All Questions
12,935 questions
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164
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Existence of directional heat equation without uniform ellipticity
I am asking for references, or for a proof idea on how to show that weak solutions of the following problem exist: search $u$ on a bounded domain $\Omega\times (0,T]$, where $\Omega\subset\mathbb{R}^d$...
- 35
1
vote
0
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88
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Schauder estimate for $f \in L^\infty$
I was reading an article where at some point the author uses the following estimate:
Let $u$ be a solution of
$$\Delta u = f \quad \text{in } B_1$$
for $f \in L^\infty$. Then $u \in C^{1,1 - \...
- 452
3
votes
0
answers
167
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Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate
If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
2
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0
answers
29
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Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
- 434
1
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2
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209
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Approximate simple function $f$ by a sequence of continuous functions on $\mathbb{R}^d$ such that $\|f_n\|_\infty\leq \|f\|_\infty$
Let $f=\sum_{i=1}^n c_i 1_{\Delta_i}$ be a simple function on $\mathbb{R}^d$, where $c_i\in\mathbb{C}$. Then we can find sequnces of continuous functions $\{f_k^{(i)}\}$ for each $i=1,\ldots,n$ such ...
- 29
1
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1
answer
152
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SOT and WOT convergence of Toeplitz operators
For the Hardy space $H^2$, every $\phi \in L^\infty (\mathbb T)$ induces a bounded Toeplitz operator $T_\phi$ on the Hardy space and $\lVert T_\phi \rVert = \lVert \phi \rVert _{\infty}$. Consequently,...
- 151
1
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248
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Solving functional analysis problems by using Algebraic geometry
I am thinking about some open problems in nonlinear functional analysis and I just wanted to know if there are any problems that have been solved by using Algebraic geometry techniques in these fields....
1
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0
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120
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Well-posedness result for a linear parabolic equation on the torus
Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$
$$ \partial_t u- ...
- 185
4
votes
1
answer
219
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Poincaré inequality on compact manifolds without boundary
The question arises from a paper The heat flow for the full bosonic string, Ann. Global Anal. Geom. 50 (2016).
In line 4, Page 362, the author claimed the following inequality which looks similar to ...
- 143
2
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0
answers
82
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What is Lipschitz constant of the radial renormalization $(X,\|\cdot\|_a) \rightarrow (X,\|\cdot\|_b)$ on a normed vector space $X$
Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping
$$
f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x, \qquad \forall x \in X,
$$
with $f(0)=0$
is a radial and maps ...
- 5,327
6
votes
1
answer
256
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Example/Existence of Positive Linear Functional which is NOT Hermitian
We know that if $\mathcal{A}$ is a unital $C^*$-algebra and if $f:\mathcal{A}\to\mathbb{C}$ is a positive linear functional then it is Hermitian. It simply follows from the fact that in $\mathcal{A}$ ...
- 173
0
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0
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93
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Orthogonalization of symmetric non-degenerate bilinear forms
It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
- 1,485
2
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1
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99
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A question on Bloch functions
Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by
$$
\|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
- 1,706
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0
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56
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What is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}$?
I have asked this here. Due to inactivity and no satisfying answers, I am asking here. Hope that's okay.
We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$
is $n$ (thanks to this ...
- 113
7
votes
1
answer
152
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Higher (BV) regularity of solutions to Poisson equation with Radon measure right-hand side?
I am trying to understand higher regularity for solutions to Poisson's equation when the right-hand side is a Radon measure. In particular:
$$\begin{cases}
\Delta u = \mu \text{ in } \Omega\\
u = 0\...
- 221
1
vote
1
answer
151
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Some operators on spheres
Let $S_2$ be the unit sphere in $\mathbb R^3$ equipped with normalized Haar measure. For a continuous function f and $\delta\in (-1,1)$ define $T_\delta f(x):=\int_{\{y:<x,y>=\delta\}}f(y)d_\...
- 1,879
2
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1
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89
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The contractivity of the time derivative of the heat semigroup in $L^p$ spaces
Let $M$ be a complete manifold. The heat semigroup $e^{-tL}$ is bounded on $L^p(M)$, for any $1 \leq p \leq \infty$;
see this for instance.
It seems that we can deduce the time derivative of the heat ...
- 121
7
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1
answer
580
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Sobolev spaces are smooth? Their dual is strictly convex?
Do you know any reference which says something about the:
Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton.
...
- 1,759
5
votes
2
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258
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Boundary value of Sobolev space
Let $D$ be a regular domain in $\mathbb R^2$. Suppose that $u \in H_0^1(D) \cap C(D)$. Does this imply $u \in C(\overline D)$ and $u|_{\partial D} = 0$?
- 177
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0
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118
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Find the maximum of an expression under the logconcave assumption
Let $F(v)$ be a cdf over $\left[0,v_{max}\right]$, $1-F(v)$ is logconcave. The corresponding density function is $f(v)$. Let $p^m$ solve $1-F(v)-f(v)v=0$ (it is a FOC of a profit maximization problem)....
1
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0
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30
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Generalization of subadditivity analogous to quasiconvexity, and variants
I am curious if there are natural generalizations of subadditivity which have been studied in the past or have been stated in the literature? I (and people that I have talked to) have not had much ...
- 21
1
vote
1
answer
165
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Calderón–Zygmund inequality for Neumann problems
My question is simple:
Let $\Omega$ be a bounded smooth domain. Does the following inequality $$\|D^2w\|_{L^q(\Omega)}\leq C\|\Delta w\|_{L^q(\Omega)},$$ hold for any function $w\in W^{2,p}(\Omega)\...
- 113
2
votes
1
answer
118
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Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius
This is a follow up from this question.
I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
- 765
5
votes
2
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364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
6
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1
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297
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Understanding exterior differential systems
Let $M$ be an $n$-dimensional smooth manifold. An exterior differential system on $M$ is by definition a graded ideal $\mathcal{I}\subset \Omega^{\bullet}(M)$ in the ring $\Omega^{\bullet}(M)$ of ...
- 2,816
1
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1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,058
1
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1
answer
76
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Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius
I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
- 765
1
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0
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55
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References for the generalized Dirichlet problem for plurisubharmonic functions on the bidisc
In the paper "On a Generalized Dirichlet Problem for Plurisubharmonic Functions and Pseudo-Convex Domains. Characterization of Silov Boundaries" Theorem 5.3, the following result is obtained ...
- 53
1
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1
answer
115
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Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
2
votes
0
answers
94
views
Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces
Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
- 1,759
7
votes
1
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253
views
Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?
In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed.
Does a Banach algebra ...
- 515
3
votes
0
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132
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Uniqueness for a nonlinear kinetic PDE-system with heat transfer coupling in one dimension
I am currently trying to understand the following article "Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit" (2021) by Favre, ...
- 185
8
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0
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177
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Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here.
Let $C^k(\mathbb{R}^n$) be the ...
- 721
1
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1
answer
62
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Integrability in the product space can follow from a property of the Nemytskii operator?
Let's say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where ...
- 1,759
8
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1
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584
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Reference request: Software for producing sounds of drums of specified shapes
Is there software that, when the input is the shape of a drum, will produce the corresponding audible sound?
0
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0
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85
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Measurable selection for the mean value theorem
When we use the mean value theorem we come across the problem of measurability of the argument. The problem is somehow like that:
Let $f:\Omega\times [0,1]\to\mathbb{R}$ be a Caratheodory function (i....
- 1,759
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0
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116
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Integral of a measurable function with parameter is measurable?
Say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ is an open set, is a function such that:
$f(x,\cdot)\in L^1_{\text{loc}}(\mathbb{R})$ for a.a. $x\in\Omega$
$f(\...
- 1,759
2
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0
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84
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Question about the Nemytsky operator on $L^p$ space
Let $\Omega\subset\mathbb{R}^N$ be a bounded open set, $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function, i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is ...
- 1,759
3
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0
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147
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Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
- 91
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1
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123
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Proving a Fourier transform inequality for functions with mixed variable bounded support
I'm working on a problem involving the Fourier transform and have encountered an inequality that I am unsure how to prove. I would greatly appreciate any help or guidance you can provide.
Let $\gamma\...
- 131
1
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175
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Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
- 47
22
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0
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869
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Can two drums almost sound the same?
Let $D\subset \mathbb R^2$ be a region and let $\Lambda=\{\lambda_1,\lambda_2,\dots\}$ be the set of eigenvalues of the Laplacian $-\Delta$ (with boundary condition $\psi=0$ on $\partial D$).
Mark Kac,...
- 3,054
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0
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79
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Is the Bures metric equivalent to the Euclidean one?
Let $K=\mathbb R$ (reall numbers) or $K=\mathbb C$ (complex numbers). Define $\mathcal M_n$ to be the space of $n\times n$ matrices $A=(a_{i,j})_{1\le i,j\le n}$, with $a_{i,j}\in K$. Let $\|\cdot\|$ ...
- 1,334
1
vote
2
answers
156
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Numerical evaluation of monomial divided differences
Suppose $f(x)=x^{n+1}$ for some $n\in\mathbb{N}$, and define the divided difference $$f[a,b]=\frac{a^{n+1}-b^{n+1}}{a-b}.$$
I am wondering about the best way to numerically evaluate $f[a,b]$ to high ...
- 45
1
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0
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52
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Making sense of constant scalar curvature metric horns
Suppose we have a compact oriented surface $S$ and we remove a point $p$ on it. We could consider a neighboorhood $U$ of the puncture $p$, so that the points in this neighboorhood are described by ...
1
vote
3
answers
220
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Solving $X$ for prescribed $\operatorname{div}(X)$ of compact support
Let $f \in C^\infty_0(\mathbb{R}^3)$ be a smooth scalar function of compact support. I would like to find a smooth vector field $X : \mathbb{R}^3 \to \mathbb{R}^3$ satisfying $\operatorname{div}(X) = ...
- 403
3
votes
1
answer
227
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Algebraic and continuous duals of an inverse limit of finite dimensional vector spaces
I have been trying to understand the following section of a paper "Revêtements du demi-plan de Drinfeld et correspondance de Langlands p-adique" by Gabriel Dospinescu and Arthur-César Le ...
- 117
5
votes
1
answer
220
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How big is the class of all closed range bounded linear operator?
Let $X$ and $Y$ be Banach spaces and let $CR(X,Y)$ denote the set $B(X,Y)$ of all bounded linear maps from $X$ to $Y$ with $T(X)$ closed in $Y$. Certainly $CR(X,Y)$ is not open in $B(X,Y)$ as given ...
- 585
2
votes
1
answer
236
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A sensible topology on the space of continuous linear maps between Fréchet spaces
Let $V_1$ and $V_2$ be Fréchet spaces. Let $\{ \lVert \cdot \rVert_{1,n} \}_{n \in \mathbb{N}}$ be a family of seminorms for $V_1$ and similarly $\{ \lVert \cdot \rVert_{2,n} \}_{n \in \mathbb{N}}$ ...
- 3,477
2
votes
0
answers
60
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Semigroup property in SPDEs
In fact, we know that a bounded linear operators on a Banach space $X$ satisfies the semigroup property, i.e. $$S(t+s)=S(t)S(s), \text{for every}\ t,s\geq 0.$$
However, in various literatures, I ...
- 57