Form Methods for Linear Evolution Problems on Hilbert Spaces

Bibliography 75 Index 77 v Introduction Evolution means the change of a certain system with respect to time. In a more suggestive language, one could talk of the " motion " of a system in time. Examples could be the movement of planets, the growth of a population, ... From a philosophical point of view (cf. G. Nickel in [9]) it turns out, that the right mathematical model for evolution is a one parameter (semi-)group. By an evolution problem, we understand the problem of obtaining information about such a motion (i.e. the semigroup) given the information how the system changes locally. In applications, this information usually arises either from observation or theoretical reasoning and might be expressed by an abstract Cauchy problem: (CP A) u ′ = A(u) u(0) = u 0 Here u 0 is the initial state of the system and belongs to some state space X, A expresses the local change of the system. In order to obtain an interesting mathematical theory, one often requires additional structure on X, e.g. that X be a Banach space. Then by a linear evolution problem we mean a Cauchy problem, where the local change A is given by a linear operator on X. Thus in the mathematical language of semigroup theory, a linear evolution problem is a problem of the form: Given a linear operator A on a Banach space X, decide whether A generates a semi-group on X (and obtain information about the semigroup from A). This problem was solved in 1948 by Hille and Yosida (in the contraction case, extendet to the general case 1952): A is the generator of a strongly continuous semigroup if and only if A is a closed, densely defined operator such that for some ω 0 the set (ω 0 , ∞) belongs to the resolvent set of A and there exists a constant M such that for all λ ∈ (ω 0 , ∞) the estimate R(λ, A) n ≤ M (Re λ − ω 0) n holds. As always with mathematical theories, there is a struggle between the generality of the theorems and the applicability. The Hille-Yosida theorem is the most general theorem concerning strongly continuous semigroups. However, the applications are very limited, for only in rather special (and rare) cases these conditions can be verified. 1 2 A characterisation which is much easier to handle is given by the Lumer-Phillips …