@inbook{2ece37296426486596b4c492904f1377,

title = "An Interpolating Inequality for Solutions of Uniformly Elliptic Equations",

abstract = "We extend an inequality for harmonic functions, obtained in Magnanini and Poggesi (Calc Var Partial Differ Equ 59(1):Paper No. 35, 2020) and Poggesi (The Soap Bubble Theorem and Serrin{\textquoteright}s problem: quantitative symmetry, PhD thesis, Universit{\`a} di Firenze, 2019), to the case of solutions of uniformly elliptic equations in divergence form, with merely measurable coefficients. The inequality for harmonic functions turned out to be a crucial ingredient in the study of the stability of the radial symmetry for Alexandrov{\textquoteright}s Soap Bubble Theorem and Serrin{\textquoteright}s problem. The proof of our inequality is based on a mean value property for elliptic operators stated and proved in Caffarelli (The Obstacle Problem. Lezioni Fermiane. [Fermi Lectures]. Accademia Nazionale dei Lincei, Rome; Scuola Normale Superiore, Pisa, 1998) and Blank and Hao (Commun Anal Geom 23(1):129–158, 2015).",

keywords = "Alexandrov Soap Bubble Theorem, Elliptic operators in divergence form, Interpolation inequality, Mean value property, Quantitative estimates, Serrin{\textquoteright}s overdetermined problem, Stability",

author = "Rolando Magnanini and Giorgio Poggesi",

note = "Funding Information: Acknowledgments The authors wish to thank Ivan Blank for useful discussions. The authors are partially supported by the Gruppo Nazionale di Analisi Matematica, Probabilit{\`a} e Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second author is member of AustMS and is supported by the Australian Laureate Fellowship FL190100081 “Minimal surfaces, free boundaries and partial differential equation” and the Australian Research Council Discovery Project DP170104880 “N.E.W. Nonlocal Equations at Work”. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.",

year = "2021",

month = mar,

day = "30",

doi = "10.1007/978-3-030-73363-6_11",

language = "English",

isbn = "978-3-030-73362-9",

volume = "47",

series = "Springer INdAM Series",

publisher = "Springer-Verlag Italia s.r.l.",

pages = "233--245",

editor = "V Ferone and T Kawakami and {Salani }, P and F Takahashi",

booktitle = "Geometric Properties for Parabolic and Elliptic PDE's",

address = "Italy",

}