Questions tagged [geometric-measure-theory]
Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.
6 questions from the last 30 days
6
votes
2
answers
354
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Can a Lipschitz function have derivative 0 on a dense set of small dimension?
Let $f: \mathbb R^n \to \mathbb R$ be Lipschitz continuous. Denote by $Z(f)$ the set on which $f$ is differentiable with derivative $0$.
Suppose $f$ is such that $Z(f)$ is topologically dense.
...
- 9,940
9
votes
1
answer
348
views
Hausdorff dimension of the stretch set of a Lipschitz map
Let $f: \mathbb R^n \to \mathbb R^m$ be a non constant Lipschitz map, for $m < n$. We denote by
$$\text{Lip}(f):= \sup_{x, y \in \mathbb R^n} \frac{|f(x) - f(y)|}{|x - y|}$$
the best Lipschitz ...
- 9,940
2
votes
1
answer
319
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Approximating the perimeter of a domain
Let $M$ be a Riemannian manifold with the volume measure $\mu$, and $\Omega$ be a bounded open subset of $M$. Assume that $\chi_\Omega$ has bounded variation, that is, $\mathrm{Per}(\Omega)<\infty$....
- 479
5
votes
1
answer
299
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Does anyone use measures that take values in real numbers and cardinal numbers?
The counting measure on $\mathbb{R}^n$ is a map that takes a subset $A$ of $\mathbb{R}^n$ and returns its cardinality if it is finite or the symbol $\infty$ if it is infinite.
So, if $A\subseteq\...
- 1,048
4
votes
1
answer
221
views
Hausdorff dimension of graphs of singular functions
Let $f: \mathbb R^n \to \mathbb R^m$ be continuous, and differentiable almost everywhere with $Df = 0$ almost everywhere.
Question: What is the maximal Hausdorff dimension of the graph of $f$?
- 9,940
0
votes
0
answers
77
views
Characteristic function of a domain to have higher order variation
For any $k\in\mathbb{N}$, the space $BV^k(\mathbb{R}^n)$ is defined as the set of all functions $f\in L^1(\mathbb{R}^n)$ such that there exists a Radon masure $D^\alpha f$ and for any $\phi\in C^\...
- 479