All Questions
Tagged with integration geometric-measure-theory
19 questions
6
votes
1
answer
400
views
Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.
Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
- 831
-1
votes
1
answer
128
views
Riesz energy for open sets in dimension $1$
This is a continuation of the question Calculation of Riesz energy for balls . As there are three questions,;I am posting a new question here. Riesz energy for a ball $B(x_0,r)$ is given by
$$I_s(B(...
- 87
3
votes
2
answers
352
views
Calculation of Riesz energy for balls
I was reading stuffs about Riesz energy which is defined for an open subset $U\in\mathbb{R}^d$ by $I_s(U)=\int_U\int_U|x-y|^{-s}\ dx\ dy$ where $dx$ and $dy$ are Lebesgue measure in $U$. Now if I take ...
- 87
8
votes
1
answer
2k
views
How badly can the Lebesgue differentiation theorem fail?
Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is integrable. Is it true that
$$
\lim_{r\to 0}\frac{\displaystyle\int_{B_r(0)}f(y)~\mathrm dy}{r^{n-1}}=0 \quad ?
$$
This is obvious if $0$ is a Lebesgue point ...
- 1,149
0
votes
1
answer
540
views
Interchange of integration and supremum
Let $u \in C^0(-T,T; L^2(B_R))$ be a measurable function, then is the following true?
$$
\int_0^R \sup_{-T<t<T} \int_{S_r} |u(\sigma ,t)|^2 \ d \sigma \ dr = \sup_{-T<t<T}\int_0^R \int_{...
- 455
6
votes
3
answers
700
views
How to estimate the integral involving the distance function
Let $\Omega\subset\mathbb{R}^n$ be an open bounded domain with smooth boundary. Consider the following integral:
$$I(t)=\int_{\Omega}e^{-\frac{d^2(y,\partial\Omega)}{t}}{\rm d}y.$$
My problem is how ...
- 561
6
votes
1
answer
1k
views
Fubini's theorem on arbitrary foliations
In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...
- 622
2
votes
3
answers
803
views
A Curved/Warped Version of Fubini's Theorem
I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...
- 622
7
votes
1
answer
625
views
Possible application of divergence Theorem?
suppose that $f \in C^1 (\mathbb{R}^{N+1},\mathbb{R})$. It's well known that if all his points are regular points i.e.
$$\nabla f (x) \neq 0 \; \; \; \forall x \in \mathbb{R}^{N+1}$$
then, for every ...
- 51
7
votes
1
answer
805
views
Can I cover a compact set by balls {B} such that {2B} has bounded overlap?
Suppose I have a compact set $K \subset B_1(0) \subset \mathbb{R}^n$. Can I always find a family of open balls $\{B_{r_j}(x_j)\}$ such that
$x_j \in K$ and $B_{r_j}(x_j) \subset B_1(0)$ for each $j$;
...
- 1,179
2
votes
0
answers
248
views
Intrinsic volume - is there a simplified formula?
I'm struggling to understand how can I compute the instrinsic volumes (or Minkowski functionals) of a submanifold $\Omega$ of $\mathbb{R}^N$. I found a formula, but I really can't understand it...
It ...
- 1,759
1
vote
0
answers
145
views
How to show that this function is continuous (Geometric Measure Theory)
I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by
$$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$
is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...
- 19
2
votes
0
answers
263
views
Is a maximal set of rectangles known for which Lebesgue’s Differentiation Theorem holds true?
Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
- 4,597
4
votes
0
answers
136
views
Integration on a family of differential forms
Let $X$ be a smooth manifold, and denote by $\Omega^*(X)$ the set of all smooth differential forms on $X$. Assume we have a family of differential forms $\omega_t \in \Omega^*(X)$, $t\in E$, ...
- 2,789
1
vote
0
answers
263
views
Does a growing manifold fixed at a point converge to its tangent plane?
Let $M$ be a smooth compact $(n-1)$ dimensional submanifold in $\mathbb{R}^n$. Let $H$ be the $(n-1)$ dimensional Hausdorff measure. Let $f(x,y,t)$ is a function for $x\in\mathbb{R}^n$, $y\in\mathbb{R}...
- 65
3
votes
1
answer
938
views
Stokes theorem for manifolds with boundary as disjoint union of submanifolds
Looking at the generalizations of Stokes theorem, I did find a version for manifold with corners, but I was surprised this generalization doesn't contain a simple example such as the cone. So my ...
- 549
2
votes
0
answers
79
views
Compute Mixed Volume with Respect to Some Regular Sets
Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
- 1,127
4
votes
1
answer
618
views
Riemannian Measures, Densities and Radon–Nikodym Theorem
If $M$ is a smooth manifold and $\mu$ is a $1$-density thereon then we may define a Borel measure (on Borel sets $A$) on $M$ as:
\begin{equation}
\nu(A) = \int_M I_A \mu.
\end{equation}
My question ...
- 5,405
32
votes
3
answers
3k
views
What are the obstructions for a Henstock-Kurzweil integral in more than one dimension?
I have recently come across the book The Kurzweil-Henstock Integral and its Differentials by Solomon Leader, in which, if I understand correctly, the HK integration process is modified in a way that ...
- 573