All Questions
3,628 questions with no upvoted or accepted answers
1
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141
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Homomorphic Commutator? Equation
So I was considering the following functional equation:
Given $H :\Bbb{C}^2 \rightarrow \Bbb{C} $ find $\theta: \Bbb{C}^2 \rightarrow \Bbb{C}$ such that
$$ \theta(H(a,b), H(c,d)) = H(\theta(a,c), ...
- 2,689
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0
answers
360
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Comparing Dirichlet energy and area of a Surface-immersion
Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...
- 1,255
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0
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95
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Construct a PDE solution from a net of approximations
Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let $...
- 5,407
1
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0
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186
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Weighted Sobolev spaces over open/closed intervals
I am escalating this question from SE, as I have been unable to obtain any guidence in relation to a possible solution.
Some context, I am working with weighted Sobolev spaces of the form $W^{m,2}(I,...
- 11
1
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0
answers
178
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Density of subspace with nonlocal/Wentzell boundary condition
Given the space $F$ defined by:
$$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$
I want to prove that the subspace $E$ of $F$ defined by $E=\...
- 111
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0
answers
62
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Reference request - Compact embedding of intermediate space
Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...
- 679
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173
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Regularity of weak solution
I have also posted the question here. Let me explain what difficulties I have. In fact, one may write
\begin{equation}
\partial_1(f-\partial_1 u)=0
\end{equation}
in $\Omega$. Then one may have the ...
- 223
1
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0
answers
125
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One-sided local $L^p$ spaces
Consider the vector space $L^p_{\text{left-loc}}$ of measurable functions $f:[0,1]\to\mathbb R$ so that for all $x\in(0,1]$ there exists $\delta>0$ so that $f|_{[x-\delta,x]}\in L^p$.
Does this ...
- 8,086
1
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0
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175
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Cuntz comparison of strictly positive elements in finite C*-algebras
Let $A$ be a finite, non-unital C*-algebra, $s\in A$ a strictly positive element and $a\in A$ a positive element that is Cuntz-equivalent to $s$, i.e. there exist sequences $\{x_n\},\{y_n\}\subset A$ ...
- 417
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0
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52
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Condition for maximizer of convex combination to be expansion mapping
I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$
$$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$
such ...
- 43
1
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0
answers
87
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pettis integral
Maybe this is rather a refernce question on Pettis integrals. Some naive questions arise:
1) Assume that $F$ is Pettis-integrable on $\Omega$ and that $\omega \subset \Omega$ is measurable. Is $f$ ...
- 53
1
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0
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207
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Vector valued Sobolev spaces
My question is in reference to this question previously asked here. As asked there, consider a function $f \in H^s_x(L^2_y) \cap L^2_x(H^s_y)$. In the notation of Lions and Magenes (Chapter 4, Vol 2), ...
- 11
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0
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29
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Counting variables to look for invariances/range conditions
A while back, I asked this question on m.se. I wasn't terribly happy with the answer, and when someone asked a very similar question which isn't getting any action, it got me thinking again. Let me ...
- 203
1
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0
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80
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Sobolev embedding on warped product
Consider the warped product $X = M \times \mathbb{R}$, with the metric $g = dr^2 + \varphi(r) g_M$, where $M$ is a compact manifold. Consider the Sobolev space $H^1(X)$ and let $H^1_{rad}(X)$ denote ...
user73978
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78
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Related to derivative of Modified Bessel I function wrt the order
I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that $Re(\dfrac{I'...
1
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0
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182
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Existence of Euler product on critical line for $L(\chi,s) L(\overline{\chi},1-s)$?
Generally there is no Euler product for Dirichlet L-functions $L(\chi,s)$ in the critical strip.(cf Is the Euler product formula always divergent for 0<Re(s)<1?)
But I would like to know if ...
- 1,199
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125
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Transformation of kernel
I have the following problem at hand.
Define the kernel
$$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$
Now, if $R(y_1,...
- 21
1
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0
answers
102
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Pertubations of self-adjoint first order operators
If we consider the self-adjoint operator $L= L_0 + h$ on the appropriate Sobolev spaces of maps from $S^1$ to $\mathbb{R}^N$, say, where $L_0$ is a first order self-adjoint operator and $h$ is a ...
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147
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Energy inequalities for Sobolev spaces of negative integer
I asked this question in mathematics stackexchange and couldn't get an answer.
Let $\phi\in H^{s}$ such that the following energy inequality is true:
$$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\...
- 101
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0
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203
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How generic are Cayley graphs of non-Abelian groups with logarithmic girth?
Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know,
For which $G$ can the ...
- 1,893
1
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0
answers
70
views
Equivalence of two definitions of weak solution (subtlety with null sets)
Consider
$$y_t - \Delta y = f$$
$$y(0) = y_0$$
with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions:
We have $y \in L^...
- 266
1
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0
answers
191
views
$L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value
Let $u \in H^1((0,T)\times S)$ be the unique solution of
$$u_{tt} + \Delta u =0$$
$$u|_{t=0}= u_0$$
$$u|_{t=T}=0$$
where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...
- 21
1
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0
answers
174
views
Estimates of entropy of functional spaces
Let $M^n$ be a compact $n$-dimensional manifold. For $k\geq 0$ let us denote by $C^k(M)$ the Banach space of $k$ times continuously differentiable functions, and $B_{C^k}$ denote the unit ball of it.
...
- 21.8k
1
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201
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Boundedness of a Hilbert space projection map
Reading this recent thread I was reminded of a related problem I still haven't solved so I post it here in hopes of a positive result.
Let $V_0 \subset H_0$ and $V_1 \subset H_1$ be separable Hilbert ...
- 11
1
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0
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129
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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
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0
answers
144
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Dual space of $l^p(\mathbb{Z},X)$
Let $X$ be a Banach space, $p \in [1,\infty)$ and $l^p(\mathbb{Z},X)$ the usual sequence space taking values in $X$. Is it always true that $(l^p(\mathbb{Z},X))^* = l^q(\mathbb{Z},X^*)$ and $(c_0(\...
- 11
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0
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90
views
Da Prato's notion of Symmetric Operator
For anyone who's familiar with G. Da Prato's books on infinite dimensional analysis, I was wondering if someone could clarify something. In, for instance, "An Introduction to Infinite Dimensional ...
- 379
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0
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383
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Topological properties of space of Radon measures
Let $M$ denote the space of signed unbounded Radon measures on $\mathbb{R}$ as is defined by Bourbaki, i.e. $M$ is the dual of $C_c$ where $C_c$ is the space of continuous functions on $\mathbb{R}$ ...
- 1,773
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460
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$H=W$ for weighted Sobolev spaces
Meyers and Serrin's $H=W$ is well known, but how does it generalize when we add weights?
Let's define $H^{m,p}(\mu_0,\dots,\mu_m)$ to be the completion of $C^\infty(\Omega)$ in the norm
$$\|u\|_{m,p,...
- 5,169
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85
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What are good bounds on ratios of subdeterminants?
Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ?
Using the ...
- 1,893
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233
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Sum-epimorphisms and prod-monomorphisms
Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
DEFINITION 1 ...
- 7,500
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0
answers
137
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A continuous choice of invertible elements
Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology.
Is there a continuous map $\alpha:A^...
- 356
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0
answers
116
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Strong solution to parabolic equation without differentiability assumption on coefficient?
Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain
$$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$
$$u|_{\partial\Omega} = 0$$
where $a$ is real-valued and satisfies
$C_1 \leq a(r) \leq C_2$ ...
- 251
1
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0
answers
120
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Fractional Poincare inequality on closed manifold
Let $u \in H^{\frac 12}(M)$ on a compact closed Riemannian manifold. Can someone refer me to a source where the inequality
$$\lVert u - \bar u \rVert_{L^{2^*}} \leq C|u|_{H^{\frac 12}}$$
is proved, or ...
- 11
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0
answers
182
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Solution of non-linear Fredholm (Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity
I have asked this question at MSE but got no response. I have rephrased it so that anyone who knows operator theory and integral equations could help me out... I faced a problem in physics which is a ...
- 11
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0
answers
98
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Topological tensor products of spaces of holomorphic functions of slow growth
Let $X$ be a Banach space, $M$ be a complex manifold, and $\Omega$ a relatively compact domain in $M$. We consider the space $\mathcal{A}^{-\infty}(\Omega, X)$ of $X$-valued holomorphic functions of ...
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0
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133
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Condition for boundedness in stationary phase theorem
I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1.
Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and $j, ...
- 93
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0
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75
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Causal (Volterra type) differential equation with local Lipschitz condition
Consider the equation
$$
u'(t) = (Fu)(t)
$$
where $F \colon L^2(0,T;\mathbb R^n) \to L^2(0,T;\mathbb R^n)$ is a causal (Volterra type)
nonlinear operator. It means that the value of $(Fu)(t_0)$ ...
- 243
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0
answers
102
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How do functions operates in a Fourier algebra $A^{q}(\mathbb T)$?
We put , $A^{q}(\mathbb T)= \{ f\in L^{q}(\mathbb T): \hat{f}\in \ell^{q}(\mathbb Z) \}.$
By Helson-Kahane-Katznelson-Rudin Theorem, it follows that,
"Let $F$ be a function on $\mathbb C$ and if $F(f)...
- 1,051
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0
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202
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Normal points of an operator and discrete eigenvalues
Let $\mathcal{H}$ and $\mathcal{L}({\mathcal{H}})$ denote a separable Hilbert space and the set of bounded linear operators on it respectively.
As a graduate student entering the field of non-...
- 145
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348
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On a property of Riemann Zeta function zeros
Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$
with $s_0$ a zero of the Riemann Zeta function in the critical strip.
This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...
- 1,199
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0
answers
270
views
Eigenvalue of product of self adjoint compact operators
Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
- 157
1
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0
answers
213
views
Harmonic extension of $L^\infty$ function is in $L^\infty$?
Let $u \in H^{\frac 12}(\Omega)$ with $\int_\Omega u = 0$ and consider the solution $v \in H^1(C)$ where $C=\Omega \times (0,\infty)$ of
$$-\Delta v(x,y) = 0$$
$$\partial_\nu v = 0$$
$$v(x,0) = u(x)$$...
- 21
1
vote
0
answers
386
views
Spectrum of convolution operator
This question was asked already on Stack Exchange under https://math.stackexchange.com/q/1114095 . It might be not on a research level, but as it could not be answered on Stack Exchange, I hope for ...
1
vote
0
answers
576
views
Perturbation of Laplacian via Kato-Rellich theorem
Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian
$$-\Delta+V(x)$$
is self-adjoint on $H^2(\mathbb{R}^3)$.
My idea is to use Kato-Rellich theorem; ...
- 11
1
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0
answers
260
views
Generating the sigma algebras on the set of probability measures
I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
- 11
1
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0
answers
206
views
What is the spectrum of $L^1(G:H)$?
Let $H$ be a compact subgroup of a locally compact topological group $G$ and
$$ L^1(G:H)=\{f\in L^1(G): R_h f=f\;\text{ a.e. }\; \forall h \in H\}$$ and $\widehat{(G:H)}=\{\xi\in \hat{G}:\xi|_H=1\}$($\...
- 91
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0
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227
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Showing existence of positive weak solution of a PDE by CoV
Given the following PDE
$$
\begin{cases}
-\Delta u+\alpha=u^q &x\in\Omega\\
u=0 &x\in\partial\Omega
\end{cases}
$$
where $\Omega\subset\mathbb R^3$ is open bounded with smooth boundary, $1<...
- 679
1
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0
answers
58
views
Measurability of eigenelements $s \mapsto (\varphi_k(s), \lambda_k(s))$ of Laplace-Beltrami on $M_s$
For each $s \in [a,b]$, let $M_s$ be a compact Riemannian manifold with no boundary.
Under what conditions on $s \mapsto M_s$ do the eigenvalues and eigenfunctions $(\varphi_k(s), \lambda_k(s))_{k \...
- 11
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0
answers
91
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A reasonable framework to study properties of operator $A \mapsto KAK$ on Banach space
Let $K$ be a continuous linear operator on $C[0,1]$ (more, precisely, it is a linear integral operator). Then $K$ defines a continous linear operator $\widehat K$ on $\mathcal L(C[0,1])$ by the rule
$$...
- 1,329