Symmetry minimizes the principal eigenvalue: an example for the Pucci's sup operator

TL;DR

This paper explicitly calculates the principal eigenvalue of the Pucci's sup operator for certain plane domains and demonstrates that among sets of fixed area, the most symmetric domain minimizes this eigenvalue.

Contribution

It provides explicit eigenvalue calculations for the Pucci's sup operator and proves symmetry leads to minimal eigenvalues for fixed-area domains.

Findings

Most symmetric domains minimize the principal eigenvalue for fixed area.

Explicit formulas for eigenvalues of the Pucci's sup operator on special domains.

Symmetry plays a key role in eigenvalue minimization.

Abstract

We explicitly evaluate the principal eigenvalue of the extremal Pucci's sup--operator for a class of special plane domains, and we prove that, for fixed area, the eigenvalue is minimal for the most symmetric set.