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13 votes
2 answers
569 views

A conjecture of De Giorgi on weighted Sobolev spaces

Let $\mu$ be a probability measure on $\mathbb{R}^d$ which is absolutely continuous with respect to the Lebesgue measure with density $\rho$. Assume that, for all $t>0$, \begin{align*} \exp \left(...
user69642's user avatar
  • 778
9 votes
2 answers
706 views

Measures whose projections are absolutely continuous

Since my question was not answered on MSE, I would like to ask it here. Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt ...
limanac's user avatar
  • 452
5 votes
2 answers
321 views

If the Hausforff dimension of the graph of a function $u$ is $N$ and $\tilde u = u$ a.e. then $\dim_H \mathrm{graph} \, \tilde u = N$ too

Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$. Let $\tilde u = u$ a.e. Is it true ...
Riku's user avatar
  • 839
5 votes
1 answer
499 views

Hausdorff dimension of the graph of a BV function

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function. Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it? Update. In an answer to this post, it ...
Riku's user avatar
  • 839
5 votes
1 answer
220 views

Alberti rank one theorem and a blow-up argument

In this paper, it is written that Alberti’s rank says that the singular part $D^s u$ with respect to $\mathcal L^d$ of the distributional derivative $Du$ of a function $u \in BV_{loc}(\mathbb R^d; \...
user avatar
5 votes
0 answers
198 views

Heuristic and graphic representation of BV functions and their singularities

This question is about some heuristics and graphs of BV functions. In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are the Heaviside function, whose ...
Riku's user avatar
  • 839
4 votes
1 answer
365 views

Lusin Lipschitz approximation in BV and Sobolev space

Theorem 5.34 in Functions of bounded variation by L. Ambrosio, N. Fusco and D. Pallara states that Let $u \in [BV(\mathbb{R}^N)]^m$. Then there exists a constant $\kappa>0$ such that for every $...
Riku's user avatar
  • 839
4 votes
0 answers
115 views

Box counting dimension of the graph of a BV function

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function. Is the box counting dimension of the graph of $u$ equal to $N$? How can we prove it? The analogous question for the ...
Riku's user avatar
  • 839
3 votes
2 answers
322 views

Hausdorff dimension of the graph of the sum of two continuous functions

How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions: Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
user avatar
2 votes
1 answer
307 views

Box counting dimension of a set and Lipschitz functions

If $f$ is Lipschitz, then the following holds for the Hausdorff dimension: $$\dim_H f(A) \le \dim_H A.$$ Is the same true for the box counting dimension?
Riku's user avatar
  • 839
2 votes
2 answers
317 views

Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative? More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of a function $$...
Riku's user avatar
  • 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 $\epsilon \to 0$...
Riku's user avatar
  • 839
2 votes
0 answers
165 views

Jacobian and Jacobian matrix of solutions of ODE with Sobolev vector field

Let $\Phi$ be the Lagrangian flow (defined as in page 6 of this paper) of the ODE $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{...
user avatar
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 ...
Riku's user avatar
  • 839
1 vote
1 answer
178 views

Growth assumption and example of finite (arbitrarily small) time blow up for ODE

Consider the following ODE initial value problem \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\ &\Phi(0,x) = x, & x \in \...
Riku's user avatar
  • 839
1 vote
0 answers
100 views

Weak estimate for difference quotient of BV function

In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ ...
Riku's user avatar
  • 839
1 vote
0 answers
47 views

Consistency of the definition of total variation for functions of one or several variables

Where can I find a proof that the definition of total variation for functions of several variables is consistent with the definition of total variation for functions of one variable?
Riku's user avatar
  • 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 ...
Riku's user avatar
  • 839
0 votes
0 answers
63 views

Coarea-like formula for BV functions (not their derivative)

Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that $$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$ Unfortunately, the formula $$f = \int_{\mathbb R} \...
Riku's user avatar
  • 839