论文标题
球中Schrödinger操作员特征值的定量不平等
Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball
论文作者
论文摘要
本文的目的是证明schrödinger操作员在球中的第一个特征值的定量不平等。更确切地说，我们优化了运算符$ \ Mathcal l_v的第一个特征值$λ（v）$：= - δ-V $与潜在的$ v $相对于$ l^1 $和$ l^1 $ and $ l^\ infty $ bongions of $ v $的潜在$ v $。该解决方案已知是中心球的特征功能，但是本文旨在证明以下形式的急剧增长率：如果$ v^*$是最小化的，则$λ（v）-λ（v^^*）\ geq c || v-v-v^*|| v-v^*|| _ {l^1（ω）}^2 $ c> 0 $ 0 $ c> 0 $。证明依靠两个衍生物的概念进行形状优化：参数衍生物和形状衍生物。我们使用参数衍生物来处理径向竞争者，并形成衍生物来处理球的正常变形。然后建立二分法以将结果扩展到所有其他电位。我们开发了一种新方法来处理径向分布和比较原理，以处理球处的二阶形状衍生物。最后，我们在这种情况下添加了有关二阶形状衍生物的强制性规范的一些评论。
The aim of this article is to prove a quantitative inequality for the first eigenvalue of a Schrödinger operator in the ball. More precisely, we optimize the first eigenvalue $λ(V)$ of the operator $\mathcal L_v:=-Δ-V$ with Dirichlet boundary conditions with respect to the potential $V$, under $L^1$ and $L^\infty$ constraints on $V$. The solution has been known to be the characteristic function of a centered ball, but this article aims at proving a sharp growth rate of the following form: if $V^*$ is a minimizer, then $λ(V)-λ(V^*)\geq C ||V-V^*||_{L^1(Ω)}^2$ for some $C>0$. The proof relies on two notions of derivatives for shape optimization: parametric derivatives and shape derivatives. We use parametric derivatives to handle radial competitors, and shape derivatives to deal with normal deformation of the ball. A dichotomy is then established to extend the result to all other potentials. We develop a new method to handle radial distributions and a comparison principle to handle second order shape derivatives at the ball. Finally, we add some remarks regarding the coercivity norm of the second order shape derivative in this context.
