Open AccessInterpolating estimates with applications to some quantitative symmetry results
Author(s) -
Rolando Magnanini,
Giorgio Poggesi
Publication year - 2022
Publication title -
mathematics in engineering
Language(s) - English
Resource type - Journals
ISSN - 2640-3501
DOI - 10.3934/mine.2023002
Subject(s) - pointwise , overdetermined system , mathematics , lipschitz continuity , function (biology) , stability (learning theory) , mathematical analysis , boundary (topology) , upper and lower bounds , pure mathematics , symmetry (geometry) , geometry , computer science , machine learning , evolutionary biology , biology
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.