Local Poincaré constants and mean oscillation functionals for $BV$ functions

Abstract

We introduce the concept of local Poincaré constant of a $BV$ function as a tool to understand the relation between its mean oscillation and its total variation at small scales. This enables us to study a variant of the BMO-type seminorms on $\varepsilon$-size cubes introduced by Ambrosio, Bourgain, Brezis, and Figalli. More precisely, we relax the size constraint by considering a family of functionals that allow cubes of sidelength smaller than or equal to $\varepsilon$. These new functionals converge, as $\varepsilon$ tends to zero, to a local functional defined on $BV$, which can be represented by integration in terms of the local Poincaré constant and the total variation. This contrasts with the original functionals, whose limit is defined on $SBV$ and may not exist for functions with a non-trivial Cantor part. Moreover, we characterize the local Poincaré constant of a function with a cell-formula given by the maximum mean oscillation of its $BV$ blow-ups. As a corollary of this characterization, we show that the new limit functional extends the original one to all $BV$ functions. Finally, we discuss rigidity properties and other challenging questions relating the local Poincaré constant of a function to its fine properties.

Figures (4)

• Figure 1: The shaded area depicts the $\tau$-contraction $\tau Q_i$ of each cube $Q_i$ ($i = 1,2,3$). Only $Q_2$ and $Q_3$ are admissible for $P_f^\tau(x,\varepsilon)$.
• Figure 2: An oriented cube $Q\subset \mathbb{R}^2$. The vectors $b_1-x_Q$ and $b_2-x_Q$ are orthogonal.
• Figure 3: The function $j=j_{a,b,\nu,c}$ of Lemma \ref{['lemma:tangents_of_SBV']} (in 2d). The function $j$ takes constant values $a,b$ and jumps from value $a$ to $b$ across the line $H_{\nu,c}$.
• Figure 4: On the left: on the interval $I_1$ of scale $r_i^{-1}k_i^{-1}$ the function is close to a jump. On the right: on the interval $I_2$ of intermediate scale $r_i^{-1}$ the function looks affine.

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Proposition 2.1
• proof
• Remark 2.2
• Lemma 3.1
• proof