Skip to main content

All Questions

Filter by
Sorted by
Tagged with
6 votes
1 answer
425 views

Lipschitz property of the symmetric rearrangement

I'm currently reading Talenti's paper "Best constant in Sobolev inequality" and am rather stuck on an argument on pg 363 (or pg 11 if you're reading the pdf). In this section of the paper, ...
Marc's user avatar
  • 457
5 votes
0 answers
140 views

Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$

Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where \begin{align}\label{eq:bounded-...
Guy Fsone's user avatar
  • 1,101
5 votes
1 answer
212 views

Can we perturb the Dirichlet boundary conditions to make harmonic maps locally invertible?

While analyzing a variational problem, I came to the following question: Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $f: \mathbb D^n \to \mathbb{R}^n$ ...
Asaf Shachar's user avatar
  • 6,741
4 votes
1 answer
266 views

Prove $\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx dist(x,\partial \Omega)^{-s}$, $s \in (0,2)$

Let $\Omega \subset \mathbb R^N$ and $s \in (0,2)$. Under what assumptions on $\partial \Omega$ do we have $$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx \mathrm{dist}(x,\partial \...
Jun's user avatar
  • 303
4 votes
1 answer
597 views

Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$? Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
user avatar
4 votes
1 answer
241 views

Is a function $u\in \mathrm{SBV}(\Omega)$ with these additional properties essentially bounded?

Some related earlier discussion can be found here. Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary, $\mathcal H^{N-1}(\partial\Omega)<\infty$ and $u\in SBV(\Omega)$. Then $$ ...
JumpJump's user avatar
  • 679
4 votes
0 answers
151 views

Estimating the size of $\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}$

Let $\Omega$ be a bounded domain in $\Bbb R^n$. Define $$ \Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}, $$ i.e. it the ring of thickness $r$ at the boundary of $\Omega$. Intuitively, ...
BigbearZzz's user avatar
  • 1,245
3 votes
0 answers
219 views

Strictly contracting solutions to the Eikonal equation on Riemannian manifolds

Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$. Question: Does there exist, on every complete ...
Nate River's user avatar
  • 6,213
2 votes
1 answer
128 views

On the existence of a complicated fractal-like set of finite perimeter

Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
BigbearZzz's user avatar
  • 1,245
1 vote
1 answer
188 views

"Schwarz symmetrization" on annulus

If $\Omega=\{x\in \mathbb R^n| 0<r_0<|x|<r_1\}$ is an annulus on $\mathbb R^n$, I am looking for a symmetrization result on $\Omega$. To be precise, for any $u \in W_0^{1,2}(\Omega)$, can we ...
Adterram's user avatar
  • 1,441
1 vote
0 answers
45 views

Decomposition of the space of Radon measures with respect fractional harmonic capacity?

It is well know that there is a generalization of Lebesgue decomposition theorem in the following way: Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
Hheepp's user avatar
  • 371
1 vote
0 answers
92 views

Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case

By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user avatar