Application deadline

1. september

The course will only be available if there are students at the IVT-faculty who have the course in their PhD-plans for that particular term, and there are resources available for lecturing the course.

This course is organized as an intensive course with concentrated teaching.

Type of course

The subject can be taken as a singular course.

• The PhD course is primarily open to UiT's PhD students (Category 1).

One or more of these categories of course students may also

be admitted:

• Category 2: Participants in the Associate Professor programme that fulfil the educational requirements.

• Category 3: Doctoral students from other universities.

• Category 4: People with a minimum of a master¿s degree (or equivalent), who have not been admitted to a doctoral programme.

Admission requirements

Obligatory prerequisites

Course content

The objective of the course is to give candidates a thorough understanding of the limit behavior of solutions to elliptic and parabolic operators with random statistically homogeneous rapidly oscillating coefficients.

Homogenization of divergence form elliptic operators with stationary almost periodic or random coefficients. Random evolution equation with lower order terms. Effective stochastic partial differential equations. G-convergence of differential operators and Gamma-convergence of functionals, homogenization of nonlinear variational problems, spectral problems of homogenization theory, boundary value problems in perforated random domains, homogenization and percolation, etc. Examples and solutions to important physical problems such as diffusion in random media (e.g. oil recovery), elasticity problems in perforated or stratified domains (e.g. bridge structures or structures on oil platforms) are presented.

Objectives of the course

Knowledge:

After passing the course, the student is expected to be able to

• Demonstrate understanding of the main concepts of stochastic homogenization.
• Apply the basic results in the theory and reach a high level in the forefront of knowledge of homogenization theory.

Skills:

After passing the course, the student is expected to be able to

• Formulate relevant research problems within the stochastic homogenization.
• Solve advanced problems connected to the theory of high international standard.

Habits of mind:

After passing the course, the student is expected to be able to

• Formulate and solve complicated problems in a way that enables communication of research on high level in the theory of stochastic homogenization.

Language of instruction and examination

Teaching methods

Assessment

Mandatory tasks: Essay

Exam: Oral exam.

The grades are Pass/Fail.

A re-sit exam will be arranged for this course

Recommended reading/syllabus

Monographs:

• V.Jikov, S.Kozlov, O.Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag, 1994.

Papers:

• Piatnitski, Andrey Homogenization of random non stationary parabolic operators. Topics on concentration phenomena and problems with multiple scales, 209--230, Lect. Notes Unione Mat. Ital., 2, Springer, 2006.

 Kleptsyna, M. L.; Pyatnitski¿, A. L. Averaging of a random nonstationary convection-diffusion problem. Russian Math. Surveys 57 (2002), no. 4, 729¿751.