Interpolating estimates with applications to some quantitative symmetry results

Rolando Magnanini; Giorgio Poggesi; Rolando Magnanini; Giorgio Poggesi

doi:10.3934/mine.2023002

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-21. doi: 10.3934/mine.2023002

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Interpolating estimates with applications to some quantitative symmetry results

Rolando Magnanini ¹,
Giorgio Poggesi ^{2
,
,}

1.
Dipartimento di Matematica ed Informatica "U. Dini", Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
2.
Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA 6009, Australia

Received: 09 September 2021 Revised: 06 December 2021 Accepted: 07 December 2021 Published: 10 January 2022

We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $ L^p $ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.
- interpolating estimates,
- Serrin's overdetermined problem,
- Alexandrov's Soap Bubble Theorem,
- constant mean curvature,
- stability,
- quantitative estimates
Citation: Rolando Magnanini, Giorgio Poggesi. Interpolating estimates with applications to some quantitative symmetry results[J]. Mathematics in Engineering, 2023, 5(1): 1-21. doi: 10.3934/mine.2023002

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Abstract

We prove interpolating estimates providing a bound for the oscillation of a function in terms of two $ L^p $ norms of its gradient. They are based on a pointwise bound of a function on cones in terms of the Riesz potential of its gradient. The estimates hold for a general class of domains, including, e.g., Lipschitz domains. All the constants involved can be explicitly computed. As an application, we show how to use these estimates to obtain stability for Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value problem. The new approach results in several novelties and benefits for these problems.

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