标题： 一维非均匀势能的Sobolev型不等式的两步最小化方法 显示英文标题

标题： Two-step minimization approach to Sobolev-type inequality with inhomogeneous potential in 1D

摘要： 我们提出了一种新方法，用于确定包含非均匀势项范数的一维Sobolev型嵌入的最佳常数。这个问题与具有非均匀势的Schrödinger算子的Green函数密切相关。在Sobolev空间中一个类似Rayleigh商的最小化问题给出了Sobolev嵌入的最佳常数。我们将最小化问题分解为两个子最小化问题，并证明Green函数提供了第一个最小化问题的最小值。然后，这使我们能够推导出非均匀有界势情况下的最佳常数和函数的新精确估计。作为应用，我们给出了一些非均匀势的例子，其Sobolev型嵌入的最佳常数和函数被明确确定。此外，我们考虑了有界势的$L^p$或测度值扰动，并利用比较公式和两步最小化方法对其进行分析。 显示英文摘要

摘要： We present a new method to determine the best constant of the Sobolev-type embedding in one dimension with a norm including an inhomogeneous potential term. This problem is closely connected to the Green function of the Schr\"{o}dinger operator with inhomogeneous potential. A minimization problem of a Rayleigh-type quotient in a Sobolev space gives the best constant of the Sobolev embedding. We decompose the minimization problem into two sub-minimization problems and show that the Green function provides the minimizer of the first minimization problem. Then, it enables us to derive a new precise estimate of the best constant and function for inhomogeneous bounded potential cases. As applications, we give some examples of the inhomogeneous potential whose best constant and function of the Sobolev-type embedding are explicitly determined. Furthermore, we consider an $L^p$- or measure-valued perturbation of the bounded potential and analyze it using a comparison formula and the two-step minimization approach.