Strong competition versus fractional diffusion
by Gianmaria Verzini
Abstract: Different phenomena can be modeled by elliptic systems involving a number of densities subject to diffusion, reaction and competitive interaction. Relevant particular cases include the Lotka-Volterra
competition, widely used in population dynamics and ecology, and the Gross-Pitaevskii one, which appears in the modeling of Bose-Einstein condensation. When competition prevails, the densities tend to segregate each other, and a typical question regards the common shared regularity (i.e. uniform HÃ¶lder bounds), uniformly w.r.t. the competition parameter.
In this talk, I will first review the theory addressing such issue in the case of standard diffusion. Next I will report on some more recent results, obtained in collaboration with Susanna Terracini (UniversitÃ di Torino) and Alessandro Zilio (EHESS-CAMS, Paris), concerning the case of anomalous diffusion, i.e. when the Laplace operator is replaced by the (nonlocal)