We consider the Dirichlet problem for partial trace operators which include the smallest and the largest eigenvalue of the Hessian matrix. It is related to two-player zero-sum differential games. No Lipschitz regularity result is known for the solutions, to our knowledge. If some eigenvalue is missing, such operators are nonlinear, degenerate, non-uniformly elliptic, neither convex nor concave. Here we prove an interior Lipschitz estimate under a non-standard assumption: that the solution exists in a larger, unbounded domain, and vanishes at infinity. In other words, we need a condition coming from far away. We also provide existence results showing that this condition is satisfied for a large class of solutions. On the occasion, we also extend a few qualitative properties of solutions, known for uniformly elliptic operators, to partial trace operators.

中文翻译：

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具有极值 Hessian 特征值的部分迹算子的 Lipschitz 估计

我们考虑包含 Hessian 矩阵的最小和最大特征值的部分迹算子的 Dirichlet 问题。它与两人零和差分游戏有关。据我们所知，没有已知的 Lipschitz 正则性结果的解决方案。如果缺少某些特征值，则此类算子是非线性的、退化的、非均匀椭圆的，既不凸也不凹。在这里，我们在非标准假设下证明了内部 Lipschitz 估计：解存在于更大的无界域中，并且在无穷远处消失。换句话说，我们需要一个来自远方的条件。我们还提供了存在结果，表明该条件对于一大类解决方案都满足。有时，我们还扩展了解决方案的一些定性属性，这些属性以一致椭圆算子而闻名，