论文标题
关于通过非局部功能的常数函数表征
On the characterization of constant functions through nonlocal functionals
论文作者
论文摘要
我们解决了H.Brezis和R.Ignat的经典开放问题，即通过涉及差异商的双积分来表征常数函数。 我们的第一个结果是对问题的全面示例。此反例需要构建一个函数，其差异商避免了一系列间隔，端点与无穷大。我们的第二个结果是对问题的积极答案，因为限制了几乎到处都是有界和近似可分化的函数，或者对具有有限变化的函数。 我们还提出了一些相关的开放问题，这些问题是由我们的积极和负面结果激励的。
We address a classical open question by H.Brezis and R.Ignat concerning the characterization of constant functions through double integrals that involve difference quotients. Our first result is a counterexample to the question in its full generality. This counterexample requires the construction of a function whose difference quotients avoid a sequence of intervals with endpoints that diverge to infinity. Our second result is a positive answer to the question when restricted either to functions that are bounded and approximately differentiable almost everywhere, or to functions with bounded variation. We also present some related open problems that are motivated by our positive and negative results.
