All Questions
Tagged with fa.functional-analysis geometric-measure-theory
100 questions
1
vote
1
answer
183
views
Metric currents on singular measures in $\mathbb R^d$
Unless I am misunderstanding a lot of works, it is my understanding that a finite and non negative measure $\mu=g\mathcal{H}^\alpha$, where $\mathcal{H}^\alpha$ is the $\alpha$-Haudorff measure, ...
- 391
5
votes
1
answer
164
views
Does quadratic asymptotic growth imply log-Sobolev inequality?
Let $f : \mathbb{R}^n \rightarrow [0,\infty)$ be a smooth function and consider $h$ s.t $h(\vec{x}) = f(\vec{x}) + \lambda \Vert \vec{x} \Vert^2$.
Does this imply that irrespective of any other ...
- 617
2
votes
0
answers
29
views
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
2
votes
1
answer
128
views
On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
- 1,245
0
votes
0
answers
94
views
When can an affine functional on the dual be represented as an element of a Banach space?
In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
- 297
1
vote
2
answers
115
views
Computation of tangent functional
In Measures Which Agree on Balls by Hoffmann-Jørgenson, the tangent functional is defined as follows.
If $x \in S$, we define the tangent functional $\tau(x,\cdot)$ at $x$ as
\begin{equation}
\...
- 297
1
vote
1
answer
343
views
Gateaux differentiability of the norm in Banach spaces
I'm struggling to understand a particular implication in the proof of Corollary 5 of this paper involving Gateaux differentiability of the norm. The claim is that Gateaux differentiability of the norm ...
- 297
1
vote
1
answer
179
views
Definition and properties of tangent functional
I am reading Measures Which Agree on Balls by Hoffmann-Jørgensen and I am somewhat confused. Here, $E$ is a Banach space, $S$ is the unit sphere, and $x \in S$.
We let $\tau(x, \cdot)$ denote the ...
- 297
4
votes
1
answer
104
views
Generalization of a bounded variation
Let $(X, d)$ be a metric space. We will say that $\gamma \colon [a,b] \to X$ is of bounded variation, if
\begin{equation}
V(\gamma) = \sup_{a=t_0 < \cdots < t_n < b} \sum_{i=1}^n d( \gamma(...
- 820
7
votes
2
answers
434
views
Vector measures as metric currents
Currents in metric spaces were introduced by Ambrosio and Kirchheim in 2000 as a generalization of currents in euclidean spaces. Very roughly, a principle idea is to replace smooth test functions (and ...
- 16.4k
0
votes
0
answers
92
views
Finding set of best approximations from a point in $c_0$ to its subspace
Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best ...
- 85
6
votes
0
answers
271
views
Existence of a limit of alpha-difference quotient for Hölder functions
Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that
\begin{equation}
\sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
- 61
2
votes
0
answers
202
views
Prove or disprove that $u=0$ a.e. on $\Bbb R^d$
Let $\Omega\subset\Bbb R^d$ be an open set. Let $k:\Bbb R^d\to [0,\infty)$ be measurable such that $0\in \operatorname{supp}k$. This implies that $\Omega\subset \Omega_k=\Omega+\operatorname{supp}k$. ...
- 1,101
5
votes
1
answer
170
views
Regular Lagrangian flow for "square root example": $\frac{d}{dt} X(t,x) = \sqrt{X(t,x)}$
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \sqrt{X(t,x)}, &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
This is the prototype of non-uniqueness ...
- 839
2
votes
2
answers
261
views
Distribution of the support function of convex bodies: beyond mean width
Let $K$ be a symmetric convex body in $\mathbb{R}^n$ (that is the unit ball of a norm). Let $h_K$ be its support function, that is $h_K(u) = \sup_{x \in K}\langle x,u \rangle$. The quantity $w(K) = \...
- 245
0
votes
0
answers
43
views
Minimal condition on set for an optimisation problem
We fix $\Omega \subset \mathbb{R}^{2}$ an open set. My question is: what are the minimum conditions we need on $E \subset \Omega$ such that the following optimisation problem:
$$
\sup\{ \int_{E}(\...
- 181
0
votes
1
answer
233
views
When does $C_b(X)$ admit a Schauder Basis?
Let $(X,d)$ be a separable and connected metric space. My question is rather short and to the point: do there exist $\{x_n\}_{n=0}^{\infty}\subseteq X$ such that
$$
\left\{d(x_n,\cdot)-d(x_0,\cdot)\...
- 195
1
vote
1
answer
172
views
A question about pushforward measures and Peano spaces
Specifically my question is the following: Let $P$ be a Peano space. If $(P,\sigma,\mu)$ and $(P,\sigma,\nu)$ are both nonatomic probability measures, does there exist a continuous function $f:P\to P$ ...
- 33
2
votes
0
answers
82
views
Estimate of Wasserstein distance and flow of vector fields under particular assumptions
Let $\mu$ be a compactly supported absolutely continuous probability measure. Let $v,u$ be Lipschitz vector fields. For a vector field $w$ recall that $\Phi_t^w$ denotes its flow.
A classical estimate ...
- 303
6
votes
1
answer
228
views
Set where the speed of convergence is uniform in Lebesgue's density theorem
Let $B \subset \mathbb R^n$ be the unit ball.
Consider a Borel measurable set $E \subset B$ with positive Lebesgue measure $|E|>0$ (say $|E| = |B|/2$).
Then, Lebesgue's density theorem, says that ...
- 393
6
votes
1
answer
268
views
Decomposition of non negative Radon measure into $L^1$ and $H^{-1}$ functions
What is a reference for the following result (which appears to be well-known in measure theory)?
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely continuous ...
- 61
1
vote
0
answers
72
views
Initial-boundary value problem for transport equation with $W^{1,p}$ velocity
Let us consider $v:\mathbb R_+ \times \mathbb R \to \mathbb R_+$ such that $v \in L^1(0,\infty, W^{1,p}(\mathbb R))$ and the transport equation
$$ \begin{cases}
u_t + v(t,x) u_x = 0 \qquad & (...
- 61
27
votes
1
answer
1k
views
The dual of $\mathrm{BV}$
$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
- 683
2
votes
0
answers
186
views
Metric on space of Borel-measurable functions
Let $(X,d_X),(Y,d_Y)$ be metric spaces and $X$ is locally-compact and fix a Borel probability measure $\nu$ on $X$. For any Borel-measurable $f:X\rightarrow Y$, let $\mathcal{K}(f,\delta)$ be the set ...
0
votes
1
answer
236
views
Estimate on total variation of composition of functions
Let $f \in BV(\mathbb R)$ and $g: \mathbb R \to \mathbb R$ be Lipschitz. How can I estimate the total variation of $f\circ g$, that is
$$
\int_{\mathbb R} \left|\frac{d}{dx}f(g(x))\right| dx \ ?
$$
...
- 303
1
vote
0
answers
141
views
On the uniform boundedness principle and the space of functions of bounded variation
Let $U$ be a bounded smooth domain of $\mathbb{R}^d$. We write $m$ for the Lebesgue measure on $U$. A function $f \in L^1(U,m)$ has bounded variation in $U$ if
\begin{align*}
V(f,U):=\sup \left\{\int_{...
- 721
13
votes
2
answers
569
views
A conjecture of De Giorgi on weighted Sobolev spaces
Let $\mu$ be a probability measure on $\mathbb{R}^d$ which is absolutely continuous with respect to the Lebesgue measure with density $\rho$. Assume that, for all $t>0$,
\begin{align*}
\exp \left(...
- 778
3
votes
0
answers
222
views
Sets of finite perimeter: intersection with an half space
I have a question regarding sets of finite perimeter. In particular I'm interested to find
$$\mu_{E \cap H_t}, \label{1}\tag{1}$$
where $E$ is a set of finite perimeter in a generic open set $\Omega \...
- 51
4
votes
0
answers
213
views
Classification of Euclidean-invariant measures?
Is there a classification of measures on $\mathbb R^n$ which are invariant under (Euclidean) isometries? Hausdorff measures of all kinds are examples -- could that be all of them? More precisely,
By ...
- 63.9k
4
votes
1
answer
378
views
Every convex set is of locally finite perimeter
I need to prove that every convex subset of $\mathbb{R}^n$ is of locally finite perimeter.
$E$ is of locally finite perimeter if there exists a vector-valued Radon measure $\mu_E$ s.t. the Gauss ...
- 73
2
votes
0
answers
92
views
First Dirichlet eigenvalue below second Neumann eigenvalue?
Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary.
I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
- 173
1
vote
0
answers
45
views
Decomposition of the space of Radon measures with respect fractional harmonic capacity?
It is well know that there is a generalization of Lebesgue decomposition theorem in the following way:
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
- 371
1
vote
1
answer
273
views
How to prove space of non-negative Radon measures is complete?
Let $\mathcal{M}^{+}(\mathbb{R}_{+})$ be space of non-negative Radon measures on $\mathbb{R}_{+}$ with bounded total variation and define the metric $\rho$ on $\mathcal{M}^{+} (\mathbb{R}_{+})$ as $$ \...
- 133
5
votes
2
answers
514
views
Concrete description of lift in Arens-Eells space
Let $X$ be a compact pointed metric subspace of the $d$-dimensional Euclidean space $(\mathbb{R}^d,d_E)$ and let $AE(X)$ denote its Arens-Eells space. Then a result of Nik Weaver shows that for every ...
- 5,405
2
votes
0
answers
90
views
Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces
In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
- 5,405
4
votes
0
answers
99
views
Fractional Hajłasz-Besov-like similar to the Korevaar-Schoen-Sobolev spaces?
Suppose that $(X,\mu,d)$ and $(Y,\nu,\rho)$ are doubling metric measure spaces. Fix $\alpha>0$ and define the space, analogously to this paper, as the collection of all measurable functions $f:X\...
- 5,405
2
votes
1
answer
258
views
Control the oscillation of a function by its total variation
Is it possible to control the oscillation of a BV vector field $u:\mathbb R^N \to \mathbb R^N$ at a point $x_0$ by the total variation of $u$?
user140746
14
votes
2
answers
2k
views
Is the composition of two nowhere differentiable functions still nowhere differentiable?
Let $f,g:\mathbb R\to\mathbb R$ be two continuous but nowhere differentiable functions. By the Denjoy–Young–Saks theorem for almost every point $x_0\in\mathbb R$ one has
$$
\limsup\limits_{x\to x_0}\...
- 938
2
votes
1
answer
307
views
Box counting dimension of a set and Lipschitz functions
If $f$ is Lipschitz, then the following holds for the Hausdorff dimension:
$$\dim_H f(A) \le \dim_H A.$$
Is the same true for the box counting dimension?
- 839
3
votes
2
answers
322
views
Hausdorff dimension of the graph of the sum of two continuous functions
How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions:
Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
user139844
4
votes
0
answers
115
views
Box counting dimension of the graph of a BV function
Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.
Is the box counting dimension of the graph of $u$ equal to $N$? How can we prove it?
The analogous question for the ...
- 839
2
votes
0
answers
71
views
Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an $s$-dimensional Hausdorff measure restricted to the Koch curve?
Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following:
Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an ...
- 839
2
votes
0
answers
73
views
Alberti rank-one theorem and irregular jump discontinuities
Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (...
- 839
0
votes
1
answer
75
views
Fractal set $E$ such that the indicator function $\mathbf{1}_E$ is BV
Is there a "fractal" set $E \subset \mathbb R^2$ such that the indicator function $\mathbf{1}_E$ is in $BV(\mathbb R^2)$?
- 839
2
votes
2
answers
317
views
Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative
What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative?
More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of
a function
$$...
- 839
2
votes
0
answers
73
views
Projection of BV function
Let $u \in [BV(\mathbb R^N)]^N$. We have
$$D^{jump} u(x) = a(x) \otimes b(x)|D^{jump}u|,$$
where $a,b \in \mathbb S^{N-1}$.
What is the projection of $D^{jump}u$ in the direction $a$? And how can ...
- 839
3
votes
0
answers
73
views
"Almost" absolute continuity of derivative of BV function if ${\rm Tr}\,D_Sf = 0$
Let $f: \mathbb R^N \to \mathbb R^N$ be a $BV$ function.
Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure: $\operatorname{div} f \ll \mathcal L^N$. This ...
- 839
1
vote
0
answers
100
views
Weak estimate for difference quotient of BV function
In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ ...
- 839
5
votes
0
answers
198
views
Heuristic and graphic representation of BV functions and their singularities
This question is about some heuristics and graphs of BV functions.
In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are
the Heaviside function, whose ...
- 839
0
votes
0
answers
63
views
Coarea-like formula for BV functions (not their derivative)
Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that
$$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$
Unfortunately, the formula
$$f = \int_{\mathbb R} \...
- 839