论文标题
反向Faber-Krahn的不平等，用于截短的Laplacian操作员
Reverse Faber-Krahn inequality for a truncated laplacian operator
论文作者
论文摘要
在本文中，我们证明了完全非线性特征的主要特征值$μ_1（ω）$的反向Faber -krahn不平等问题， \\ u＆=＆0＆\ text {on} \ partialω。 \ end {array} \ right。 \]这里$λ_n（d^2 u）$代表$ u $的Hessian Matrix的最大特征值。更准确地说，我们证明，对于开放，有限的，凸域$ω\ subset \ mathbb {r}^n $，不等式\ [μ_1（ω）\ leq \ frac \ frac {π^2} μ_1（b _ {\ text {diam}（ω）/2}），\]，其中$ \ text {diam}（ω）$是$ω$的直径为true。不平等实际上意味着更强的结果，即，在直径约束下球的最大性。 此外，我们在不同种类的约束下讨论了$μ_1（ω）$的最小化。
In this paper we prove a reverse Faber-Krahn inequality for the principal eigenvalue $μ_1(Ω)$ of the fully nonlinear eigenvalue problem \[ \label{eq} \left\{\begin{array}{r c l l} -λ_N(D^2 u) & = & μu & \text{in }Ω, \\ u & = & 0 & \text{on }\partial Ω. \end{array}\right. \] Here $ λ_N(D^2 u)$ stands for the largest eigenvalue of the Hessian matrix of $u$. More precisely, we prove that, for an open, bounded, convex domain $Ω\subset \mathbb{R}^N$, the inequality \[ μ_1(Ω) \leq \frac{π^2}{[\text{diam}(Ω)]^2} = μ_1(B_{\text{diam}(Ω)/2}),\] where $\text{diam}(Ω)$ is the diameter of $Ω$, holds true. The inequality actually implies a stronger result, namely, the maximality of the ball under a diameter constraint. Furthermore, we discuss the minimization of $μ_1(Ω)$ under different kinds of constraints.
