Bellettini, Costante;
(2023)
Hypersurfaces with mean curvature prescribed by an ambient function: Compactness results.
Journal of Functional Analysis
, 285
(10)
, Article 110125. 10.1016/j.jfa.2023.110125.
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Abstract
We consider, in a first instance, the class of boundaries of sets with locally finite perimeter whose (weakly defined) mean curvature is gν, for a given continuous positive ambient function g, and where ν denotes the inner normal. It is wellknown that taking limits in the sense of varifolds within this class is not possible in general, due to the appearance of “hidden boundaries”, that is, portions (of positive measure with even multiplicity) on which the (weakly defined) mean curvature vanishes, so that g does not prescribe the mean curvature in the limit. As a special instance of a more general result, we prove that locally uniform Lq-bounds on the (weakly defined) second fundamental form, for q > 1, in addition to the customary locally uniform bounds on the perimeters, lead to a compact class of boundaries with mean curvature prescribed by g. The proof relies on treating the boundaries as oriented integral varifolds, in order to exploit their orientability feature (that is lost when treating them as (unoriented) varifolds). Specifically, it relies on the formulation and analysis of a weak notion of curvature coefficients for oriented integral varifolds (inspired by Hutchinson’s work [7]). This framework gives (with no additional effort) a compactness result for oriented integral varifolds with curvature locally bounded in Lq with q > 1 and with mean curvature prescribed by any g ∈ C0 (in fact, the function can vary with the varifold for which it prescribes the mean curvature, as long as there is locally uniform convergence of the prescribing functions). Our Our notions and statements are given in a Riemannian manifold, with the oriented varifolds that need not arise as boundaries (for instance, they could come from two-sided immersions).
| Type: | Article |
|---|---|
| Title: | Hypersurfaces with mean curvature prescribed by an ambient function: Compactness results |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1016/j.jfa.2023.110125 |
| Publisher version: | https://doi.org/10.1016/j.jfa.2023.110125 |
| Language: | English |
| Additional information: | Copyright © 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). |
| Keywords: | Ambient-prescribed-mean-curvature; hypersurfaces; Hidden boundaries; Oriented integral varifolds; Weak second fundamental form |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10174002 |
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