论文标题
通过严格的Faber-Krahn类型不等式的第一个特征值的域变化
Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality
论文作者
论文摘要
对于$ d \ geq 2 $和$ \ frac {2d+2} {d+2} <p <\ p <\ infty $，我们证明了第一个eigenvalue $λ_1（ω）$ p $ p $ lip-lipschitz $ supschitz domain $ supset $ cumsset $ fafer-krahn类型的不平等现象，c $ p $ p $ c \ f.混合边界条件）在极化下。我们将这种不等式应用于$ω\ setMinus \ mathscr {o} $的域上的障碍物问题，其中$ \ mathscr {o} \ subset \ subset \ subsetω$是障碍。在$ω$和$ \ mathscr {o} $上的某些几何假设下，我们证明了$λ_1（ω\ setminus \ mathscr {o}）$的严格单调性，相对于某些翻译和$ \ mathscr {o} $ in $ω$的某些翻译和旋转。
For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $λ_1(Ω)$ of the $p$-Laplace operator on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$ (with mixed boundary conditions) under the polarizations. We apply this inequality to the obstacle problems on the domains of the form $Ω\setminus \mathscr{O}$, where $\mathscr{O}\subset \subset Ω$ is an obstacle. Under some geometric assumptions on $Ω$ and $\mathscr{O}$, we prove the strict monotonicity of $λ_1 (Ω\setminus \mathscr{O})$ with respect to certain translations and rotations of $\mathscr{O}$ in $Ω$.
