All Questions
7 questions
4
votes
2
answers
392
views
Lebesgue differentiation theorem at boundary points for Sobolev traces
$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...
- 5,380
8
votes
2
answers
297
views
Averaging the mass of a Sobolev function $f\in W^{1,p}(\Omega)$ near $\partial\Omega$
Recently, I asked a somewhat related question here. In the comment section, I found the formula
$$
\lim_{r\to 0}\frac{1}{r}\int_{\Omega_r} f(x)\,dx = \int_{\partial \Omega}f(\sigma)\,d\mathcal{H}^{n-1}...
- 1,245
1
vote
0
answers
118
views
Convergence in unbounded domains
Lemma. Let $\mu$ be a measure in $\mathcal{M}(\Omega)$ and let $(v_{n})$ be a sequence of functions in $W^{1,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$ converging to a function $v$ in the weak topology of ...
- 43
2
votes
0
answers
279
views
Relationship between $p$-capacity and Riesz $s$-capacity of a set
What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
- 839
2
votes
1
answer
487
views
Difference quotient for functions of bounded variation
Let $u:\mathbb{R}^N \to \mathbb{R}^N$, $u \in BV(\mathbb{R}^N)$, be a function of bounded variation.
We have that the following holds
$$(\ast) \qquad \frac{1}{|B_r(0)|}\int_{B_r(0)} \frac{|u(x+z)-...
user60665
5
votes
1
answer
248
views
Approximation of monotone Sobolev functions
Let $f\in W_{loc}^{1,2}(\mathbb R^2)$ be a continuous monotone (real valued) function (monotone in the sense that the maximum and minimum of $f$ in a precompact open set are attained at the boundary)....
4
votes
1
answer
239
views
sub and super-levelset regularity for Sobolev functions
I'm wondering if there are known results about the "regularity" (in some sense to be determined) of sub and super levelsets of Sobolev functions $u\in W^{1,p}(\mathbb{R}^d)$. More precisely:
Assume $...
- 5,380