2020/12/23 by Frederic Weber, Weber, Frederic, Rico Zacher +1 · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2012.12974
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)≤ C/t, where β∈ (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.