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Monotonicity formulas in potential theory

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Abstract

Using the electrostatic potential u due to a uniformly charged body \(\Omega \subset {\mathbb {R}}^n\), \(n\ge 3\), we introduce a family of monotone quantities associated with the level set flow of u. The derived monotonicity formulas are exploited to deduce a new quantitative version of the classical Willmore inequality.

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Acknowledgements

The author are grateful to G. Crasta, A. Farina, I. Fragalà, C. Mantegazza, J. Metzger, M. Novaga, and D. Peralta-Salas for useful comments and discussions during the preparation of the paper. The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA), which is part of the Istituto Nazionale di Alta Matematica (INdAM), and partially funded by the GNAMPA project “Principi di fattorizzazione, formule di monotonia e disuguaglianze geometriche”. The authors would like to thank the anonymous referee for the careful reading of the manuscript as well as for his/her valuable suggestions.

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Correspondence to Lorenzo Mazzieri.

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Communicated by A. Malchiodi.

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Agostiniani, V., Mazzieri, L. Monotonicity formulas in potential theory. Calc. Var. 59, 6 (2020). https://doi.org/10.1007/s00526-019-1665-2

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