In this article, we establish a Lyapunov-type inequality for the following extremal Pucci’s equation: $$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}{\mathcal{M}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC}}^{+}(D^{2}u)+b(x)|Du|+a(x)u=0 & \text{in}~\unicode[STIX]{x1D6FA},\\ u=0 & \text{on}~\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$ where $\unicode[STIX]{x1D6FA}$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 2$. This work generalizes the well-known works on the Lyapunov inequality for extremal Pucci’s equations with gradient nonlinearity.

中文翻译：

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极值 PUCCI 方程的 LYAPUNOV 型不等式

在本文中，我们为以下极值 Pucci 方程建立了 Lyapunov 型不等式：$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}{\mathcal{M}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC }}^{+}(D^{2}u)+b(x)|Du|+a(x)u=0 & \text{in}~\unicode[STIX]{x1D6FA},\\ u= 0 & \text{on}~\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$在哪里$\unicode[STIX]{x1D6FA}$是一个光滑的有界域$\mathbb{R}^{N}$,$N\geq 2$. 这项工作概括了关于具有梯度非线性的极值 Pucci 方程的 Lyapunov 不等式的著名工作。