2019
DOI: 10.4171/rmi/1146
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Towards a reversed Faber–Krahn inequality for the truncated Laplacian

Abstract: We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for the very degenerate elliptic operator \mathcal{P}^+_{1} mapping a function u to the maximum eigenvalue of its Hessian matrix. The aim is to show that, at least for square type domains having fixed volume, the symmetry of the domain maximizes the principal eigenvalue, contrary to what happens for the Laplacian.

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Cited by 13 publications
(23 citation statements)
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“…We finally remark that sort of reversed Faber-Krahn and Lieb inequality have been recently proved for the first eigenvalue of a degenerate operator called truncated Laplacian, see [2].…”
Section: Giovanni Cupini and Eugenio Vecchimentioning
confidence: 81%
See 3 more Smart Citations
“…We finally remark that sort of reversed Faber-Krahn and Lieb inequality have been recently proved for the first eigenvalue of a degenerate operator called truncated Laplacian, see [2].…”
Section: Giovanni Cupini and Eugenio Vecchimentioning
confidence: 81%
“…then Ω must be a ball. We refer to [ We end this section with a brief recap on the optimization eigenvalue problem (2). We refer to [3]…”
mentioning
confidence: 99%
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“…Problem () is the evolution version of the elliptic problem {λjfalse(D2z(x)false)=0,inΩ,z(x)=g(x),onΩ,which was extensively studied in [1, 3–7, 11, 12, 22, 23]. In particular, for j=1 and j=N, problem () is the equation for the convex and concave envelope of g in normalΩ, respectively, that is, the solution z is the biggest convex (smallest concave) function u, satisfying ug (ug) on Ω, see [22, 23].…”
Section: Introductionmentioning
confidence: 99%