All Questions
Tagged with geometric-measure-theory soft-question
13 questions
0
votes
1
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189
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Visualization of the disintegration theorem [closed]
Where can I find a picture that gives a visualization of the disintegration theorem?
If such reference does not exist, what would a nice visualization of this fundamental result look like?
- 109
1
vote
1
answer
140
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Meaning of "quantitative result" [closed]
Recently I've begun reading on metric measure spaces and I keep seeing statements containing the phrase ", quantitatively". What does this mean, I googled it and couldn't find a rigorous answer.
- 5,405
2
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0
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71
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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
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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
2
votes
2
answers
317
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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
3
votes
0
answers
73
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"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
5
votes
0
answers
198
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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
3
votes
0
answers
141
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Partially BV vector fields and renormalization
Why does the approach used to prove Theorem 4.1 in the paper by Le Bris and Lions on Renormalized solutions of some transport equations with partially $W^{1,1}$ velocities and applications not work ...
user123672
4
votes
1
answer
365
views
Lusin Lipschitz approximation in BV and Sobolev space
Theorem 5.34 in Functions of bounded variation by L. Ambrosio, N. Fusco and D. Pallara states that
Let $u \in [BV(\mathbb{R}^N)]^m$. Then there exists a constant $\kappa>0$ such that for every $...
- 839
5
votes
1
answer
220
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Alberti rank one theorem and a blow-up argument
In this paper, it is written that Alberti’s rank
says that the singular part $D^s u$ with respect to $\mathcal L^d$ of the distributional derivative $Du$ of a function $u \in BV_{loc}(\mathbb R^d; \...
user123457
6
votes
0
answers
171
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The distributional gradient of the closest isometry to the differential of a smooth map
The setting-a "linear algebra" fact:
Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \...
- 6,741
14
votes
2
answers
1k
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Category theory & geometric measure theory?
My background is essentially Geometric Measure Theory and its application to partial differential equations (e.g. linear and non-linear hyperbolic conservation laws). These are currently my research ...
- 980
1
vote
0
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259
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Is there a precise relationship between ``Geometric Functional Analysis" and high-dimensional probability/information theory?
The 2009 course on GFA by Roman Vershynin (https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf) introduced the subject with this line on the course page, "...
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