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WS 2023/24 09.10.2023 - 02.02.2024
Liste der zur Klausur zugelassenen Studierenden
1.Klausur Fr. 09.02.2024 10-12 V2-205
Matr.Nr.
K1 Note
4076974 1,0
2883551 2,7
3155240 1,7
3993979 2,0
4068713 1,0
3996025 1,0
3886333 1,7
3946514 1,0
3960693 1,0
4071881 1,0
4081099 1,3
2791415 1,0
2.Klausur Do. 28.03.2024 10-12 H10
Matr.Nr.
K2 Note
2883551 2,3
2864037 5,0
2293093 2,7
3991658 3,0
Deadline for submission of homeworks is always Friday.
Joint submission of homework by groups of students is not allowed.
At least 50% of points for homework is required for admission to the exam.
This percentage is calculated as A/M, where A is the number the
earned points and M is the maximum possible number of points.
The exercises marked with * are taken into account in A, but not in M.
Blatt 0 - keine Abgabe
Blatt 1 - Abgabe bis 20.10.23
Blatt 2 - Abgabe bis 27.10.23
Blatt 3 - Abgabe bis 03.11.23
Blatt 4 - Abgabe bis 10.11.23
Blatt 5 - Abgabe bis 17.11.23
Blatt 6 - Abgabe bis 24.11.23
Blatt 7 - Abgabe bis 01.12.23
Blatt 8 - Abgabe bis 08.12.23
Blatt 9 - Abgabe bis 15.12.23
Blatt 10 - Abgabe bis 05.01.24
Blatt 11 - Abgabe bis 12.01.24
Blatt 12 - Abgabe bis 19.01.24
Blatt 13 - Abgabe bis 26.01.24
0. Introduction
Elliptic equations of 2nd order in divergence and non-divergence
forms.
The divergence theorem. Physical origin of the equations in divergence form.
Origin of the equations in non-divergence form and generators of diffusion processes.
1. Weak Dirichlet problem for divergence form
Distributions and Sobolev spaces.
Weak Dirichlet problem for operators
with measurable coefficients.
Solvability of the weak Dirichlet problem.
Weak Dirichlet problem for operators with lower order terms (uniqueness and
existence).
Sobolev inequality.
Theorem of Lax-Milgram.
The Fredholm alternative.
Estimates of Lµ-norm
of solutions.
2. Higher order derivatives of weak solutions
Existence of 2nd order weak derivatives.
Existence of higher order weak derivatives.
Equations with lower order terms.
Sobolev embedding theorem and existence of classical derivatives.
Strong solutions for equations in non-divergence form.
3. Hölder continuity for equations in divergence form
Faber-Krahn inequality.
Mean value
inequality for subsolutions.
Poincare inequality.
Weak Harnack inequality for positive supersolutions.
Oscillation inequality.
Hölder continuity of weak solutions (Theorem of de Giorgi).
Weak solutions of inhomogeneous equations.
Application to quasi-linear equations.
A fixed point theorem of Leray-Schauder.
4.
Boundary behavior
Hölder continuity up to a flat boundary.
Boundary as a graph.
C1
boundary.
Solvability of the classical Dirichlet problem.
5.* Harnack inequality for equations in
divergence form
Derivation of the Harnack inequality (Theorem of Moser) from the weak Harnack inequality.
6*.
Hölder estimates for equations in non-divergence form
Strong and classical solutions.
Maximum principle of Alexandrov-Pucci.
Weak Harnack inequality.
Oscillation lemma and Hölder continuity (Theorem of Krylov-Safonov).