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Questions tagged [bounded-variation]

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Construction of a "Dirac" jump process

We work on a probability space $(\Omega,\mathscr{F},\mathbb{P})$ endowed with a filtration $\mathbb{F}$, and consider the positive line $[0,\infty)$. I am wondering if one can make sense of the ...
Daneel Olivaw's user avatar
1 vote
0 answers
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How can we calculate the Euler-lagrange equations?

In this paper https://arxiv.org/pdf/1907.09605.pdf \ let $\Omega \subset \mathbb{R}^n$ with $n \geq 1$ be a bounded Lipschitz domain with boundary $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
Mohamed's user avatar
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5 votes
1 answer
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Arens regularity of $\mathrm{BV}(\mathbb{R})$

$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of ...
Tobias Fritz's user avatar
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4 votes
1 answer
98 views

Generalization of a bounded variation

Let $(X, d)$ be a metric space. We will say that $\gamma \colon [a,b] \to X$ is of bounded variation, if \begin{equation} V(\gamma) = \sup_{a=t_0 < \cdots < t_n < b} \sum_{i=1}^n d( \gamma(...
Kacper Kurowski's user avatar
2 votes
1 answer
165 views

Status of the Bressan conjecture

Let me first recall what is the Bressan conjecture. Take a $BV\cap L^\infty$ vector field $X$ on some open subset of $\mathbb R^n$ such that there exists an $L^\infty$ function $\alpha\ge 1$ so that $ ...
Bazin's user avatar
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4 votes
0 answers
306 views

If a derivative is defined everywhere and $\pm1$ almost everywhere, is it constant?

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that the set $A:=\{x\in\mathbb{R};f'(x)\not\in\{1,-1\}\}$ has measure $0$. Does this imply that $f'$ is constant? Context: I was ...
Saúl RM's user avatar
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7 votes
2 answers
731 views

What happens if we consider functions of bounded variation that are not in $L^1$?

A function $f \in L^1(\mathbb R^n)$ is said to be of bounded variation if there exists a constant $C \geq 0$ such that $$ \int_{\mathbb R^n} f(x) \operatorname{div} \phi(x) \; dx \leq C \sup_{ x \in \...
shuhalo's user avatar
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27 votes
1 answer
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The dual of $\mathrm{BV}$

$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
Gary Moon's user avatar
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