Macro Perspective on Microscopic Structures

Henrik Shahgholian will receive funding from the Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Harmonic Analysis and Partial Differential Equations Group at the Department of mathematics, KTH Royal Institute of Technology, Stockholm.

The goal of the planned project is to study the mathematical process of the so-called homogenization, which is of central importance in modeling inhomogeneous materials. Composite materials may consist of several ingredients with different properties. An exact mathematical model may be impossible to implement because of the complexity caused by the inhomogeneity of the material.

In order to describe quantitatively the properties of the composite material, a simplification is necessary. It is done through averaging over the volume of the material. Such a theoretical study may result in finding the best mixture of the ingredients in order to obtain the desired properties of the composite.

Such approximations often assume periodic or random pattern in the structure of the composite. One can think of a herd of zebras observed from a large distance. It is difficult to distinguish a single animal. The further the observer is, the more uniform the pattern seems to be.

Similar averaging procedures can be applied in other applications. For example, the study of fluid flow in a porous material, which gives rise to the so-called perforation problem. For example, one tries to find the best averaging method to model porous underground reservoirs. Such mathematical methods are important in models used by natural oil producers, paper manufacturers and in other manufacturing industries.

Theoretical problems encountered in the averaging process extend over several branches of mathematics. The project will study averaging for certain types of equations with focus on theoretical aspects of the procedure.