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Limit of minimizers of a class of functionals

Assume that $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n $. Consider a functional $$ \mathcal{F}(u)=\int_\Omega(|\nabla u|^2+h^{-1}|u-u_0|^2) \, dx $$ where $ h>0 $ is a parameter and $ ...
Luis Yanka Annalisc's user avatar
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1 answer
96 views

A question about filterbasis

K. Hardy and R. G. Wood assert that the family in line 4 is a filterbase. I couldn't show it.
Mehmet Onat's user avatar
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0 votes
1 answer
135 views

When integrating by part produces a singularity

I'm currently interesting in the following operator: $$O[f](x):=f(x)-2xe^{x^2}\int_x^{+\infty}dt \, e^{-t^2} f(t)$$ for all $x\in\mathbb{R}$ and $f$ smooth and decaying at infinity fast enough with ...
BlueCharlie's user avatar
0 votes
1 answer
154 views

Finite dimensionality of a subspace

Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds: $$ \...
Ali's user avatar
  • 4,115
0 votes
1 answer
247 views

Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e

The Riemann-Liouville integral is defined by $$ I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t $$ where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
Grandes Jorasses's user avatar
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1 answer
120 views

A property of the canonical dual frame in a Hilbert space

Let $\{ g_n \} $ be a frame in a separable Hilbert space $H$. Then the frame operator $S:H\to H$ defined as \begin{equation} S f := \sum_{n=1}^\infty (f,g_n)g_n \end{equation} is a Hilbert space ...
an_ordinary_mathematician's user avatar
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1 answer
281 views

Roots of linear combination of $x \sin x$

Let $\theta=(\theta_1,\theta_2,\cdots \theta_n)$, and $a_{ij}$ are constants. There is no condition on the positiveness of $a_{ij}$. Under which condition on $\theta$, such that the following function ...
tony's user avatar
  • 405
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1 answer
143 views

Differential form of the multidimensional "orthogonal dilation" operator

For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion. ...
Kanghun Kim's user avatar
0 votes
1 answer
188 views

A question about uniformities generated by pseudometrics

Suppose that for all $n$ natural numbers, $d_{n}$ is a pseudometric on set $X $. Define $d=\sum_{n=1}^{\infty }a_{n}\frac{d_{n}}{1+d_{n}}$, where $\left( a_{n}\right) $ is a sequence of positive ...
Mehmet Onat's user avatar
  • 1,367
0 votes
1 answer
93 views

Property stronger than $T_1$ and weaker than regularity

Recently I got interested in the following property of topological spaces: $(X,\mathcal{T})$ satisfies (P) if the following holds: For any nonempty closed subsets $F$ and $G$ with $F\ne G$, there are ...
Ig.topolg's user avatar
0 votes
2 answers
101 views

Minimal set of functions to characterize a distribution

In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:...
stats_model's user avatar
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1 answer
75 views

The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$

Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix. $$...
ABB's user avatar
  • 4,058
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1 answer
170 views

When some Fourier coefficients are fixed, can we control the extremals of the function?

Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f :\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold? 1- $-n\lneqq f_{\min}$ (where $f_{\...
ABB's user avatar
  • 4,058
0 votes
1 answer
294 views

Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?

I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. THEOREM 1. Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\infty$, and $X$ be a Banach ...
Akira's user avatar
  • 825
0 votes
1 answer
328 views

Deduce that a function is zero on interval $[0,M]$

I have been thinking about this for the last few days but I was not able to produce a definitive answer. Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (...
Grandes Jorasses's user avatar
0 votes
1 answer
291 views

Tensor product is complete?

Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be Banach spaces and let the norm $\|\cdot\|_{V\otimes W}$ on the tensor product space $V\otimes W$ be admissible in the following sense: for $v\in V, w\in ...
Martin Geller's user avatar
0 votes
1 answer
243 views

Condition for set of the type $\{(a,b)|a \in A, \ b = f(a)\}$ to have empty interior if $A$ has empty interior [closed]

Let us consider $$\mathcal X = \{(a,b)|a \in A, \ b = f(a)\}, $$ where $A \subset L^1(\mathbb R)$ has empty interior and $f:L^1 \to L^1$ is a bijective map. Does $\mathcal X$ also have empty interior? ...
Riku's user avatar
  • 839
0 votes
1 answer
130 views

Riesz transform after linear transformation

I am encountering the term $\partial_x \mathcal{R}_x(f(x,y))$. I needed to do the following linear transformation $$x' = a x+ by,\,\,\,\,\, y'=ax-by,\,\,\, and \,\,f(x,y)=g(x',y') $$ I ended up with ...
Mr. Proof's user avatar
  • 159
0 votes
1 answer
166 views

Relationship between elliptic and parabolic problems and their discretizations

Let us consider the fully nonlinear problem $$ \begin{cases} F(x,u,Du,D^2 u) = 0 & \text{ in } \Omega \\ u=0 & \text{ in } \partial \Omega \end{cases} $$ Suppose that we know that the ...
user485442's user avatar
0 votes
1 answer
126 views

Bounding integral expression with Sobolev norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
user avatar
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1 answer
101 views

"Project" an operator outside of a von Neumann Algebra into it

Suppose $W$ is a proper von Neumann Algebra contained in $B(H)$ and the identity in $W$ is the identity mapping of $H$ (namely, $W$ does not have non-trivial null space). Given a self-adjoint $T\in W$...
Sanae Kochiya's user avatar
0 votes
1 answer
83 views

Do the weakly null sequences in a Banach module factor?

Let $A$ be a Banach algebra with a bounded approximate identity, and let $E$ be a Banach left $A$-module. Suppose neither $A$ nor $E$ has the Schur property. Question: Given a weakly null sequence $(...
Onur Oktay's user avatar
  • 2,605
0 votes
1 answer
93 views

References and updates on a $L^p$ Factorization theorem by Maurey

In the "Exposé XV" of the 1972-1973 of the Maurey-Schwartz Seminar of functional analysis ("Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans un espace $L^p(\Omega,...
Gamabunto's user avatar
0 votes
1 answer
159 views

Partial orders on downward closed sets [closed]

Let $P = (V, \sqsubseteq)$ be a partial order and $\mathfrak{D}(P)$ denote the class of downward-closed subsets of the partial order $P$ (i.e, the class of $A \subseteq V$ such that $y\in A \;\&\; ...
user65526's user avatar
  • 639
0 votes
1 answer
43 views

Exhaustions of product subsets by smaller product subsets

Let $X$ be a compact metric space, $A,B\subset X$ be subsets and $f\colon X\times X\to \mathbb{R}$ a continuous function that is strictly positive on $A\times B$. Do there exist increasing sequences ...
Federico Vigolo's user avatar
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1 answer
233 views

Borel sigma algebra coming from the weak topology on TVS

Let $(X,\tau)$ be a topological vector space. Suppose that, there is a sequence of subsets $X_n\subseteq X$ with, For every $n\in \mathbb{N}$, the topology $\tau$ on $X_n$ is second countable and ...
ABB's user avatar
  • 4,058
0 votes
1 answer
236 views

Is this a contraction mapping for small $T$?

Let $G$ be the heat kernal, i.e. for $0\le t<s$ and $x,y\in\mathbb R$ $$G(t,x;s,y):=\frac{1}{\sqrt{4\pi(s-t)}}\exp\left(-\frac{(y-x)^2}{4(s-t)}\right).$$ For $T>0$, let $\mathcal H_T:=\{h:[0,T]\...
GJC20's user avatar
  • 1,334
0 votes
1 answer
79 views

The operator of exponential derivative applied in quotients

I have an other question for a function different to the example given before in the link below: Exponential derivative operator and continuous functions We define for instance a function as: $$H(y)=\...
Adam Hammam's user avatar
0 votes
1 answer
254 views

Continuity of Kakutani fixed points

Let $X$ be a compact and convex space and let $T=[0,1]$ be some parameter space. Let $F:X\times T\rightrightarrows X$ be a correspondence that is compact-valued, convex, and upper-hemicontinous. By ...
tsm's user avatar
  • 229
0 votes
1 answer
158 views

Abelian twisted reduced group C*-algebra

Let $G$ be an abelian discrete group. Then is $C_r^*(G, \sigma)$ abelian?
Peg Leg Jonathan's user avatar
0 votes
1 answer
417 views

Application of Green function for non linear PDE [closed]

In the case of linear PDE, say $$Lu=0$$ if we have its green function say $G(x,y)$ then using that one can give solution of non homogenous PDE i.e. $Lu_f=f$ where $u_f=G*f$. Is the same thing hold for ...
Curious student's user avatar
0 votes
1 answer
163 views

Regarding socle of a C* algebra

I wanted to know if the socle of a complex C*-algebra is essential? Can anyone suggest a text where the socle is studied in detail. I tried reading it from the book by Bernard Aupetit, A Primer in ...
user531706's user avatar
0 votes
1 answer
310 views

Regularity properties of conditional distributions

Let $(X,Y)\in\mathbb{R}^n\times\mathbb{R}^m$ be a pair of random variables with joint density $p(x,y)$. I am interested in the regularity properties of the conditional densities $p(y|x)$ and $p(x|y)$ (...
user19200's user avatar
0 votes
1 answer
248 views

Approximating arbitrary probability measures by discrete ones

Let $H$ be a separable Hilbert space and let $\mu$ be an arbitrary probability measure on $H$. I would like to approximate this distribution by by a finitely supported discrete distribution is the ...
TOM's user avatar
  • 2,288
0 votes
1 answer
98 views

Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?

Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$ where the coefficient $a$ are smooth and bounded and $D$ is a bounded and smooth domain of $\mathbb R^d$ $$ \begin{...
Kernel's user avatar
  • 446
0 votes
1 answer
147 views

The relationship between the first eigenfuntions and the second eigenfuntions on sphere [closed]

Recently I considered the following question: If we give a second eigenfuntions $g$ on sphere, then can we construct a first eigenfuntions $f$ by $g$? Is there any relationship between the first ...
管山林's user avatar
0 votes
1 answer
116 views

Fractional Laplacian and support

Let $u:\mathbb [-1,1] \to \mathbb R$ such that $\mathrm{supp}(u) \subset B_{1/2}(0)$. Under what assumptions on $u$ does it hold $$\mathrm{supp}\Big((-\Delta)^s u\Big) \subset B_{1/2}(0),$$ where $(-\...
Lao's user avatar
  • 217
0 votes
1 answer
76 views

Reverse Hölder type inequality for the Laplacian raised to a power

I am studying integrals of the form $\int (\Delta \rho)^{\alpha} f^{\beta}$ where $0<\alpha < 1, \beta \geq 0$ and $\rho, f \in C_c^{\infty}(\mathbb{R}^n).$ My goal is to obtain lower bound on ...
Student's user avatar
  • 537
0 votes
1 answer
971 views

Is the pointwise supremum of a continuous function continuous?

Suppose $f(x , y)$ is continuous in both variables. For any $\epsilon > 0$ and some $y_0$, let $h_{\epsilon}(x) = \max_{y^{'}: \| y^{'} - y_0 \| \leq \epsilon} f(x , y^{'})$. It seems to me that $...
Saeed's user avatar
  • 3
0 votes
2 answers
238 views

Fractional Laplacian of $(a-x)_+^\alpha$ in $(0,1)$

How can I compute the spectral fractional Laplacian of $(a-x)_+^\alpha$ on $\Omega = (0,1)$? Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
Jay's user avatar
  • 109
0 votes
1 answer
225 views

Convergence in compact-open topology on the Sierpiński space

Question: Equip $\{0,1\}$ with the Sierpiński topology $\{\{1\},\{0,1\},\emptyset\}$, let $X$ be a compact metric space, and equip $C(X,\{0,1\})$ with the compact-open topology. Let $\{B_n\}_{n=1}^{\...
ABIM's user avatar
  • 5,405
0 votes
1 answer
120 views

Density property fractional heat kernel

Let us consider $$p_t^{(n+2)}(\tilde x) , $$ where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (x_1, \ldots, x_n, 0, 0) \in \mathbb R^{n+2}$, and $p_t^{(n)}(x)$ is the heat kernel for $(-\...
Jay's user avatar
  • 109
0 votes
1 answer
236 views

Estimate on total variation of composition of functions

Let $f \in BV(\mathbb R)$ and $g: \mathbb R \to \mathbb R$ be Lipschitz. How can I estimate the total variation of $f\circ g$, that is $$ \int_{\mathbb R} \left|\frac{d}{dx}f(g(x))\right| dx \ ? $$ ...
Jun's user avatar
  • 303
0 votes
1 answer
86 views

Lattice of functions and their minimal separating set upto topological equivalence

There is a very wide series of questions I have been thinking about and I am wondering if there is any literature on this type of structures. Let's start with the set of all functions $F: \mathbb{R} \...
Heraiwa's user avatar
  • 39
0 votes
1 answer
129 views

what is the geometry interpretation of fractional differintegral?

By some analogy, the integral and differential can be extend to factorial differintegral, see https://en.wikipedia.org/wiki/Differintegral My question is, what is the he geometry interpretation of ...
XL _At_Here_There's user avatar
0 votes
1 answer
74 views

What is the source to find cardinal invariants for a function space C(X, Y), equipped with uniform or fine topology?

I would like to know about the technique to check the cardinality properties for the function space C(X, Y), where X is a tychonoff space and Y a metric space, equipped with uniform or fine topology.
Mir Aaliya's user avatar
0 votes
1 answer
373 views

Proof of the analytic Fredholm theorem in Borthwick

I've stumbled across a proof of the analytic Fredholm theorem given in Theorem 6.1 in Spectral Theory of Infinite-Area Hyperbolic Surfaces by David Borthwick (see below). Given the notion of being &...
0xbadf00d's user avatar
  • 167
0 votes
1 answer
89 views

Does $\int_0^t \Vert u_x(s,\cdot) \Vert_{L^2} ds \le C$ imply $\Vert u_x (t,\cdot) \Vert_{L^2(\mathbb R)} \le C$ in the heat equation?

For the parabolic equation $$u_t + f(u)_x - u_{xx} = 0$$ one has $$\Vert u(t,\cdot) \Vert_{L^2(\mathbb R)} + 2\int_0^t \Vert u_x(s,\cdot) \Vert_{L^2} ds \le \Vert u(0,\cdot) \Vert_{L^2(\mathbb R)}.$$...
user avatar
0 votes
1 answer
142 views

Some multivariate Taylor series and corresponding smoothness balls

Suppose I have a multivariate function $f$ from $\mathbb{C}^d$ to $\mathbb{C}$ that accepts a Taylor expension of the form $$f(\mathbf x) = \sum\limits_{\mathbf k \in \mathbb N^d} a_{\mathbf k} \...
lrnv's user avatar
  • 686
0 votes
1 answer
106 views

Existence of uniform approximator that also approximates derivative

Let $S$ be a subset of $C^1([0, 1], \mathbb{R})$. It is a well-known fact that given a function $f\in C^1([0, 1], \mathbb{R})$ and a sequence $\{f_n\}\subset C^1([0,1], \mathbb{R})$ such that $f_n\to ...
potionowner's user avatar

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