This is a home page of a course on the calculus of variations. The topic of this course is the theory of variational integrals with linear growth on the Euclidean and more general metric measure spaces. Examples include minimal surfaces, the total variation, the area functional and the total variation flow. We discuss definitions. existence, regularity and integral representations of the minimizers. Sobolev spaces and functions of bounded variation are used extensively in the arguments. The learning objective of the course is to get to know cutting edge methods in the calculus of variations that are needed in doing research in this field of mathematics.
Schedule: The meetings will be on Mon at 14-16 and Thu at 14-16 in M2. The first meeting will be on Monday 7 January 2017 at 14-16 in M2. The course will be run in the 3rd and 4th teaching periods (January-April 2019).
Course practices: This is interactive course and it consists of lectures, project works, presentations and workshops. The participants are expected to do a project work and give a presentation during the course. The are many interesting research topics for master's and doctoral theses. The course practices will be discussed in the first meeting.
Prerequisites: Participants are expected to take a course on Sobolev spaces before attending this course. Nonlinear partial differential equations and geometric measure theory are also recommended.
Grading: There is no exam for this course. The grading will be based on participation and attendance.
Instructor: Juha Kinnunen
Schedule: The lectures will run until Thursday 7 March 2019. After that there will be project work presentations.
Mon 11 March at 10-12 in M2: Nicholas Dietrich (Project 13)
Thu 14 March at 14-16 in M2: Kari Väisänen (Project 11)
Mon 18 March at 14-16 in M2: Cintia Pacchiano (Project 11)
Mon 25 March at 14-16 in M2: Timo Takala (Project 12)
Tue 26 March at 10-12 in M3: Matias Vestberg (Project 2)
Wed 27 March at 14-16 in M3: Valter Lillberg (Project 4)
Thu 28 March at 14-16 in M2: Julian Weigt (Project 6)
Mon 1 April at 14-16 in M2: Emma-Karoliina Kurki (Project 16)
Tue 2 April at 10-12 in M3: Lien Tran (Project 1)
Mon 8 April at 14-16 in M3: Ivan Yashchuk (Project 15)
Tue 9 April at 10-12 in M3: Anton Vavilov (Project 6)
Wed 10 April at 14-16 in M3: Armando Gutierrez (Project 10)
Thu 11 April at 14-16 in M3: Kristian Moring (Project 17)
L. Ambrosio and P. Tilli, Topics in analysis in metric spaces, Oxford University Press, 2004.
L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, 2000.
F. Andreu, V. Caselles, and J.M. Mazon, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Progress in Mathematics. Birkhauser, 2004.
A. Björn and J. Björn, Nonlinear potential theory on metric spaces, European Mathematical Society, 2011.
M. Giaquinta, G. Modica, and J. Souček, Cartesian currents in the calculus of variations I. Cartesian currents, Springer, 1998.
M. Giaquinta, G. Modica, and J. Souček, Cartesian currents in the calculus of variations II. Variational integrals, Springer, 1998.
J. Heinonen, Lectures on analysis on metric spaces, Springer, 2001.
J. Heinonen, P. Koskela, N. Shanmugalingam, J. Tyson, Sobolev spaces on metric measure spaces. An approach based on upper gradients, Cambridge University Press, 2015.