All Questions
12,935 questions
3
votes
1
answer
278
views
$L^{\infty}$ estimate for heat equation with $L^2$ initial data
Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem:
$$\begin{...
- 1,759
0
votes
0
answers
29
views
Advection diffusion equation with position dependent advection
I am trying to solve a equation that looks like $$u_t=f(x)u_x+Du_{xx}$$
any insight to whether an solution might exist would be very much appreciated!
-1
votes
1
answer
168
views
A question in functional analysis about selfadjoint operator [closed]
In Hilbert space $u$, Let $T_1$,$T_2$ is selfadjoint operator, if exit $c>0$ such that $cI\le T_1\le T_2$, prove $T_1$,$T_2$ have a bounded inverse operator and $c^{-1}I\ge T_1^{-1}\ge T_2^{-1}$.
I ...
- 1
1
vote
1
answer
249
views
Sufficient condition such that the set of zeros of an analytic function $f:\mathbb{R}^n \to \mathbb{R}$ contains only isolated points
Consider a real- analytic function $f: \mathbb{R}^n \to \mathbb{R}$. We know that zeros of $f$, roughly speaking, live in the low dimensional manifolds.
My question: Does a 'reasonable' sufficient ...
- 671
0
votes
0
answers
94
views
The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
- 852
0
votes
0
answers
51
views
Rescaling of cosine families
First of all, the best wishes for 2024. Recently, I got aware of cosine operator families (in the framework of evolution equations). It is well-known, that operator semigroups can be rescaled (see for ...
2
votes
0
answers
162
views
Does integration by parts formula hold in $H^1(0,T,L^2(\Omega))$?
Let $\Omega$ be an open set from $\mathbb{R}^N$. How can we prove that if $u,v\in H^1(0,T,L^2(\Omega))$ (in Bochner sense) then $(u\cdot v)'\in L^2(0,T,L^1(\Omega))$ with $(uv)'=u'v+v'u$ and the ...
- 1,759
1
vote
1
answer
99
views
Convergence of ODE with uniform $L^\infty \cap L^1$ bound on nonlinearity
Consider the IVP
$$
\left\{
\begin{aligned}
\frac{d}{dt} \Phi_n(t,x) &= f_n(\Phi_n(t,x)) && \forall t \in \mathbf{R}_+ \\
\Phi_n(0,x) &= x && \forall x \in \mathbf{R}
\end{...
- 73
0
votes
0
answers
52
views
Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
- 697
1
vote
0
answers
141
views
$L^{p}$ estimate for $\frac{|\nabla f|^{2}}{f}$
I’m trying to obtain an $L^{p}$ estimate under certain conditions. Suppose $f\in C^{2}(B_{2})$ is a function satisfying $0<f<1$ and $f\Delta f>\frac{1}{n} |\nabla f|^{2}$, where $B_{2}\subset\...
- 155
1
vote
2
answers
271
views
An integral inequality?
Let $v \in C^\infty(\mathbb R)$ such that $1 \ge v \ge 0$ and $\int_{\mathbb R} v \, dx = 1$.
I want to show that if
$$\int_{\mathbb R} v |v''|^2 \, dx < + \infty. \tag{$\star$}$$
then
$$ \int_{\...
- 83
1
vote
0
answers
63
views
Nitsche's method for p-Laplace equation
My question is about how you impose Dirichlet boundary conditions for the p-Laplace equation.
The minimization form of this problem is to find the function $u$ in $W_1^p(\Omega)$ that minimizes the ...
- 840
1
vote
0
answers
82
views
For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$
For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
- 825
2
votes
0
answers
103
views
Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$
Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...
- 90
1
vote
0
answers
45
views
Characterization of the Picard's condition for integral equation
Picard's condition (Thm. 15.18, Kress et al. 1989) is essential to study the existence of a solution of a Fredholm integral equation of the first kind. Specifically, consider (the univariate case) the ...
- 143
16
votes
1
answer
784
views
The determinant as a differential operator
According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
- 52.3k
4
votes
0
answers
149
views
Isomorphic copies of $c_0$ in the projective tensor products
There exist Banach spaces $X$ such that the projective tensor product $X\mathbin{\hat{\otimes}}_\pi X$ contains an isomorphic copy of $c_0$ [BourgainPisier1983]. Moreover, $X$ is an $\mathcal{L}_\...
- 2,605
1
vote
1
answer
150
views
Unable to understand an application of Minkowski's inequality
Consider the following exerpt from the paper "Non-linear Quantum Processes" by Segal:
with the norm $\|F\|=\left(\int\|F(x)\|^p \, d x\right)^{1 / p}$, then the operator $T_1^{\prime}: F \...
- 90
7
votes
0
answers
164
views
Nontrivial examples of locally compact quantum groups
What are some families of locally compact quantum groups that are neither groups, duals of groups, compact, nor discrete?
- 3,816
2
votes
0
answers
139
views
Multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
How to rigorously demonstrate that multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
I am utilizing the exponential type definition established by Elias Stein in the book 'Fourier ...
- 21
7
votes
3
answers
909
views
Using the Stone-Weierstrass theorem to solve an integral limit
The following question was posted on math stack exchange here but it got no answers
Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
- 331
2
votes
0
answers
78
views
Converse of existence of minimizers
Let $(V,\|\cdot\|)$ be a real normed linear space. $V$ has the property that given any nonempty convex, closed subset $K$, there exists a unique $v_0\in K$ such that $\|v_0\| \leq \|v\|, \forall v\in ...
- 131
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
2
votes
1
answer
65
views
Existence of solution to Cauchy boundary value problem in Lipschitz class of functions
For a research question I have run into the following problem that seems intuitively true but I do not know how to prove it and am not sure in which generality.
Let $\Omega\subset \mathbb{R}^2$ be a ...
- 41
3
votes
1
answer
180
views
Are the paths of the Brownian motion contained in a suitable RKHS?
Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are not contained in $H_B$.
But is ...
- 31
1
vote
1
answer
138
views
Can functions with "big" discontinuities be in $H^1$?
How can I prove that the function:
$$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...
- 1,759
2
votes
0
answers
172
views
AQFT from a Lagrangian
In physics, the fundamental description of physical theories frequently revolves around the concept of a Lagrangian. My expertise encompasses diverse algebraic formulations within the domain of ...
- 424
1
vote
0
answers
40
views
System of equations with one integral equation
Start with a system of three equations such that two of the equations are ordinary or partial differential equations, but one of them is an integral equation as follows:
$C = \int_{0}^{\infty} X \: ...
- 5,146
1
vote
0
answers
114
views
Global existence for large data in $H^{-1/2}(\mathbb R)$ of viscous Burgers' equation with external forcing
First, a quick summary of what to know about viscous (or dissipative) Burgers' equation
$$ u_t-u_{xx}=(u^2)_x. \tag{1}\label{1}$$
Recall that $\dot H^{-1/2}(\mathbb R)$ is a scaling-critical Sobolev ...
- 1,075
7
votes
0
answers
177
views
What is the current status of research on the von Neumann's inequality for $n \ge 3$?
Problem
Let $(T_1, \ldots, T_n)$ be a tuple of commuting contractions in Hilbert space $H$.
Does a constant $C_n \ge 1$ exist, for which it would be true, that:
$$\forall_{p \in \mathbb{C}[x_1, \ldots,...
- 63
4
votes
3
answers
869
views
Can these identities for the Euler-Mascheroni constant be proven?
I stumbled upon these 4 limit/integral identities involving Euler's constant aka gamma (~0.5772). They appear to be valid based on inspection but I have no idea how to prove them. In addition, I have ...
- 194
3
votes
0
answers
89
views
Cylindrical Wiener processes or SPDE that can make use of Banach valued rough paths?
Rough paths theory has an often advertised perk that it mostly works for general Banach spaces. I am trying to think of some nice examples that actually use this feature, and am coming up stuck.
The ...
- 193
0
votes
0
answers
170
views
Extreme confusion over the tautology of the solution for PDEs
Let $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth divergence free vector field. Then, it is well-known from the theory of Navier-Stokes equation that there exists some $T \in (0,\infty]$ and a ...
- 3,477
0
votes
0
answers
88
views
How to solve with FEM a semilinear elliptic equation?
I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
- 1,759
0
votes
0
answers
135
views
Relative bounds for vorticity
Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
- 317
1
vote
0
answers
102
views
If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?
Definitions
Representation
Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$.
We call $\...
- 63
1
vote
1
answer
191
views
What is kth vortex formula?
I want to study the kth vorticity equation. The NS equation is provided as
\begin{align}\label{eq1}
&\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \right)\textbf{u} = ...
- 317
6
votes
1
answer
249
views
Syndetic sets and Banach limits: reference request
First of all, let us give a few definitions. Suppose that $A$ is a subset of natural numbers. We say that $A$ is syndetic if there is a constant $M$ such that every set of $M$ consecutive natural ...
- 7,193
5
votes
2
answers
327
views
Regularity of stochastic heat equation
For the heat equation:
$$\frac{\partial u}{\partial t} = \Delta u + \zeta(t,x)$$
where $u:\mathbb{R}^{+}\times \mathbb{R}^{n} \rightarrow \mathbb{R}$ and $\zeta$ is a space-time white noise.
I'm ...
- 51
4
votes
0
answers
111
views
Flatness of $C_0(S)$-module $L_\infty(S,\mu)$
Let $S$ be a locally compact Hausdorff space. By $C_0(S)$ we denote the space of continuous functions vanishing at infinity. Let $\mu$ be a finite Borel regular measure om $S$, then consider $L_\infty(...
- 1,697
2
votes
0
answers
87
views
A question on the maximum principle of second order elliptic equations
Let $Lu=a^{ij}u_{ij} + b^i u_i$ be an elliptic operator of second order in a bounded domain $\Omega$. Assume that $a^{ij}$ is uniformly elliptic. Then it's well known that the following maximum ...
- 438
3
votes
2
answers
615
views
A problem about how dominated convergence is used in the analysis of variation
I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds and get stuck on Lemma6. When $\lambda>\Lambda_1$, with $\Lambda_1=(2 m-1) ! \operatorname{vol}\left(S^{2 ...
- 809
2
votes
0
answers
97
views
Is the Leray projection continuous with respect to the Frechet topology of smooth periodic vector fields in $3$ dimensions?
Let $\mathbb{T}^3:=(\mathbb{R}/\mathbb{Z})^3$ be the $3$-torus and $C^\infty(\mathbb{T}^3,\mathbb{R}^3)$ be the Frechet space of smooth periodic vector fields on $\mathbb{T}^3$.
By Helmholtz ...
- 3,477
0
votes
0
answers
192
views
Reference request: an introduction to nuclear spaces
I am looking for a short introduction to nuclear spaces and nuclear operators. I am interested in these spaces as they often arise in mathematically rigorous quantum field theories. I have read the ...
- 721
1
vote
0
answers
87
views
$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?
Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
- 309
4
votes
1
answer
147
views
Embeddings of the maximal domain for the Laplacian
Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function:
$$D = \left\{ f \in L^2(\...
- 41
0
votes
0
answers
56
views
Why is von Neumann inequality important for equivalence of $\forall_j \ T_j^n\rightarrow 0$ in A-topology and abs continuity of $(T_1,\ldots, T_N)$?
The whole theorem goes as follows:
Let $(T_1, \ldots, T_N)$ be a tuple of commuting operators in Hilbert space $H$ satisfying:
$$\exists_{M > 0} \ : \ \forall_{p \in \mathbb{C}[z_1, \ldots, z_N]} \ ...
- 63
8
votes
2
answers
675
views
Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well
It is known that an entire function that is nowhere zero must be the exponential of another entire function.
Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...
- 286
2
votes
1
answer
196
views
Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\operatorname{div}g(x)\...
3
votes
0
answers
140
views
Trace class operators
There is a notion of trace class operator in a Hilbert space.
Is there a notion of trace class operator in arbitrary Banach space? locally convex space?
A reference will be helpful.
- 21.8k