标题： 一种二维最小化方法用于一维有界势的Sobolev型不等式 显示英文标题

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

摘要： 我们提出了一种新方法，用于确定包含有界非均匀势项的单变量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空间中一个类似于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，并表明Green函数提供了第一个最小化问题的极小值。然后，这使我们能够得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。 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 a bounded inhomogeneous potential term. This problem is closely connected to the Green function of the Schr\"odinger 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.