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Institutes

Institute for Analysis

Workgroup Applied Analysis

Classical methods for Partial Differential Equations

Classical methods for Partial Differential Equations

The lecture notes of each lecture will be posted after the lecture. In case that you find any typos please write an email to ioannis.anapolitanos@kit.edu

Lecture 1 16.10.2017
Subjects: A general motivation for partial differential equations, physical derivation of the heat equation, the Poisson equation, the Laplace equation.

Lecture 2 17.10.2017
Subjects: Mean-value property for harmonic functions, maximum principle for harmonic functions.

Lecture 3 23.10.2017
Subjects: Proof of Maximum principle for harmonic functions and applications, fundamental solution of the Laplace equation and properties.

Lecture 4 24.10.2017
Subjects: Solving the Poisson equation with the help of the fundamental solution of Laplace equation, $C^k$ and Lipschitz domains

Lecture 5 30.10.2017
Subjects: Green's representation formula, Green's functions, Greens' function for the hapf space, Green's function for the sphere.
A corrected version was uploaded on 06.11.2017

Lecture 6 06.11.2017
Subjects: Green's function of the sphere, existence and calculation of harmonic extensions of functions continuous on the boundary of the sphere.
Note:pages 3 and 4 were exchanged, page 2 has a correction of the lecture

Lecture 7 07.11.2017
Subjects: Elliptic operators and Maximum Principles

Lecture 8 13.11.2017
Subjects: Weak maximum principles for elliptic operators and Hopf's lemma

Lecture 9 14.11.2017
Proof of Hopf's lemma, Strong maximum principle and proof for $c=0$ .

Lecture 10 20.11.2017
Subjects: Strong maximum principle for $c \geq 0$ . Homogeneous heat equation in $\mathbb{R}^n$ , self-similar solution, classification of all self-similar spherically symmetric solutions of the heat equation in $\mathbb{R}^n$ .
Correction: The Hopf's lemma requires only that $u(x_0) \geq 0$ and no strict inequality. Therefore, it can indeed be applied to obtain the strong maximum principle of $c \geq 0$ .

Lecture 11 21.11.2017
Subjects: Fundamental solution of the heat equation and its properties. Solving the homogeneous heat equation in $\mathbb{R}^n$ with the help of the heat equation.
Note: A corrected version was uploaded on the 27th of November
Correction: In the second last page in the forth line g(y) is missing.

Lecture 12 28.11.2017
Subjects: The inhomogeneuos heat equation in $\mathbb{R}^n$ .

Lecture 13 29.11.2017
Subjects: The inhomogeneuos heat equation in $\mathbb{R}^n$ (continuation of previous lecture) and short introduction to separation of variables.

Lecture 14 04.12.2017
Subjects: Reminder of properties of the Fourier series. An application of Fourier series and separation of variables to the heat equation

Lecture 15 05.12.2017
Subjects: Separation of variables for the heat and string equations. Derivation of the string equation

Lecture 16 11.12.2017
Subjects: Separation of variables for the string equation (continued). Transport equation in $\mathbb{R}^n$ .

The following two animations show some separated solutions of the string equation.

string1.m
string2.m

The following four animations illustrate the method of characteristics in the simple case of linear transport equation.

characteristics1.m
characteristics2.m
characteristics3.m
characteristics4.m

Lecture 17 12.12.2017
Subjects: The wave equation on the real line, the wave equation on the half real line with Dirichlet boundary condition

Lecture 18 19.12.2017
Subjects: The wave equation on $\mathbb{R}^n$ . Reduction of the wave equation in $\mathbb{R}^3$ to wave equation
in the half real line.

Lecture 19 20.12.2017
Subjects: The wave equation in three dimensions: Derivation of the solution and properties.

Lecture 20 08.01.2018
Subjects: Fourier transformation and elementary properties. Schwartz functions.

Lecture 21 09.01.2018
Subjects: Riemann-Lebesgue Lemma, Fourier inversion formula, Plancherel Theorem.

Lectures 22 23 15-16.01.2018
Subjects: Applications of Fourier transform to PDEs, Maximum principle for heat equation, Classification of second order linear PDEs (elliptic, hyperbolic, parabolic), motivation for Sobolev spaces.

Lecture 24 22.01.2018
Subjects: Weak derivatives, Sobolev spaces, completeness of Sobolev spaces, Leibnitz rule for Sobolev spaces

Lecture 25 23.01.2018
Subjects: Approximation with smooth functions
The proof of Lemma 8.6 has been simplified and the sketch of the graph of $\chi_R$ correcte. The proof of Lemma 8.7 (1) has also been slightly simplified. In the 4th page refference to the skript of analysis 3 was added.

Lecture 26 24.01.2018
Subjects: Weak convergence, Banach Alaoglou Theorem (special case), compact operators, the space $W_0^{1,p}$ .

Lecture 27 29.01.2018
Subjects: In this lecture we studied different Properties of the spaces $W_0^{1,p}(U)$ , which will help us prove next time the Rellich Kondrachov compactness theorem.

Lecture 28 05.02.2018
Subjects: Rellich Kondrachov compactness Theorem, construction of the lowest eigenvalue of the Laplacian on $W_0^{1,2}(U)$ , $U \subset \mathbb{R}^n$ open bounded.

Lecture 29 06.02.2018
Subjects: Construction of an orthonormal basis of eigenfunctions of the Laplacian in $W_0^{1,2}(U), \quad U \subset \mathbb{R}^n$ open bounded.
The last two pages have some hints for the exam. The list of the subjects is not necessarily exhaustive.