MATS235 Sobolev spaces (9 cr)

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Study level:
Advanced studies
Grading scale:
0-5
Language:
English, Finnish
Responsible organisation:
Department of Mathematics and Statistics
Curriculum periods:
2024-2025, 2025-2026, 2026-2027, 2027-2028

Description

Sobolev spaces are an important tool in modern analysis and in applied mathematics. The course contains the essential parts of the theory of Sobolev spaces like
  • the convolution approximation
  • weak (distributional) derivatives
  • partition of unity and approximation of Sobolev functions by smooth functions
  • Sobolev inequalities
  • the ACL-charterization of Sobolev functions
  • weak and strong convergence in L^p- and Sobolev spaces
  • p-capacity

Learning outcomes

In the course, the basic properties of Sobolev spaces are studied. After the course, the student can use the definition of the weak derivative and its properties, Sobolev inequalities, approximation of Sobolev functions by smooth functions and different characterizations of Sobolev spaces.

Additional information

Lectured in the fall 2024 and fall 2026.

Description of prerequisites

Measure and integration theory 1&2

Study materials

lecture notes

R. A. Adams (and J. J. F. Fournier), Sobolev Spaces

W.P. Ziemer, Weakly Differentiable Functions; ISBN: 978-0-387-97017-2

Literature

  • L.C. Evans & R.F. Gariepy, Measure Theory and Fine Properties of Functions; ISBN: 9781482242386
  • G. Leoni, A first course in Sobolev spaces; ISBN: 978-0821847688

Completion methods

Method 1

Evaluation criteria:
Course exam points and exercise points. Depending on the implementation, there may be two midterm exams instead of the course exam.
Select all marked parts

Method 2

Evaluation criteria:
final exam points
Select all marked parts
Parts of the completion methods
x

Teaching (9 cr)

Type:
Participation in teaching
Grading scale:
0-5
Language:
English, Finnish

Teaching

x

Exam (9 cr)

Type:
Exam
Grading scale:
0-5
Evaluation criteria:
final exam points
Language:
English, Finnish
No published teaching