Evolutionary Inclusions

Abstract

AbstractThis 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 $$\displaystyle (u,f)\in \overline {\partial _{t,\nu }M(\partial _{t,\nu })+A}, $$ ( u , f ) ∈ ∂ t , ν M ( ∂ t , ν ) + A ¯ , where $$f\in L_{2,\nu }(\mathbb {R};H)$$ f ∈ L 2 , ν ( ℝ ; H ) is given and $$u\in L_{2,\nu }(\mathbb {R};H)$$ u ∈ L 2 , 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.