I will consider a class of non-local equations, assuming that their solutions have a global maximum, and discuss estimates on their position. Such points are of special interest as they describe where hot-spots settle on the long run or, in a probabilistic description, the region around which paths typically concentrate. I will address, on the one hand, the interplay of competing parameters whose compromise the resulting location is, and the impact of geometric constraints, on the other.
This is joint work with A. Biswas, based on the recent papers
[1] Universal constraints on the location of extrema of eigenfunctions of non-local Schrödinger operators, J Diff Equations,
https://doi.org/10.1016/j.jde.2019.01.007,
2019
[2]
Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for
non-local Schrödinger equations with exterior conditions, to appear in
SIAM J Math Anal, 2019 (arXiv:1711.09267)
[3] Maximum principles for time-fractional Cauchy problems with spatially non-local components, Fract Calc Appl Anal 21, 1335–1359, 2018
https://doi.org/10.1515/fca-2018-0070
[4] Ambrosetti-Prodi type results for Dirichlet problems of the fractional Laplacian, arXiv:1803.08540, 2018
[5] Hopf's lemma for viscosity solutions to a class of non-local equations with applications, arXiv:1902.07452, 2019
Toaldo Bruno Department of Mathematics University of Torino Via Carlo Alberto 10 10123 Torino, Italy
tel: +39 0110912850