Teaching
Overview
The course Partial differential equations 2 takes place at the University of Helsinki starting on 13.03.24 and until 03.05.24.
Please consult the official page on the site of the University of Helsinki for all practical information related to the course. On the Moodle page you will find all of the assignments and the model solutions for the exam, which took place on Wednesday 8 May.
Should you have questions related to the lectures or the exercises, please contact me at giovanni.covi@helsinki.fi.
Please find the lecture notes of the course in the Materials section. The official, redacted version will appear soon after the course.
Contents
We plan to continue the discussion of the wave equation begun in the previous course Partial differential equations 1, considering the problem in higher dimension. We will introduce the Fourier transform, and apply the technique of Fourier transformation to solve PDE´s. Moreover, we will define the fundamental concept of weak solution, which will need some preliminaries from Functional analysis. Finally, we will study elliptic PDE´s of second order, showing the existence of weak solutions and some results for higher regularity.
Modalities of exam
Evaluation for the course can be obtained in one of the three following ways:
100% from final - You only take part to the final exam, and your grade is the grade of the exam.
50% from weekly assignments, 50% from final - You solve the weekly assignments and you take part to the final exam. Your grade is the average of the two. Each problem from the assignments is given 2 points (perfect or almost perfect solution), 1 point (honest attempt) or 0 points (no solution, or very wrong solution). The grade from the assignments is obtained as percentage of points obtained over all available points.
Essay - You write an essay on a topic related to the content of the course, and are evaluated for that. Should you be interested in this modality, please write to me in order to discuss a possible topic. The deadline is for the first week of May!
Schedule of the lectures
Here is a tentative schedule of the lectures (changes may still occur):
Wed 13.3, 12:15 - 14:00 - Lecture 1 - Solution of the 1D wave equation by separation of variables. Domains of influence.
Wed 13.3, 14:15 - 16:00 - Lecture 2 - Generalized solutions. Duhamel's method.
Wed 20.3, 12:15 - 14:00 - Lecture 3 - Special solutions to the 3D wave equation: plane, cylindrical and spherical waves.
Wed 20.3, 14:15 - 16:00 - Practice 1 - Old assignment from PDE 1 due on 13.3
Fri 22.3, 12:15 - 14:00 - Lecture 4 - Existence and uniqueness for the 3D wave equation.
Wed 27.3, 12:15 - 14:00 - Lecture 5 - Transform methods.
Wed 27.3, 14:15 - 16:00 - Practice 2 - Old assignment from PDE 1 due on 26.3
Fri 5.4, 12:15 - 14:00 - Lecture 6 - Weak derivatives and Sobolev spaces.
Wed 10.4, 12:15 - 14:00 - Lecture 7 - Second order elliptic PDEs. Weak solutions. Lax-Milgram theorem.
Wed 10.4, 14:15 - 16:00 - Practice 3 - Assignment 1
Fri 12.4, 12:15 - 14:00 - Lecture 8 - First existence theorem for weak solutions: energy estimates.
Wed 17.4, 12:15 - 14:00 - Lecture 9 - Fredholm alternative.
Wed 17.4, 14:15 - 16:00 - Practice 4 - Assignment 2
Fri 19.4, 12:15 - 14:00 - Lecture 10 - Fredholm alternative (continued) and Second existence theorem for weak solutions.
Wed 24.4, 12:15 - 14:00 - Lecture 11 - Third existence theorem for weak solutions. Boundedness of the inverse.
Wed 24.4, 14:15 - 16:00 - Practice 5 - Assignment 3
Fri 26.4, 12:15 - 14:00 - Lecture 12 - Upgrading weak solutions to strong solutions: the case of the Laplacian.
Fri 3.5, 12:15 - 14:00 - Practice 6 - Assignment 4
Wed 8.5, usual place and time (that is, Room B321 in Exactum, from 12:00 to 16:00) - EXAM!
S2B1 - Hauptseminar Funktionalanalysis - Elliptische Partielle Differentialgleichungen (Elliptic PDE)
Overview
The seminar Hauptseminar Funktionalanalysis - Elliptische Partielle Differentialgleichungen took place at the University of Bonn during the Winter semester 2023/2024.
The official announcement can be found here, and the detailed list of topics that were discussed can be found here. The preliminary meeting has taken place on Wednesday 26 July at 10 (c.t.), in Seminar room 0.003.
Should you have questions related to the material, please contact me at giovanni.covi@uni-bonn.de.
News
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Please visit the eCampus page of the seminar. Here is the schedule of the lectures:
12 October - 1 - Preliminaries from functional analysis: Hölder spaces (Giovanni Covi)
19 October - 2 - Preliminaries from functional analysis: Sobolev spaces (Giovanni Covi)
26 October - 3 - The Laplace equation (Marc Sowa)
2 November - 4 - Harnack's inequality for the Laplace equation (Giovanni Covi)
9 November - 5 - Harnack's inequality for the Poisson equation; Schauder estimates for the Laplacian (Giovanni Covi)
16 November - 6 - Schauder estimates for operators in non-divergence form (Mija Delija)
23 November - 7 - Schauder estimates for operators in divergence form (Giovanni Covi)
30 November - 8 - Existence and uniqueness of solutions (Lorenzo Nastase)
7 December - 9 - De Giorgi's first step (Daniel Linn)
14 December - No lecture
21 December - 10 - De Giorgi's second step (Hendrik Baers)
11 January - 11 - Comparison principle for viscosity solutions (Giovanni Covi)
18 January - 12 - Perron's method (Marlon Huestege)
25 January - No lecture
1 February - 13 - The method of continuity (Giovanni Covi)
References
Here you can download some useful references for the seminar:
Xavier Fernández-Real, Xavier Ros-Oton. Regularity Theory for Elliptic PDE. arxiv
Connor Mooney. A proof of the Krylov-Safonov theorem without localization. arxiv
S4B2 - Topics in inverse problems for nonlocal operators
Overview
The seminar Topics in inverse problems for nonlocal operators took place at the University of Bonn during the Summer semester 2023.
The official page of the seminar on eCampus can be found here. The detailed list of topics which were discussed can be found here. The slides of the preliminary meeting are here.
Should you have questions related to the material, please contact me at giovanni.covi@uni-bonn.de.
References
Here you can download some useful references for the seminar:
Inverse problems for a fractional conductivity equation. pdf
Nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. pdf
Hitchhiker’s guide to the fractional Sobolev spaces. pdf
The Calderòn problem for the fractional Schrödinger equation. pdf
On instability mechanisms for inverse problems. pdf