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

Abstract

We investigate shape optimization for the principal eigenvalue of the Pucci extremal operator \[ \left\{ \begin{aligned} -\mathcal{M}^+_{λ,Λ}(D^{2}u)&=μ^{+}_{1}(Ω)u &&\text{in }Ω,\\ u &=0 &&\text{on }\partialΩ, \end{aligned} \right. \] in dimension three. Since ${\mathcal{M}}_{\hat{I}»,\hat{I}›}^{+}$ 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 $\{\hat{I}{\text{c◯}}^{\ddot{I}}{‰}_{{\hat{I}}^3,a}\}$ parametrized by an anisotropy factor ${\hat{I}}^3\in \left[\frac{1}{\sqrt{\ddot{I}}‰},\sqrt{\ddot{I}}‰\right]$ and an affine shear parameter $a\in (-\ddot{I}€,\ddot{I}€)$, under fixed ellipticity ratio $\ddot{I}‰=\hat{I}›/\hat{I}»\ge 1$. 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 $({\hat{I}}^3,a)=(1,0)$ among domains in the family $\{\hat{I}{\text{c◯}}^{\ddot{I}}{‰}_{{\hat{I}}^3,a}\}$. 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 $γ\mapsto γ^{-1}$ in the associated volume functional and establish strict monotonicity away from the self-dual point ${\hat{I}}^3=1$. In particular, for $\ddot{I}‰>1$, 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.

25 Pages, 1 figure

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