Shape Optimization for the Principal Eigenvalue of the Pucci Operator in Three Dimensions

TL;DR

This paper studies how the shape of three-dimensional domains affects the principal eigenvalue of the nonlinear Pucci operator, revealing that symmetric, unsheared, and isotropic shapes minimize this eigenvalue within a specific family.

Contribution

It introduces a new family of three-dimensional domains and proves that the symmetric, unsheared shape uniquely minimizes the principal eigenvalue under volume constraint.

Findings

Symmetric, unsheared domains minimize the eigenvalue.

Anisotropy or shear increases the eigenvalue.

Extends planar symmetry results to three dimensions.

Abstract

We investigate shape optimization for the principal eigenvalue of the Pucci extremal operator \[ \left\{ \begin{aligned} -\mathcal{M}^+_{\lambda,\Lambda}(D^{2}u)&=\mu^{+}_{1}(\Omega)u &&\text{in }\Omega,\\ u &=0 &&\text{on }\partial\Omega, \end{aligned} \right. \] in dimension three. Since $\mathcal{M}^+_{\lambda,\Lambda}$ is fully nonlinear, in non-divergence form, and non-variational, classical symmetrization and rearrangement methods are not available. We introduce a three-dimensional family of double--pyramidal domains $\{\Omega^\omega_{\gamma,a}\}$ parametrized by an anisotropy factor $\gamma \in \left[\frac{1}{\sqrt{\omega}},\sqrt{\omega}\right]$ and an affine shear parameter $a\in(-\pi,\pi)$, under fixed ellipticity ratio $\omega=\Lambda/\lambda \ge 1$. Within this family and under a fixed-volume constraint, we prove that the volume-normalized principal eigenvalue is uniquely minimized at the symmetric unsheared configuration $(\gamma,a)=(1,0)$ among domains in the family $\{\Omega^\omega_{\gamma,a}\}$. The proof combines an explicit construction of positive eigenfunctions on seven patches with a lower bound under affine shear deformations. Using the homogeneity and orthogonal invariance of the Pucci operator, we identify an involutive symmetry $\gamma\mapsto \gamma^{-1}$ in the associated volume functional and establish strict monotonicity away from the self-dual point $\gamma=1$. In particular, for $\omega>1$, any nontrivial anisotropy or shear strictly increases the normalized principal eigenvalue. This reveals a genuinely three-dimensional rigidity mechanism for a fully nonlinear spectral problem and extends to dimension three the symmetry-minimization phenomenon previously known in the planar case.