Lecturer.
Dr. Yana A. Kinderknecht , Building E2 4, Room 435, Tel.: 302-4743Abstract.
The evolution in time of a physical system is in general defined through a system of differential equations and can usually be rewritten as a so called Cauchy problem. So, one is interested in studying well-posedness of the Cauchy problem together with the properties of its solution. This can be done in many important applications via the theory of operator semigroups which yields a solution to the Cauchy problem in terms of a semigroup of linear operators. This theory has become an essential tool in many areas of modern Mathematical Analysis such as Functional Analysis, Partial Differential Equations, Stochastics and Mathematical Physics. The aim of the course is first to develop the general theory of strongly continuous semigroups of linear operators on Banach spaces. It is planned to discuss the questions of existence, construction and approximation of semigroups and their perturbations. Further, the theory will be applied to investigate different classes of evolution equations. It is supposed to discuss second order parabolic equations, the Schrödinger equation, the wave equation, pseudo-differential equations related to Feller(--Markov) stochastic processes. Not only classical, but also some recent results will be presented. Connections with Stochastic Analysis and Quantum Mechanics will be mentioned.Conditions.
Schedule: Lectures: Thursday, 14:00 -16:00, SR 8 Build. E2 4. Beginning: 17.04.2014. Exercises: each second Wednesday, 14:00 - 16:00, SR 9 Build.E2 4 . Beginning: 07.05.2014.
Language: The course language is English.
Audience: The course is suitable for students specializing in mathematics, physics, computer science.
Preliminary knowledge: Analysis I---III, Functional Analysis I (or Functional Spaces).
Certificate.
Credits: 4,5 LP.
Award of certificate: In case of active participation in the exercises, minimum 1/2 of all homework points and successful passing the (oral or written) exam.
Literature.
[1] A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.
[2] K.J. Engel, R. Nagel. A Short Course on Operator Semigroups, Springer, 2006.
[3] K.J. Engel, R. Nagel. One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.
[4] N.Jacob. Pseudo-differential operators and Markov processes. Vol.I. Imperial College Press, 2001.
[5] J. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, 1985.
[6] M. Reed, B. Simon. Methods of Modern Mathematical Physics. Vol. I---II. Academic Press, 1975.
[7] K. Yosida. Functional Analysis. Springer, 1968.
[8] E. Hille, R.S. Phillips. Functional Analysis and Semi-Groups. AMS, 1957.
Exercises.
Tutor: Dr. Yana A. Kinderknecht.
Content of lectures.
Lecture 1. An Abstract Cauchy Problem. Definitions of a semigroup, an infinitesimal generator, a uniformly continuous semigroup, a strongly continuous semigroup of bounded linear operators on a Banach space. A continuity of a semigroup at 0 is equivalent to the continuity everywhere. Translation semigroup. Integral of a function taking values in a Banach space.
Lecture 2. Uniformly continuous semigroups. Isomorphism between uniformly continuous semigroups and bounded linear operators (their generators). Uniqueness of the semigroup generated by a bounded linear operator. Standard properties of uniformly continuous semigroups. Multiplication semigroup.
Lecture 3. Strongly continuous semigroups. Norm estimate. Contraction semigroups. Standard properties of strongly continuous semigroups. Generators are closed and densely defined operators. Translation semigroup on different Banach spaces.
Lecture 4. Strongly continuous semigroups are uniquely defined by generators. The well-posedness theorem. Example of a non-semigroup solution of a Cauchy problem for an evolution equation. Examples of strongly continuous semigroups: multiplication semigroup, translation semigroup, heat semigroup. Convolution and Fourier transform, their properties.
Lecture 5. Strong continuity of the heat semigroup on the space L^2. Core criterium. Pseudo-differential operators and their symbols. Convolution and Fourier transform of measures, their properties. Bochner + Levy-Khinchin Theorem. Continuous negative definite functions, examples.
Lecture 6. Convolution semigroups, their connection with continuous negative definite functions. Operator semigroups associated with convolution semigroups, their strong continuity and view of generators. Feller semigroups.
Lecture 7. Resolvent set and resolvent of a closed operator. Resolvent of a generator of a strongly continuous contraction semigroup as the Laplace transform of the semigroup, norm estimate of the resolvent. The resolvent equation. Neumann-series representation of the resolvent. Resolvent set is open. Resolvent of a generator of a semigroup on L^2 associated to a convolution semigroup.
Lecture 8. Dissipative operators, examples. The closedness of the range of (\lambda - L) is equivalet to the closedness of the dissipative operator L. The intersection of the resolvent set of a closed dissipative operator with the positive ray is either empty, or the ray itself.
Lecture 9. Yosida approximations of a closed densely defined dissipative operator, their properties, examples. Lemma on estimate of the difference of twocontraction semigroups genearated by bounded commuting operators. The Hille-Yosida Theorem in the form of Lumer-Phillips.
Lecture 10. The Hille-Yosida Theorem in the form of Lumer-Phillips for closable operators. The Hille-Yosida Theorem for contraction semigroups and for the general case. Examples on existense results for parabolic equations in the spaces C_\infty and L^p.
Lecture 11. Generation results for groups and adjoint semigroups. Relations between groups and semigroups, conditions of possibility to embed a strongly continuous semigroup into a strongly continuous group. The Hille-Yosida Theorem for groups. Adjoint operator. Generation of a strongly continuous contraction semigroup by a dissipative operator whose adjoint is also dissipative. Adjoint semigroup. Strong continuity of the adjoint semigroup of a strongly continuous semigroup in the case of a reflexive Banach space. The Stone Theorem, its applications to quantum mechanics.
Lecture 12. Approximation and representation of semigroups based on the inversion of the Laplace transform. First and Second Trotter-Kato Approximation Theorems.
Lecture 13. The Chernoff Theorem and its corollaries. The Post-Widder Inversion Formula. The Lie-Trotter Formula. The Feynman formula for multiplicative perturbations of Feller semigroups.
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