Hamburg University of Technology / Institute of Mathematics / 18th Internet Seminar / Phase 1: The Lectures

# Phase 1: The Lectures

From the middle of October to the middle of February we provide a lecture each week. Our aim is to give a thorough introduction to the field, at a speed suitable for master or Ph.D. students. The weekly lecture will be accompanied by exercises, of which the solutions will be provided by the participants.

## Lectures & Exercises

### Lecture 1 - $C_0$-semigroups

Description

Dear Participants,

this, finally, is the beginning of this year's Internet Seminar. In this lecture we give an introduction to strongly continuous semigroups on Banach spaces. We introduce their generators and derive fundamental properties. In an "interlude" we treat some topics from operator theory and integration of Banach space valued functions.

We kindly ask the team of Wuppertal to produce solutions for the exercises of this lecture. They will be made public on the homepage.

We encourage all of you to comment on the lectures and on the solutions of the exercises, ask your questions in case there are problems with understanding and quite generally take part in contributing to the discussion board.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 2 - Characterisation of generators of $C_0$-semigroups

Description

Dear Participants,

according to the title, the main issue of the lecture is to present a characterisation of generators of $C_0$-semigroups. To keep things simple we restrict our main focus to the case of quasi-contractive semigroups. This is also in view of the applications which will come up later. The last issue is an exponential formula for $C_0$-semigroups, which will turn out to be important for several purposes. We start with an interlude on operators as well as on some more integration.

We kindly ask the team of Voronezh to provide solutions for the exercises of this lecture. They will be made public on the homepage.

We repeat our encouragement to use the discussion board.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 3 - Holomorphic semigroups

Description

Dear Participants,

the main issue of the lecture is to give an introduction to holomorphic semigroups and their generators. This is the second part of the lecture, and it will be combined with an introduction to the relation between contraction semigroups and accretive operators on Hilbert space.

In order to give some more information on Banach space valued holomorphic functions we start by explaining (and proving) the equivalence of different notions of holomorphy for these functions.

Concerning the web page of the ISem, you may have noticed that above the links to the lectures there is also a link to Table of Contents'. We will keep this link updated according to the progress of the lectures. We think that you may find it helpful for easy orientation.

We kindly ask the team of Darmstadt to provide solutions for the exercises of this lecture. They will be made public on the homepage.

We repeat our encouragement to use the discussion board.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 4 - The Sobolev space $H^1$, and applications

Description

Dear Participants,

the lecture culminates in the introduction of the Dirichlet Laplacian as the generator of a holomorphic semigroup. In order to achieve this goal, we first have to define Sobolev spaces (not in ultimate generality). Again, this definition will be prepared by the introduction of distributional derivatives.

As an aside for the curious: the treatment of forms will start in Lecture 5.

We kindly ask the team of Lecce (Università del Salento) to provide solutions for the exercises of this lecture. They will be made public on the homepage.

We repeat our encouragement to use the discussion board.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 5 - Forms and operators

Description

Dear Participants,

in this lecture we start the treatment of forms in Hilbert spaces. The important issue is that forms can be used to define operators that are of the kind appearing in the previous lectures. The link establishing the connection between forms and operators are representation theorems, which will be treated first. The theory will be applied, in particular, to the classical Dirichlet form, and the Dirichlet Laplacian will be obtained as the associated operator.

We kindly ask the team of Karlsruhe to provide solutions for the exercises of this lecture. They will be made public on the homepage.

As always, we encourage the participants to use the discussion board.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 6 - Adjoint operators, and compactness

Description

Dear Participants,

we continue the investigation of the interplay between forms and operators. We start by defining the adjoint of an operator in a Hilbert space and then define self-adjoint operators. As a consequence of the spectral theorem for compact self-adjoint operators we will obtain the conclusion that the Dirichlet Laplace operator on a bounded open set always has an orthonormal basis of eigenfunctions, with the sequence of eigenvalues tending to infinity. It should not come as a surprise that this kind of result cannot be obtained by pure operator theory, but has to be supplemented by hard' facts from analysis. These consist, in this lecture, in the compactness of the embedding of $H^1_0$ into $L_2$, for bounded Omega.

This lecture is somewhat longer than the previous ones. We think that, in a first reading, you might skip the proof of the spectral theorem for compact self-adjoint operators (because you certainly know the finite-dimensional version), and you might also just accept the compactness theorem mentioned above and come back to it at a later time.

We kindly ask the team of Marrakesh to provide solutions for the exercises of this lecture. They will be made public on the homepage.

As always, we encourage the participants to use the discussion board.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 7 - Robin boundary conditions

Description

Dear Participants,

in this lecture we continue the treatment of the Laplacian on a bounded open subset of $\mathbb{R}^n$, by introducing other kinds of boundary conditions. This can only be done on open sets with a sufficiently nice boundary. We start by describing different qualities of the boundary, and we state and discuss Gauss' theorem. In order to describe the boundary conditions we have in mind, we need further results from the theory of the Sobolev space $H^1(\Omega)$: denseness, trace, and compactness of the embedding. The required properties are treated in an interlude. With these tools one can define the Neumann Laplacian and the Robin Laplacian and obtain properties of these operators.

Clearly, the theory of forms lives' from the applications. So, it is unavoidable that one has to use the results concerning the Sobolev space $H^1$. In the present lecture, the interlude makes up about the half of the material. As said previously, a strategy to work through the lecture might be to just accept the results of the interlude and come back to it at a later time.

If you think that Lectures 6 and 7 are rather long, you can look forward to Lecture 8, which will be substantially shorter.

We kindly ask the team of Stuttgart to provide solutions for the exercises of this lecture. They will be made public on the homepage.

As always, we encourage the participants to use the discussion board.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 8 - The Dirichlet-to-Neumann operator

Description

Dear Participants,

in this lecture we investigate another application of the theory of forms, the Dirichlet-to-Neumann operator. It will become clear why we used the mapping $j$ in Lecture 5 and why we did not require it to be injective. The Dirichlet-to-Neumann operator acts on boundary values of functions: for a harmonic function on Omega, its trace is mapped to its normal derivative. In this way one obtains a self-adjoint operator in $L_2(\partial\Omega)$. This is treated along with an important generalisation.

From the Notes as well as from the quoted references you will see that with this lecture we are in an area of recent work and ongoing research and development.

We kindly ask the team of Tübingen to provide solutions for the exercises of this lecture. They will be made public on the homepage.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 9 - Invariance of closed convex sets

Description

Dear Participants,

after the examples and applications presented in the last lectures we return to a more structural topic concerning forms and semigroups. The treatment is an outgrowth and a generalisation of the question how to recognise that the semigroup associated with a form on an $L_2(\mu)$ is positive. This question is treated in a more general version, namely asking for criteria that a closed convex subset of the Hilbert space is invariant under the semigroup. In this form the criteria are also applicable for characterising $L_\infty$-contractivity and the sub-Markovian property of semigroups. For the application of the criteria to the Laplacian in $L_2(\Omega)$ it turns out that lattice properties of $H^1(\Omega)$ are needed, and these are presented in an interlude.

We kindly ask the team of Ulm to provide solutions for the exercises of this lecture. They will be made public on the homepage.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 10 - Interpolation of holomorphic semigroups

Description

Dear Participants,

this lecture starts with an interlude on interpolation of operator valued functions on $L_p$-spaces. The aim is to obtain $L_p$-properties of $C_0$-semigroups on $L_2$ which are also sub-Markovian (or substochastic). The last section of the lecture illustates the interplay between invariance, interpolation and duality for $C_0$-semigroups on $L_2$-spaces.

We kindly ask the team of Salerno to provide solutions for the exercises of this lecture. They will be made public on the homepage.

Quite clearly, we will not post a lecture on December 24. Instead, we wish you all a Merry Christmas and a Happy New Year. And also, we wish you a recreative and relaxing Christmas break. You will have to wait for the next lecture until January 7.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 11 - Elliptic Operators

Description

Dear Participants,

welcome back after the Christmas break!

In this lecture we apply the theory presented in previous lectures to elliptic operators with measurable coefficients. It will become apparent that many of the topics presented so far enter the treatment of these operators. Nevertheless we will also need additional order properties of the Sobolev space $H^1(\Omega)$, presented in an interlude.

We kindly ask the team of Chemnitz to provide solutions for the exercises of this lecture. They will be made public on the homepage.

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 12 - Sectorial forms

Description

Dear Participants,

in this lecture we present a slightly more general approach to forms. In a way, it is an amalgamation of the French approach' used so far and the `Kato approach' to forms. It will be shown that up to a certain point these approaches are equivalent. However, the new version will also allow the treatment of non-complete forms. This will be illustrated by two important examples: Robin boundary conditions and the Dirichlet-to-Neumann operator for rough domains.

We kindly ask the team of Delft to provide solutions for the exercises of this lecture. They will be made public on the homepage.

Have fun and enjoy reading. (And use the Discussion Board!!)

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 13 - The Stokes operator

Description

Dear Participants,

this lecture deals with the Stokes operator, a topic coming from fluid dynamics. It arises from some linearisation in the (non-linear) Navier-Stokes equation; but this is not even touched upon in the lecture. The main new features with respect to the previous lectures is that the Hilbert space where the Stokes operator lives is a subspace of a $\mathbb{K}^n$-valued $L_2$-space, and that for the treatment it is necessary to introduce a Sobolev space of order $-1$. A substantial part of the lecture is devoted to the discussion of properties of the spaces arising in the treatment.

We kindly ask the team of Sydney to provide solutions for the exercises of this lecture. They will be made public on the homepage.

This lecture is the last but one lecture of this year's ISem; the 14th and last lecture will come out next week. We will have to start the project phase soon.

Have fun and enjoy reading. (And use the Discussion Board!!)

Best wishes from the ISemTeam,

Wolfgang Arendt
Ralph Chill
Christian Seifert
Hendrik Vogt
Jürgen Voigt

### Lecture 14 - Non-autonomous equations

Description

Dear Participants,

in this lecture we present a treatment of non-autonomous inhomogeneous equations $u'(t) + A(t)u(t) = f(t)$, where each of the operators $A(t)$ is associated with a form. The treatment requires the use of Gelfand triples and Hilbert space valued $L_2$-spaces and Sobolev spaces. These topics will be presented first. The main abstract auxiliary tool is Lions' representation theorem. In the final section the existence and uniqueness of solutions will be treated.

We kindly ask the teams of Dresden and Hamburg to provide solutions for the exercises of this lecture. They will be made public on the homepage.

This lecture is the last lecture of this year's ISem. Thank you for participating, solving exercises, and contributing to the discussion board. We hope you enjoyed the material we presented. For us it was a mixture of enthusiasm, very hard work and great pleasure to prepare the lectures.

We are preparing the project phase now; you will be informed as soon as we have the projects to which you can then apply.

During February we will prepare and post on the ISem page a corrected version of the lecture notes in a single pdf with hyperlinks. The posting will be communicated by an email.