Local Poincaré constants and mean oscillation functionals for $BV$ functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916128524075008 |
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| author | Arroyo-Rabasa, Adolfo Bonicatto, Paolo Del Nin, Giacomo |
| author_facet | Arroyo-Rabasa, Adolfo Bonicatto, Paolo Del Nin, Giacomo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_10794 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local Poincaré constants and mean oscillation functionals for $BV$ functions Arroyo-Rabasa, Adolfo Bonicatto, Paolo Del Nin, Giacomo Analysis of PDEs 26B30, 26D10 (primary), 49Q20 (secondary) 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. |
| title | Local Poincaré constants and mean oscillation functionals for $BV$ functions |
| topic | Analysis of PDEs 26B30, 26D10 (primary), 49Q20 (secondary) |
| url | https://arxiv.org/abs/2402.10794 |