Partial Differential Equations II
(Lecture course in Summer Term 2019)

Instructors: Thomas Schmidt (lecture, exercise class).

Lecture (first on April 2nd):

• Tue, 8-10, H5 and Thu, 12-14, H3

Exercise classes (first on April 10th):

• Wed, 10-12, Room 142, NN

Relevance: The course is eligible as a part of the master in Mathematics, Mathematical Physics, Industrial Mathematics, or Economathematics and is recommended to everyone with interests in analysis. Clearly, other interested participants are also very welcome.

Prerequisites: The lecture builds on basic knowledge in analysis (including the theory of Lebesgue integration) and linear algebra. Some experience with either PDEs or analysis is general is recommended. It is not absolutely necessary, however, to have attended the preceding PDE lecture.

ECTS points: The module (consisting of the lecture and the exercise class) has a worth of 12 ECTS points.

Contents: It is planned to cover the following topics:

• maximum principles for linear differential operators (maximum principles and estimates, Hopf boundary point lemma, variants for non-linear equations),
• weak differentiability (definition and characterization of weak derivatives, examples, calculus rules),
• Sobolev spaces (definition, functional-analysis properties, Meyers-Serrin theorem, Sobolev embeddings, Poincaré inequalities, Rellich theorem),
• L²-theory for linear elliptic PDE systems (abstract description of boundary values problems, coercivity, Lax-Milgram lemma, L² existence theory),
• regularity theory for linear elliptic PDEs and PDE systsms (selected topics, as time permits).

Literature: Common books (of different scope) are:

• R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Elsevier, 2003,
• D.R. Adams, L.I. Hedberg, Functions Spaces and Potential Theory, Springer, 1996,
• L.C. Evans, Partial Differential Equations, American Mathematical Society, 1998,
• L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 2015,
• D. Gilbarg, N.E. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001,
• E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003,
• J. Jost, Partial Differential Equations, Springer, 2013,
• G. Leoni, A First Course in Sobolev Spaces, American Mathematical Society, 2009,
• V. Maz'ya, Sobolev Spaces, Springer, 2011,
• J. Rauch, Partial Differential Equations, Springer, 1991,
• F. Sauvigny, Partial Differential Equations (2 volumes), Springer, 2012,
• E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970,
• M.E. Taylor, Partial Differential Equations (3 volumes), Springer, 1996,
• W.P. Ziemer, Weakly Differentiable Functions, Springer, 1989.