All Questions
12 questions with no upvoted or accepted answers
3
votes
0
answers
73
views
"Almost" absolute continuity of derivative of BV function if ${\rm Tr}\,D_Sf = 0$
Let $f: \mathbb R^N \to \mathbb R^N$ be a $BV$ function.
Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure: $\operatorname{div} f \ll \mathcal L^N$. This ...
- 839
3
votes
0
answers
860
views
decreasing rearrangements: why the asymmetry of measure-preserving maps?
Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
- 16.6k
2
votes
0
answers
162
views
$\int_{\mathbb{R}^{N}\setminus\Omega}\vert x-z\vert^{-N-\alpha} dz = c \ \forall x\in\partial U$ implies $dist(x,\partial\Omega)=c, x \in \partial U$?
Let $\alpha \in \mathbb R_+$, $\Omega \subset \mathbb R^N$ and $U \subset \Omega$. Is it true that if
$$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-\alpha} dz = \text{constant} \quad \text{for all ...
- 61
2
votes
0
answers
100
views
What is the weak limit of $f_n \ \mathrm{sign}(f_n - 1)$ if $f_n \to f$ weakly in $L^p([0,1])$?
Let $f_n: [0,1] \to \mathbb R$ be a uniformly bounded sequence in $L^p$. Then there exists a subsequence such that $f_{n_k} \to f$ weakly in $L^p([0,1])$. What is the weak limit of the sequence of ...
- 217
2
votes
0
answers
71
views
Example of BV vector field $c$ without bounded divergence such that $u$ is bounded where $u_t + div(cu) = 0$
What is an example of vector field $c: \mathbb R_+ \times \mathbb R^N \to \mathbb R^N$ with $c \in L^1(\mathbb{R}_+, BV(\mathbb R^N))$ without bounded divergence $div_x c$ but such that there exists a ...
- 839
2
votes
0
answers
73
views
Projection of BV function
Let $u \in [BV(\mathbb R^N)]^N$. We have
$$D^{jump} u(x) = a(x) \otimes b(x)|D^{jump}u|,$$
where $a,b \in \mathbb S^{N-1}$.
What is the projection of $D^{jump}u$ in the direction $a$? And how can ...
- 839
2
votes
0
answers
199
views
Convergence of the difference quotient of a BV function
Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
What can be said about the difference quotient
$$
\frac{u(x+\epsilon y)-u(x)}{\epsilon}
$$
regarding its convergence as $\epsilon \to 0$...
- 839
1
vote
0
answers
107
views
Level sets of a BV function and its derivative
Given $u \in BV(\Omega; \mathbb{R}^M)$, where $\Omega \subset \mathbb{R}^N$, what is the relationship between its level sets and its distributional derivative $Db$?
More specifically, does Alberti ...
- 839
1
vote
0
answers
283
views
Density of Dolean exponentials in L2 and Wiener Measure
Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...
- 19
0
votes
0
answers
57
views
Projection measure and an integral formula for Lipschitz functions
Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as
$$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
- 75
0
votes
0
answers
94
views
When can an affine functional on the dual be represented as an element of a Banach space?
In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
- 297
0
votes
1
answer
440
views
Variation on Fatou's lemma for Sobolev norms
Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions
$$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$
If I am not ...
- 1