Acta mathematica scientia,Series B ›› 2025, Vol. 45 ›› Issue (1): 143-152.doi: 10.1007/s10473-025-0111-0

Previous Articles     Next Articles

LOG-CONCAVITY OF THE FIRST DIRICHLET EIGENFUNCTION OF SOME ELLIPTIC DIFFERENTIAL OPERATORS AND CONVEXITY INEQUALITIES FOR THE RELEVANT EIGENVALUE

Andrea Colesanti   

  1. Dipartimento di Matematica & Informatica 'U. Dini', Universitˊa degli Studi di Firenze, Viale Morgagni 67/A-50134, Firenze, Italy
  • Received:2024-08-30 Published:2025-02-06
  • About author:Andrea Colesanti, E-mail,: andrea.colesanti@unifi.it
  • Supported by:
    Disuguaglianze analitiche e geometriche, funded by the Gruppo per Analisi Matematica la Probabilità e le loro Applicazioni.

Abstract: Given an open bounded subset Ω of Rn we consider the eigenvalue problem {Δuu,V=λVu,u>0in Ω,u=0on Ω, where V is a given function defined in Ω and λV is the relevant eigenvalue. We determine sufficient conditions on V such that if Ω is convex, the solution u is log-concave. We also determine sufficient conditions ensuring that λV, as a function of the set Ω, verifies a convexity inequality with respect to the Minkowski addition of sets.

Key words: eigenvalue, log-concavity, elliptic operator, Brunn-Minkowski inequality, convex body

CLC Number: 

  • 35E10
Trendmd