Existence Families, Functional Calculi and Evolution Equations / by Ralph deLaubenfels.

Format:

Book

Author/Creator:

DeLaubenfels, Ralph, 1951- author.

Contributor:

SpringerLink (Online service)

Series:

Lecture Notes in Mathematics, 0075-8434 ; 1570.

Lecture Notes in Mathematics, 0075-8434 ; 1570

Language:

English

Subjects (All):

Global analysis (Mathematics).

Analysis.

Local Subjects:

Analysis.

Physical Description:

1 online resource (XVI, 244 pages).

Contained In:

Springer eBooks

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1994.

System Details:

text file PDF

Summary:

This book presents an operator-theoretic approach to ill-posed evolution equations. It presents the basic theory, and the more surprising examples, of generalizations of strongly continuous semigroups known as 'existent families' and 'regularized semigroups'. These families of operators may be used either to produce all initial data for which a solution in the original space exists, or to construct a maximal subspace on which the problem is well-posed. Regularized semigroups are also used to construct functional, or operational, calculi for unbounded operators. The book takes an intuitive and constructive approach by emphasizing the interaction between functional calculus constructions and evolution equations. One thinks of a semigroup generated by A as etA and thinks of a regularized semigroup generated by A as etA g(A), producing solutions of the abstract Cauchy problem for initial data in the image of g(A). Material that is scattered throughout numerous papers is brought together and presented in a fresh, organized way, together with a great deal of new material.

Contents:

Intuition and elementary examples

Existence families

Regularized semigroups

The solution space of an operator and automatic well-posedness

Exponentially bounded (Banach) solution spaces

Well-posedness on a larger space; Generalized solutions

Entire vectors and entire existence families

Reversibility of parabolic problems

The cauchy problem for the Laplace equation

Boundary values of holomorphic semigroups

The Schrödinger equation

Functional calculus for commuting generators of bounded strongly continuous groups

Petrovsky correct matrices of generators of bounded strongly continuous groups

Arbitrary matrices of generators of bounded strongly continuous groups

More examples of regularized semigroups

Existence and uniqueness families

C-resolvents and Hille-Yosida type theorems

Relationship to integrated semigroups

Perturbations

Type of an operator

Holomorphic C-existence families

Unbounded holomorphic functional calculus for operators with polynomially bounded resolvents

Spectral conditions guaranteeing solutions of the abstract Cauchy problem

Polynomials of generators

Iterated abstract Cauchy problems

Equipartition of energy

Simultaneous solution space

Exponentially bounded simultaneous solution space

Simultaneous existence families

Simultaneous existence families for matrices of operators

Time dependent evolution equations.

Other Format:

Printed edition:

ISBN:

9783540483229

Access Restriction:

Restricted for use by site license.