论文标题
通过还原为热内核的一般非本地扩散方程的li-yau不平等现象
Li-Yau inequalities for general non-local diffusion equations via reduction to the heat kernel
论文作者
论文摘要
我们建立了一个还原原则，以在非常一般的框架中引起非本地扩散问题的li-yau不等式，该框架涵盖了离散和连续的设置。我们的方法不是基于曲率维度不平等，而是基于溶液的热内核表示，而在于将问题减少到热核中。作为一个重要的应用，我们通过获得$（在太空中）的正溶液（在太空中）的li-yau不等式来解决一个长期的开放问题，形式$（ - δ）^{β/2}（\ log u）\ leq c/t $，$β\ in（0,2）$。我们还通过在离散设置中的一个示例来说明我们的一般结果，证明了在完整图上扩散的尖锐的li-yau不平等。
We establish a reduction principle to derive Li-Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li-Yau inequality for positive solutions $u$ to the fractional (in space) heat equation of the form $(-Δ)^{β/2}(\log u)\leq C/t$, where $β\in (0,2)$. We also illustrate our general result with an example in the discrete setting by proving a sharp Li-Yau inequality for diffusion on a complete graph.
