We carry out an analysis of the size of the contact surface between a Cheeger set E and its ambient space Omega subset of R-d. By providing bounds on the Hausdorff dimension of the contact surface partial derivative E boolean AND partial derivative Omega, we show a fruitful interplay between this size itself and the regularity of the boundaries. Eventually, we obtain sufficient conditions to infer that the contact surface has positive (d -1) dimensional Hausdorff measure. Finally we prove by explicit examples in two dimensions that such bounds are optimal. (C) 2021 Elsevier Masson SAS. All rights reserved.
Caroccia, M., Ciani, S. (2022). Dimensional lower bounds for contact surfaces of Cheeger sets. JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES, 157, 1-44 [10.1016/j.matpur.2021.11.010].
Dimensional lower bounds for contact surfaces of Cheeger sets
Ciani, S
2022
Abstract
We carry out an analysis of the size of the contact surface between a Cheeger set E and its ambient space Omega subset of R-d. By providing bounds on the Hausdorff dimension of the contact surface partial derivative E boolean AND partial derivative Omega, we show a fruitful interplay between this size itself and the regularity of the boundaries. Eventually, we obtain sufficient conditions to infer that the contact surface has positive (d -1) dimensional Hausdorff measure. Finally we prove by explicit examples in two dimensions that such bounds are optimal. (C) 2021 Elsevier Masson SAS. All rights reserved.| File | Dimensione | Formato | |
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2005.06439.pdf
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