Abstract
For a complete noncompact Riemannian manifold with bounded geometry, we prove a “generalized” compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit manifolds at infinity. We extend previous results contained in Nardulli (Asian J Math 18(1):1–28, 2014), in such a way that the main theorem is a generalization of the generalized existence theorem, i.e., Theorem 1 of Nardulli (Asian J Math 18(1):1–28, 2014). We replace C2,α locally asymptotic bounded geometry with C0 locally asymptotic bounded geometry.
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Acknowledgments
The authors would like to aknowledge Christina Sormani, Pierre Pansu, Andrea Mondino, Michael Deutsch, Frank Morgan, Manuel Ritoré, Luigi Ambrosio for their useful comments and remarks. A special thank goes to Reinaldo Resende de Oliveira for his careful proof reading of the final version.
Funding
The first author has been partially supported by Capes, Brazil (grant no 920551). The second author has been partially sponsored by Fapesp (2018/22938-4) and by CNPq (302717/2017-0), Brazil.
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Muñoz Flores, A.E., Nardulli, S. Generalized Compactness for Finite Perimeter Sets and Applications to the Isoperimetric Problem. J Dyn Control Syst 28, 59–69 (2022). https://doi.org/10.1007/s10883-020-09517-y
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DOI: https://doi.org/10.1007/s10883-020-09517-y