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14 votes
3 answers
1k views

Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?

Recall that we say that a bounded measurable set $S\subset\mathbb R^n$ is said to be Caccioppoli if the indicator function $1_S$ is BV, and we set $$ \operatorname{perim}(S)=\| \nabla 1_S\|_{TV} $$ ...
Dominic Wynter's user avatar
11 votes
1 answer
451 views

Does every smooth map of rank at most d factor through a d-manifold?

Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map whose rank at any point of $\R^m$ is at most $d$. Here and below, smooth means infinitely differentiable. Can we ...
Dmitri Pavlov's user avatar
10 votes
1 answer
872 views

Current vs Varifold

I know the basic definitions concerning current and varifold, and they are generalization of submanifolds. What are their respective pros and cons? What are their crucial similarities and differences?
JSCB's user avatar
  • 1,630
6 votes
1 answer
802 views

Approximation of a Sobolev function that has vanishing trace on the reduced boundary of a Caccioppoli (i.e. finite perimeter) set

For $\Omega\subset\mathbb{R}^N$ open and bounded, let $W^{1,p}(\Omega)$ denote the usual Sobolev space of $L^p(\Omega)$ functions with weak partial derivatives in $L^p(\Omega)$ and $W_0^{1,p}(\Omega)$ ...
Elgrimm's user avatar
  • 143
5 votes
0 answers
273 views

Is there any geometrical/homological intuition behind symmetrized gradient?

The gradient/differential/exterior differential/divergence/curl are all strictly related first order differential operators. As far as I understood, they are the base of (co)homological theories in ...
Romeo's user avatar
  • 980
5 votes
0 answers
240 views

The boundary integral of a harmonic function

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary and $f$ be a harmonic function on $\Omega.$ It is known that $$ \limsup_{\varepsilon\rightarrow0^{+}}\intop_{\partial\Omega_{...
Han Ju's user avatar
  • 53
2 votes
2 answers
643 views

Estimating the Hausdorff measure of a subset of the sphere

Let $f: S^{n-1}\to \mathbb{R}$ be a continuous function ($S^{n-1}\subset \mathbb{R}^n$ is the unit sphere), $f(a)>0$ and $f(b)<0$ for certain points $a,b\in S^{n-1}$. By continuity these ...
Florian's user avatar
  • 2,270
2 votes
1 answer
741 views

Continuity of the perimeter of level sets w.r.t. level function

Working with the level set method introduced by Osher & Sethian in shape optimization I came across a simple question that I did not succeed to prove. It mainly asserts that the perimeter of the ...
Bogdan's user avatar
  • 1,759
1 vote
1 answer
146 views

Is a locally invertible weak limit of injective maps injective almost everywhere?

This is a cross-post. Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries. Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be Lipschitz injective maps ...
Asaf Shachar's user avatar
  • 6,741
1 vote
1 answer
160 views

Differentiability of an integral of geodesic distance

Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$. Q1: Define $$ g(t)=\...
Hengchao Chen's user avatar