Summaries of all lectures

This file contains the summaries of all lectures. The purpose of it is to create a file that has all the theorems and definitions of the course as well as to enumarate them appropriately so that it is easier to cite them.

**New: For a summary of the coming week 12 see below.**

Brief summaries of the lectures of the first week

Subjects: Some example of boundary value problems and eigenvalue problems, test functions, weak derivatives, definition and elementary properties of Sobolev spaces, approximation of functions of Sobolev spaces by smooth functions.

Brief summaries of the lectures of the second week

Subjects: Proof of approximation by smooth functions, weak convergence, Banach-Alaoglu Theorem, Compact Operators, the spaces , Rellich-Kondrachov compactness Theorem.

**Corrections:** The definition of and the statement of theorem 0.5 have been corrected.

Brief summaries of the lectures of the third week

Subjects: The eigenvalue problem of the Laplacian on bounded domains. Fourier transform. Fractional Sobolev spaces and imbedding theorems. An Application of the imbedding theorems.

Brief summaries of the lectures of the fourth and fifth week

Fractional Sobolev spaces and imbedding theorems. An Application of the imbedding theorems.

(Most of this was in the summary of last week, but it has been changed a little bit). Trace theorem and Poincare inequalities.

**The last three theorems of the summary have been corrected on 09.05 a little bit before midnight. The Poincare inequality was corrected once again on 16.05**

Brief summaries of the lectures of the sixth week

Subjects: Elliptic differential operators, related boundary value problems and definitions of weak solutions. Lax-Millgram Theorem.

** For a brief summary of the lectures of week 7, see summaries of all lectures**

Subjects: Theorems for existence and uniqueness of weak solutions of elliptic boundary value problems. Fredholm alternative.

** For a brief summary of the lectures of week 8, see summaries of all lectures**

Subjects: More on the existence of weak solutions of boundary value problems. Interior regularity of weak solutions and regularity to the boundary.

**Please note that the inequalities (13) and (14) in the summary were corrected.**

**Weak 9** We will continue with the regularity theory and we will discuss the weak maximum principles (see section 7 of the summaries)

**Weak 10** We will discuss Hopf's lemma and the strong maximum principles. We will also discuss properties of the eigenfunctions and eigenvalues of symmetric elliptic operators on (see Chapter 8 of the summaries).**Addition:** The spactral theorem for compact self-adjoint operators has been added in the beginning of Chapter 8.

**Note:** The theorems 8.1, 8.2 (i) of the summaries have been slightly modified.

**Weak 11**: We are going to finish Chapter 8. After that we will state some questions of recent research, and we will also state some open eigenvalue problems. Time permitting we will start explaining how to prove Theorem 6.9 in the case of non-constant coefficients (until now we have proven it only for the case of constant coefficients ).

**Weak 12:** Last week we prepared a discussion of open eigenvalue problems in quantum mechanics. We are going to discuss them. After that we are going to discuss how to prove Theorem 6.9 for coefficients that are not constant (Corollary 6.10 follows then from Theorem 6.9 with the same proof as in the case of constant coefficients ). Time permitting, we will start the proof of Theorem 6.11.

**Weak 13:** We will continue the proofs of theorems 6.9, 6.10 6.11.