Reverse Faber-Krahn inequality for a truncated Laplacian operator

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Enea Parini
Aix-Marseille Université
https://orcid.org/0000-0003-3740-8045
Julio D. Rossi
Universidad de Buenos Aires. Departamento de Matemática
https://orcid.org/0000-0001-7622-2759
Ariel Salort
Universidad de Buenos Aires. Departamento de Matemática
https://orcid.org/0000-0002-2106-7031

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.

Paraules clau
truncated Laplacian, reverse Faber–Krahn inequality, spectral optimization

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Parini, Enea et al. «Reverse Faber-Krahn inequality for a truncated Laplacian operator». Publicacions Matemàtiques, 2022, vol.VOL 66, núm. 2, p. 441-55, http://raco.cat/index.php/PublicacionsMatematiques/article/view/402225.
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