Questions tagged [bounded-variation]
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14 questions
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About the first variation of total variation of BV functions
Let $u v\in W^{1,1}(\Omega)$ and consider the gradient functional
$$
J(u) = \int_\Omega |\nabla u(x)| \, dx
$$
and the perturbation $u_\varepsilon = u + \varepsilon v$, with $\varepsilon > 0$.
On ...
- 1,165
2
votes
1
answer
116
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Perimeter of $E \cup B_{\tau\rho}$ in $B_\rho$
Let $E \subset \mathbb{R}^n$ be a set of finite perimeter. Fix $\rho>0$ and a center $x_0\in\mathbb{R}^n$, and write
$
B_r := \{x\in\mathbb{R}^n:\ |x-x_0|<r\}, \qquad \partial B_r := \{x:\ |x-...
- 181
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2
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297
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Total variation of $\hat F_n -F$
Let $F$ be the cumulative distribution function and $\hat F_n$ of $F$ its empirical analogue. I want to estimate the Lebesgue-Stieltjes integral $$I_n:=\int_{-\infty}^\infty G\,\mathrm d\mu_n,$$ where ...
- 49
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Coarea formula for bounded variation functions on manifolds
We consider an $n$-dimensional Riemannian manifold $M$ (not necessarily compact) without boundary. Following the paper ''Heat semigroup and functions of bounded variation on Riemannian
manifolds'' by ...
- 31
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99
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Restriction of BV functions to hyperplane
Let's say that $f$ is a $\mathbb{R}$-valued function of bounded variations on $\Omega = (a,b)\times(c,d)\times\mathbb{R}$. Is it true that for almost all $x\in]a,b[$, the restriction
$$
\left\lbrace\...
- 23
2
votes
1
answer
132
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Weak limit of difference quotients of BV functions
I would need a reference for the following result (or a counterexample, as I am not 100% sure this is true for $BV$, while I know it is for Sobolev). Let $f: \mathbb{R}^d \to \mathbb{R}$ be a $BV$ ...
- 911
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Structure of the discontinuity set $D_f$ for functions $f$ of bounded variation
Let $f:[0,1]\rightarrow \mathbb{R}$ have total variation bounded by $K\in \mathbb{N}$. The set of discontinuity points $D_f$ equals $\cup_k D_k$ where
$$\textstyle
D_k:= \{ x\in [0,1] : |f(x)-f(x+)|&...
- 4,630
1
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0
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103
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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 $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
- 11
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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 ...
- 6,876
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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(...
- 830
2
votes
1
answer
293
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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
$
...
- 16.7k
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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 ...
- 13.1k
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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 \...
- 5,525
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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 ...
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