Evolutionary inclusions

Abstract

This chapter is devoted to the study of evolutionary inclusions. In contrast to evolutionary equations, we will replace the skew-selfadjoint operator A by a so-called maximal monotone relation A ⊆ H × H in the Hilbert space H. The resulting problem is then no longer an equation, but just an inclusion; that is, we consider problems of the form (u,f)∈∂t,νM(∂t,ν)+A¯, (u,f)∈ ∂ t,ν M(∂ t,ν )+A, where f∈ L2,ν(ℝ; H) is given and u∈ L2,ν(ℝ; H) is to be determined. This generalisation allows the treatment of certain non-linear problems, since we will not require any linearity for the relation A. Moreover, the property that A is just a relation and not neccessarily an operator can be used to treat hysteresis phenomena, which for instance occur in the theory of elasticity and electro-magnetism.