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

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.