arXiv subjects

Sebastian Kräss

Publications and source records attributed to Sebastian Kräss.

3 recordsLinked to original sources

arXiv2023

Li-Yau type and Harnack estimates for systems of reaction-diffusion equations via hybrid curvature-dimension condition

We prove Li-Yau and Harnack inequalities for systems of linear reaction-diffusion equations. By introducing an additional discrete spatial variable, the system is rewritten as a scalar diffusion equation with an operator sum. For such operators in a mixed continuous and discrete setting, we introduce the hybrid curvature-dimension condition $CD_{hyb} (κ,d)$, which is a combination of the Bakry-Émery condition $CD(κ,d)$ and one of its discrete analogues, the condition $CD_Υ(κ,d)$. We establish a hybrid tensorisation principle and prove that under $CD_{hyb} (0,d)$ with $d<\infty$ a differential Harnack estimate of Li-Yau type holds, from which a Harnack inequality can be deduced by an integration argument.

math.AP↗

arXiv2023

Aronson-Bénilan and Harnack estimates for the discrete porous medium equation

We consider the porous medium equation (PME) on a locally finite graph and identify suitable curvature-dimension (CD) conditions under which a discrete version of the fundamental Aronson-Bénilan estimate holds true for positive solutions of the PME. We also show that these estimates allow to prove Harnack inequalities which are structurally similar to the continuous case. The new CD conditions are illustrated with several concrete examples, e.g.\ complete and chain-like graphs.

math.AP↗

arXiv2022

Li-Yau and Harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete Laplacian

We consider operators of the form $L u(x) = \sum_{y \in \mathbb{Z}} k(x-y) \big( u(y) - u(x)\big)$ on the one-dimensional lattice with symmetric, integrable kernel $k$. We prove several results stating that under certain conditions on the kernel the operator $L$ satisfies the curvature-dimension condition $CD_Υ(0,F)$ (recently introduced by two of the authors) with some $CD$-function $F$, where attention is also paid to the asymptotic properties of $F$ (exponential growth at infinity and power-type behaviour near zero). We show that $CD_Υ(0,F)$ implies a Li-Yau inequality for positive solutions of the heat equation associated with the operator $L$. The Li-Yau estimate in turn leads to a Harnack inequality, from which we also derive heat kernel bounds. Our results apply to a wide class of operators including the fractional discrete Laplacian.

math.AP↗