Stefanie Sonner

Hardy type inequalities for Δλ-Laplacians

We derive Hardy-type inequalities for a large class of sub-elliptic operators that belong to the class of -Laplacians and find explicit values for the constants involved. Our results generalize previous inequalities obtained for Grushin-type operators which were proved to be sharp.

Mathematical analysis of a quorum sensing induced biofilm dispersal model and numerical simulation of hollowing effects

We analyze a mathematical model of quorum sensing induced biofilm dispersal. It is formulated as a system of non-linear, density-dependent, diffusion-reaction equations. The governing equation for the sessile biomass comprises two non-linear diffusion effects, a degeneracy as in the porous medium equation and fast diffusion. This equation is coupled with three semi-linear diffusion-reaction equations for the concentrations of growth limiting nutrients, autoinducers, and dispersed cells. We prove the existence and uniqueness of bounded non-negative solutions of this system and study the behavior of the model in numerical simulations, where we focus on hollowing effects in established biofilms.

A note on a derivation method for SDE models: Applications in biology and viability criteria

ABSTRACT We discuss a method, which was popularized by E. J. Allen and that is frequently used in applications to construct SDE models. The derivation procedure is based on information about the elementary processes involved in the dynamics and their corresponding probabilities. We formulate criteria for the viability of the resulting models. In particular, explicit necessary and sufficient conditions are deduced for the non-negativity and/or boundedness of solutions. Moreover, we show that the class of deterministic models for which the construction leads to an admissible SDE extension is strongly limited. Several examples are presented to illustrate the implications of our results.