Syllabus for Non-Linear Partial Differential Equations

Ickelinjära partiella differentialekvationer

Syllabus

• 10 credits
• Course code: 1MA338
• Education cycle: Second cycle
• Main field(s) of study and in-depth level: Mathematics A1F
• Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
• Established: 2022-03-03
• Established by: The Faculty Board of Science and Technology
• Applies from: Autumn 2022
• Entry requirements:

120 credits including 90 credits in mathematics. Participation in Partial Differential Equations. Proficiency in English equivalent to the Swedish upper secondary course English 6.

• Responsible department: Department of Mathematics

Learning outcomes

On completion of the course, the student should be able to:

• define viscosity solutions and describe the basic properties and techniques used to study viscosity solutions,
• give an outline of Perron's method,
• describe comparison principles and apply the method of doubling the variables to prove comparison principles,
• give an account of Control theory and derive the optimality condition in terms of a Hamilton-Jacobi-Bellman equation,
• describe the main ideas in the proof of Harnack's inequality and apply Harnack's inequality to prove Hölder continuity.

Content

Calculus of variations, Newtonian potentials, Estimates for the Poisson equation, Schauder estimates, Non-variational techniques, Hamilton-Jacobi-Bellman equations, Viscosity solutions, Perron's method, Maximum and Comparison Principles, Existence and Uniqueness, Hopf 's lemma, Harnack's inequality, Elliptic estimates, ABP-estimates. Conservation laws. Ishii's lemma. Alexandrov's theorem for convex functions.

Instruction

Lectures and problem solving session.

Assessment

Written assignments during the course combined with an oral follow-up examination at the end of the course (10 credits).

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Applies from: Autumn 2022

Some titles may be available electronically through the University library.

• Evans, Lawrence C. Partial differential equations

2nd ed.: Providence, R.I.: American Mathematical Society, 2010

Find in the library

• Koike, A Beginner's Guide to the Theory of Viscosity Solutions [Elektronisk resurs]

The Mathematical Society of Japan, 2014

Find in the library

• Caffarelli, Luis A; Cabré, Xavier Fully nonlinear elliptic equations

Providence, R.I.: American Mathematical Society, cop. 1995

Find in the library