Reverse Faber-Krahn inequality for a truncated Laplacian operator

• Parini, Enea ^[1] ; Rossi, Julio D. ^[2] ; Salort, Ariel ^[2]
  1. [1] Aix-Marseille University

     Aix-Marseille University

     Arrondissement de Marseille, Francia

     
  2. [2] Universidad de Buenos Aires

     Universidad de Buenos Aires

     Argentina

     
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 66, Nº 2, 2022, págs. 441-455
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper we prove a reverse Faber–Krahn inequality for the principal eigenvalue µ1(Ω) of the fully nonlinear eigenvalue problem (−λN (D2u) = µu in Ω,u = 0 on ∂Ω.Here λN (D2u) stands for the largest eigenvalue of the Hessian matrix of u. More precisely, we prove that, for an open, bounded, convex domain Ω ⊂ RN , the inequality µ1(Ω) ≤π2[diam(Ω)]2= µ1(Bdiam(Ω)/2), where diam(Ω) is the diameter of Ω, holds true. The inequality actually implies a stronger result, namely, the maximality of the ball under a diameter constraint. Furthermore, we discuss the minimization of µ1(Ω) under different kinds of constraints.

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