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Institutes

Institute for Analysis

Workgroup Applied Analysis

Boundary and Eigenvalue Problems

Boundary and Eigenvalue Problems

Lecture 1 April 18th
Subjects: Illustration of some boundary and eigenvalue problems, explanation of there importance, and motivation to introduce spaces of functions that are not differentiable.

Lecture 2 April 19th The notes include both lectures of Tuesday in the first block and the fourth block.
Subjects: Weak derivatives, Sobolev spaces and elementary properties of Sobolev spaces (completeness, Leibnitz rule). Approximation with infinitely differentiable functions.

Lecture 3 April 25th
Subjects: Proof of the approximation of $W^{k,p}(\mathbb{R}^n)$ functions by $C_c^\infty(\mathbb{R}^n)$ functions. Weak convergence and its properties, Banach alaoglou Theorem.

Lecture 4 April 26th
Subjects: Compact operators, special case of the Rellich-Konrachov compactness theorem, with proof.

Lectures 5,6 May 2-3rd
Subjects: Existence of minimizer of $\int_U |\nabla u|^2$ on $W_0^{1,2}(U)$ and proof that it is an eigenfunction of the Laplacian. Construction of an orthonormal basis of $L^2(U)$ consisting of eigenfunctions of the Laplacian. Short explanation of the connection with the ansatz of separation of variables. Regularity of a boundary and a second part of the Rellich compactness theorem. Fourier transformation and its basic properties.
Corrections to the notes: On page 8 in the definition of the Fourier transformation there is a minus missing in the exponent. In page 9 the approximating sequence $f_m$ needs correction. $f_m=f \chi_{\{x \in \mathbb{R}^n: |x| \leq m\}}$ , where $\chi_A$ denotes the characteristic function of $A$ .

Lecture 7 May 9th
Subjects: Fourier inversion formula, Fractional Sobolev spaces in $\mathbb{R}^n$ ( $H^s(\mathbb{R}^n)$ ), A Sobolev imbedding theorem, application to eigenfunctions of Schödingen operators with smooth potential.

Lecture 8, May 10th
Note that part of the lecture is in the lecture notes of lecture 7.

Subjects: Regularity of the eigenfunctions of the Schrödinger Hamiltonian of the hydrogen atom, two Poincare inequalities.

Lecture 9, May 17th
Subjects: Remarks on the two Poincare inequalities and proof of a special case, Trace Theorem and Trace of a function, functions of trace zero.

Lectures 10-11, May 23-24th
Subjects: More elements of the proof of trace theorem, Heat Reaction-diffusion equations, Motivation and definition of elliptic operators, weak solutions of associated boundary value problems, Lax-Millgram Theorem.

Lectures 12-13, May 30-31st
Subjects: Energy estimates, first Theorem for existence of weak solutions, adjoint of a linear bounded operator, Fredholm alternative.

Lectures 14-15, June 6-7th
Subjects: Second and third existence theorem for weak solutions, Interior regularity of week solutions
In the formulation of theorem 6.9 there are a few typos. See the next lecture for correction, where everything is formulated from the beginning.

Lectures 16-17, June 13-14th
Subjects: Interior regularity of weak solutions of boundary value problems, regularity up to the boundary, weak maximum principles.

Lecture 18, June 20th
Subjects: Rest of the proof of the weak maximum principle for $c \geq 0$ Hopf's lemma and its proof.
Correction and remarks: On page 4 equation two holds on $\partial B(0,r)$ and not on $R$ . See the beginning of page 5, the proof of step 1 has been a little bit simplified. Also see page 3 the remark about what Hopf's lemma physically says, something that I forgot to mention during the class.

Lecture 19, June 21st
Please note that the beginning of the lecture is in the last page of the pdf file of the previous lecture. Subjects: Strong maximum principles, some properties of eigenvalues and eigenfunctions of symmetric elliptic operators.

Lecture 20, June 27th
Please see the correction of (*) in page 11 (same as the last psage of last lecture) Subject: Eigenfunctions and eigenvalues of symmetric elliptic operators.

Lecture 21, June 28th
Proof of positivity and uniqueness (up to a multiplicative constants) of eigenfunctions of symmetric elliptic operators to the principal eigenvalue, Can you hear the shape of the drum? Preparations of discusssion of open problems in quantum mechanics.

Lecture 22, July 4th
Subjects: Some open eigenvalue problems in quantum mechanics. Please note that the discussion in the class was more detailed than the one in the notes.
Corrections: In page 25 in the inequality of the theorem the $\|u\|_{L^2(V)}$ should be replaced with $\|u\|_{H^{m+2}(V)}$

Lectures 23-24, July 5, July 11
Subjects: Proof of theorem 6.9 for general elliptic operators in the case $m=0$ .
Attention: If you has downloaded the old file of lecture 23 please delete it. The old proof has a mistake and some typos that I fixed. I am going to explain the corrections in the class. Also small steps that were not explained detailed enough in the lecture are underlined with red twice.

Lecture 25, July 12
Subjects: Proof of theorem 6.9 for general m. Beginning of the proof of theorem 6.11. Note that part of the lecture is at the end of the previous file.