In this paper we present a comprehensive existence theory for linear and nonlinear reaction random walk systems. The methods are based on semigroup theory for solutions of differential equations on Banach spaces. The solution properties on a bounded domain sensitively depend on the choice of boundary conditions. For Neumann or for periodic boundary conditions, singularities are transported alon...

We prove a Filippov type existence theorem for solutions of a fractional differential inclusion defined by a nonconvex set-valued map with Dirichlet boundary conditions. The method consists in application of the contraction principle in the space of selections of the set-valued map instead of the space of solutions.

We prove that the two dimensional free magnetic Schrr odinger operator, with a xed constant magnetic eld and Dirichlet boundary conditions on a planar domain with a given area, attains its smallest possible eigenvalue if the domain is a disk. We also give some rough bounds on the lowest magnetic eigenvalue of the disk. Running title: Magnetic Rayleigh-type inequality.