Analytic semigroup

An analytic semigroup, sometimes called holomorphic semigroup, is a family of bounded linear operators on a real or complex Banach space into itself, with a complex-valued sector and an angle. Analytic semigroups are a special type of strongly continuous semigroups, which are used in the analysis, in order to prove existence and uniqueness of solutions of partial differential equations such as the heat equation.

Interestingly, the study of analytic semigroups is mainly because of their smoothing properties: So the solution of the associated Cauchyproblems is about always infinitely differentiable in and is always positive in the domain of the generator.

Definition

A family is called analytic semigroup if the following applies to an angle:

  • .
  • For everyone.
  • The mapping is analytic on.
  • The picture is on for strongly continuous.

If for each is additionally restricted, limited analytic semigroup is called (but: a bounded strongly continuous semigroup which is analytic in general is not limited analytic semigroup ).

Infinitesimal generator

Analogous to strongly continuous semigroups we consider the operator with

And

The operator ( infinitesimal ) called producer or generator and is densely defined and closed.

Properties

  • Creates an analytic semigroup, then exist and for all. If the semigroup restricted, can be selected.
  • Exists, so that a limited analytic semigroup generated.
  • Applies to all.
  • Matches the inverse Laplace transform of the resolvent with the semigroup, ie for and a suitable way in.

Examples

  • Is a normal operator ( such as a self-adjoint operator), thus creating a limited analytic semigroup.
  • Generates a strongly continuous semigroup, then the generator of an analytic semigroup with angle.
  • If a domain with smooth boundary, then the Laplace operator with Dirichlet boundary condition generated, ie, A bounded analytic semigroup.

The Cauchy problem

Creates a bounded analytic semigroup, the abstract Cauchy problem

For the initial value and a Hölder - continuous function by the function

Solved.

Closed operator Hermitian adjoint Laplace-Operator
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