arXiv 2005.074172020-05-15math.APmath.OC

球上薛定谔算子的eigenvalue的定量不等式

Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball

Idriss Mazari

摘要：

本文的目的是证明球中薛定谔算子第一特征值的一个定量不等式。更准确地说，我们优化的第一特征值$λ(V)$的运营商$\mathcal L_v:=-Δ-V$与Dirichlet边界条件相对于潜在的$V$，$L^1$和$L^\infty$约束下$V$。该解决方案已被称为是一个中心球的特征函数，但本文旨在证明以下形式的急剧增长率：如果$V^*$是一个极小，然后$λ(V)-λ(V^*)\geq C ||V-V^*||_{L^1(Ω)}^2$的一些$C>0$。证明依赖于两个概念的衍生物形状优化：参数衍生物和形状衍生物。我们使用参数导数来处理径向竞争对手，形状导数来处理球的法向变形。一个二分法，然后建立推广的结果，所有其他的潜力。我们发展了一种新的方法来处理径向分布和比较原则来处理二阶形状导数在球。最后，我们添加了一些关于在这种情况下的二阶形状导数的davity范数的备注。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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