**IMPORTANT ANNOUNCMENT:** There will be office hours on 16.08, 17.08, 18.08 starting at 13.00.

Clarifacation about the exercises. You might be asked for an exercise only if one of the following holds:

1. It was used in the lecture to prove some theorem

2. It indicates why an assumtion of a theorem is necessary by giving counterexample when the assumption fails (sheet 1 ex 2.2, sheet 4 ex 2, sheet 6 ex 2.2)

3. It is a direct physical application. This refers to the hydrogen atom (including sheet 2 ex 2.1 that was used for it) and that the Laplace in does not have any eigenvalues.

# Boundary and Eigenvalue Problems (Summer Semester 2016)

Lecturer: | Dr. Ioannis Anapolitanos |
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Classes: | Lecture (0157500), Problem class (0157600) |

Weekly hours: | 4+2 |

Lecture: | Monday 8:00-9:30 | SR 3.68 |
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Tuesday 8:00-9:30 | SR 3.68 | |

Problem class: | Tuesday 14:00-15:30 | SR 3.69 |

Lecturer | Dr. Ioannis Anapolitanos |
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Office hours: Tuesday 13-14 | |

Room 2.025 Kollegiengebäude Mathematik (20.30) | |

Email: ioannis.anapolitanos@kit.edu | |

Problem classes | Jonas Hirsch |

Office hours: | |

Room 2.035 Kollegiengebäude Mathematik (20.30) | |

Email: jonas.hirsch@kit.edu |

# A few words about the course

A boundary value problem is a differential equation imposed on some domain together with additional conditions regarding the value of its solution at the boundary. A very typical example is the Poisson equation for a body, where we know the potential at its boundary and the charge density and we want to find the potential inside the body. The boundary value problems are often static (this means that there is no time involved in the problem) but they can be time dependent as well (for example in the case of the heat equation). The importance of the boundary value problems relies on the fact that often in physics or in engineering we are able to measure or impose properties on the boundary and then we want to know what happens inside the body/domain.

Given a linear operator A on an appropriate space X of functions, we are are looking for and such that . This is an eigenvalue problem. It can possibly be combined with a boundary value problem, where we look for solutions of with certain properties for on the boundary. There are several reasons that the eigenvalue problems are important. One of them is that they arise in minimization or maximazation problems with a constraint. Another reason comes from the fact that if X is a separable Hilbert space and A is a compact and self-adjoint operator, then the normalised eigenfunctions of A form an orthonormal basis of X. This means that we can express everything in terms of eigenfunctions. Therefore, if we now the eigenfunctions and eigenvalues of A then we know -in a sense- everything about A. This explains, for example, why the separation of variables for linear differential equations works often.

In the course we will study several different boundary value problems and eigenvalue problems. We will study existence properties as well as properties of the solutions. For that purpose we will discuss and use certain tools like theory of Sobolev spaces (including Compactness and Imbedding Theorems and trace theory), Elliptic Differential operarors, the Lax-Millgram Lemma, the Fredholm Alternative, the maximum principles, the min-max formulas for the eigenvalues of an operator.

The course will be accompanied by an exercise session. The participation in it is highly recommended.

# Prerequisites

The Analysis I-III and Linear Algebra I-II courses are important prerequisites for the course. In addition, it will be assumed that you have already heard at least one of the courses *Differential Equations and Hilbert Spaces* or *Functional Analysis*. In particular, familiarity with Hilbert spaces, Banach spaces, spaces, compact operators and the spectral theorem for compact self-adjoint operators will be assumed. If you do not have some of the prerequisites but still want to visit the course, please write to me an email so that we can talk personally.

# Summaries of the lectures

Posting of brief summaries of the lectures is planned to be done till Saturday night of the weak before them. These files give a very dry text having only the notation, definitions and theorems, without any proofs, examples, remarks or motivation. The purpose of this is that you get the possibility to see a little bit what will come in the next week each time. It is highly recommended that you read it and think about it even for a short of time before coming to the lecture. Such a small effort from your side might make you understand much more during the lecture than you would understand without any preparation. You can find a brief summary of the lectures in the following link

http://www.math.kit.edu/iana1/edu/bevp2016s/page/summaries/

# Lecture notes

In the following link you find informal lecture notes for the course. Some additional small explanations given in the lecture might not always be here but the most important parts are written here. If you find any typos please send an email to ioannis.anapolitanos@kit.edu

http://www.math.kit.edu/iana1/edu/bevp2016s/page/lecturenotes/

# Feedback for the course

Your feedback for the course (suggestions, difficulties, critisism) is highly appreciated. You can either talk to us, or if you are too shy to do this, you can anonymously write in the following link.

https://docs.google.com/forms/d/1-bVeh40IYAJ0UFc9Nl8nUpdr94SHFYDq4zKgedz7bI8/viewform

# Exercise sheets

The exercise sheets are going to be posted one week before they are solved. It is highly recommend that you try to solve the exercises alone before attending the exercise session.

Exercise sheet 1

Exercise sheet 2

Exercise sheet 3

Exercise sheet 4

Exercise sheet 5

Exercise sheet 6

Exercise sheet 7

Exercise sheet 8

Exercise sheet 9

Exercise sheet 10

Exercise sheet 11

Exercise sheet 12

# References

L. C. Evans: Partial Differential Equations, American Mathematical Society 1998, (this is the most important referrence for the course).

R. A. Adams, J. F. Fournier: Sobolev spaces, Academic Press 2003.

Gilbarg and Trudinger: Elliptic Partial Differential Equations of Second Order, Springer 1998.

W. Strauss: Partial differential equations. An introduction. Second edition (this refference will rarely be used).