We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for the very degenerate elliptic operator $\mathcal{P}^+_{1}$ mapping a function $u$ to the maximum eigenvalue of its Hessian matrix. The aim is to show that, at least for square type domains having fixed volume, the symmetry of the domain maximizes the principal eigenvalue, contrary to what happens for the Laplacian.

中文翻译：

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截断拉普拉斯算子的费伯-克拉恩不等式正相反

我们考虑具有非常退化的椭圆算子$ \ mathcal {P} ^ + _ {1} $的非线性特征值问题，该函数具有Dirichlet边界条件，将函数$ u $映射到其Hessian矩阵的最大特征值。目的是表明，至少对于具有固定体积的正方形型畴，该畴的对称性使本征值最大化，这与拉普拉斯算子相反。