Reverse Faber-Krahn inequality for a truncated Laplacian operator (Q2154811)

The authors provide an affirmative answer to a conjecture raised in [\textit{I. Birindelli} et al., Rev. Mat. Iberoam. 36, No. 3, 723--740 (2020; Zbl 1445.35158)] regarding the validity of the reversed Faber-Krahn inequality for the first eigenvalue \(\mu_1(\Omega)\) of the eigenvalue problem \[ \begin{cases} -\lambda_N(D^2 u)= \mu u &\text{ in }\Omega,\\ u = 0 & \text{ on }\partial \Omega, \end{cases} \] where \(\lambda_N(D^2 u)\) is the \(N\)-th eigenvalue of the Hessian matrix \(D^2 u\), and \(\Omega \subset \mathbb{R}^N\) is a bounded convex domain, \(N \geq 2\). Even stronger, the authors prove that \(\mu_1(\Omega) \leq \mu_1(B_{\mathrm{diam}(\Omega)/2})\) and the ball is a nonunique maximizer of \(\mu_1(\Omega)\) under the diameter constraint. In the proof, the authors use the explicit value of \(\mu_1(R)\) for rectangles \(R\) established in [loc. cit.]. Moreover, it is shown that, under the \(C^2\)-smoothness assumption on the first eigenfunction, the Reuleaux triangle \(\mathcal{R}\) satisfies \(\mu_1(\mathcal{R}) < \mu_1(B_{\mathrm{diam}(\mathcal{R})/2})\). Several minimization problems for \(\mu_1(\Omega)\) under various constraints are also discussed and several conjectures are made.