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Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.

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Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,...
pyridoxal_trigeminus's user avatar
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4 votes
1 answer
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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\...
Cosine's user avatar
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4 votes
1 answer
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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 $f$?
Nate River's user avatar
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6 votes
2 answers
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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. ...
Jan Nienhaus's user avatar
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8 votes
2 answers
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Let $f:\mathbb{R}^2\to \mathbb{R}$ be $L$-Lipschitz. Let $f_\varepsilon:=f*\eta_\varepsilon$ be its smooth $\varepsilon$-mollification, where $\eta_\varepsilon(x)=\frac{1}{C\varepsilon^2}\eta(|x|/\...
Saúl RM's user avatar
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8 votes
1 answer
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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 ...
Martin Sleziak's user avatar
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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^\...
Ribhu's user avatar
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2 votes
1 answer
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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, $\mathrm{Per}(\Omega)<\infty$....
mlk's user avatar
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5 votes
2 answers
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In the article "Funzioni BV e tracce" by Anzellotti and Giaquinta (MR555952, Zbl 0432.46031), at page 6 you can read (assume $\Omega \subset \mathbb{R}^n$ open): "The following example ...
No-one's user avatar
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3 votes
1 answer
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Motivation: As a consequence of this question, every function $u$ in $W^{1,2}(\mathbb{R}^n,\mathbb{R})$ has a representative $u^*$ such that there is a null set $E\subset \mathbb{R}^n$ with the ...
Nate River's user avatar
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5 votes
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$\DeclareMathOperator{\graph}{\operatorname{graph}}$ I would like to know if, given $f\in W^{1,2}(\mathbb{R}^n,\mathbb{R})$, it is true that we can always cover $\graph(f)\subset\mathbb{R}^{n+1}$ with ...
No-one's user avatar
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4 votes
2 answers
295 views

Let $u:\mathbb{R}^n\to\mathbb{R}$ be a function in $W^{1,2}$ and let $u^*(x)=\lim_{r\to 0} \frac{1}{\omega_n r^n} \int_{B_r(x)} u(y) dy$ be the fine representative of $u$. From Evans-Gariepy Theorem 4....
No-one's user avatar
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0 votes
1 answer
806 views

At present I work with tools that involves doubling metric space, my definition of DME is: A metric space $X$ is called doubling with constant $N$, where $N \geq 1$ is an integer, if, for each ball $...
Community's user avatar
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4 votes
1 answer
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Suppose that $\Omega_1, \Omega_2 \subseteq \mathbb R^n$ are convex domains. We assume that they contain the origin. Then the radial projection $P : \partial\Omega_1 \rightarrow \partial\Omega_2$ ...
mlk's user avatar
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-2 votes
1 answer
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I have the following question: given $\omega\subset \mathbb{R}^d$ a bounded open set and $\eta\in (0,1)$, can I find an open set $\omega_\eta\subset\subset \omega$ with Lipschitz boundary such that $\...
Ayman Moussa's user avatar
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