave Function a "Smooth" Function of the Potential?
Consider the Schroedinger equation in a spherically symmetric system. In the unit system under which energy is measured in Hartree and length in Bohr radius $a_0$, the schroedinger equation can be ...
- 183
2
votes
0
answers
148
views
Green's functions on linear subspaces and relations to boundary conditions
Consider the Laplacian $-\Delta$ on (in a suitable sense) twice differentiable functions subject to homogeneous Dirichlet boundary conditions $\mathscr{H}=\{f : f(0)=f(1)=0\}$. We can identify the ...
- 151
2
votes
0
answers
79
views
Point Spectrum of a Second Order System of Differential Equations
Consider the following operator acting on $H^1(\mathbb{R})$
$$
\mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) \...
- 516
2
votes
0
answers
110
views
Is there any weighted sobolev embedding with non-decaying weight
Is there any weighted sobolev embedding like
$$(\int_{R^d}(1+|x|)^s |u|^pdx)^{1/p}\leq C||\nabla^a u||_{L^q}$$?
Here $s>0$, and for some appropriate $p, q$.
- 31
1
vote
0
answers
37
views
Unique smallest degree algebraic solution to polynomial ODE
Let's assume we are given a degree $d$ polynomial VF as a map $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$
$$f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a^j_{i_{1},\dots,i_{n}}x_{1}^{i_{1}}\dots x_{n}^{i_{n}}$$...
- 247
1
vote
0
answers
89
views
Domain where the fractional Laplacian operator is a closed operator
Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$
Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\...
- 111
1
vote
0
answers
70
views
Continuity in the uniform operator topology of a map
I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...
- 39
1
vote
0
answers
161
views
The norm of Sobolev space involving the time
Question. Is the following way of writing the norm of a Sobolev space involving the time correct? I would be grateful for any help.
Let's assume we have a function
$$
\mathbf{u} (\mathbf{x}; t) = \...
1
vote
0
answers
119
views
On some integral equation
Let $M$ be the set of continuous and increasing functions $h: [0,1)\to\mathbb R_+$ s.t. $h(0)=0$ and $h(1-)=+\infty$. Consider the equation as follows:
$$
1-t= \int\limits_0^\infty p(x)\Phi\left(\frac{...
- 1,399
1
vote
0
answers
97
views
Reference for unique classical solution to quasilinear uniformly parabolic PDEs
In this post, the author mentioned that "we know there is a unique classical solution (see the references below, for example)". I have tried to read the two references the author provided, ...
- 11
1
vote
0
answers
235
views
Fredholmness of elliptic operator on Hölder spaces
Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...
- 163
1
vote
0
answers
131
views
Uniqueness of solution of Volterra Integral Equation with deviating argument
In the context of a physics problem, I am looking at a linear integral equation 2nd kind Volterra equation with deviating (centrosymmetric) argument in the unknown $u(t) \in L^2[a,b]$:
\begin{equation}...
- 259
1
vote
0
answers
147
views
Property of Fixed Point Function
Given an operator $\mathcal{T}$ that maps from a function $f: \mathbb{R}^d\rightarrow \mathbb{R}$ to another function $f': \mathbb{R}^d\rightarrow \mathbb{R}$, we are interested in the fixed point $f^*...
- 1,127
1
vote
0
answers
154
views
Lyapunov stability for nonlinear PDEs
Where can I find a theorem about Lyapunov stability for the equation in Hilbert space?
$$
y' = Fy,
$$
where $F : V \to V'$ is a nonlinear operator , $y' \in L^2(0,T,V')$, $V$ is a Hilbert space.
...
- 243
1
vote
0
answers
93
views
inverse problem to resolution of the identity
Suppose $f_\lambda(x),\lambda \in \mathbb{R}$ is a continuous/smooth family of bounded function on $\mathbb{R}$. Given a measurable set $\mathfrak{m}$ on $\mathbb{R}$, define $$E_\mathfrak{m}:L^2\...
- 587
1
vote
0
answers
360
views
Comparing Dirichlet energy and area of a Surface-immersion
Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...
- 1,255
1
vote
0
answers
182
views
Solution of non-linear Fredholm (Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity
I have asked this question at MSE but got no response. I have rephrased it so that anyone who knows operator theory and integral equations could help me out... I faced a problem in physics which is a ...
- 11