Several critical physical properties of a material are controlled by its geometric construction. Therefore,
analyzing the effect of a material’s geometric structure can help to improve some of its beneficial physical
properties and reduce unwanted behavior. This leads to the study of boundary value problems in complex
domains such as perforated domain, thin domain, junctions of the thin domain of different configuration,
domain with rapidly oscillating boundary, networks domain, etc.
In this thesis colloquium, we will discuss various homogenization problems posed on high oscillating domains.
We discuss in detail one of the articles (see, Journal of Differential Equations 291 (2021): 57-89.), where
the oscillatory part is made of two materials with high contrasting conductivities. Thus the low contrast
material acts as near insulation in-between the conducting materials. Mathematically this leads to the study
of degenerate elliptic PDE at the limiting scale. We also briefly explain another interesting article (see,
ESAIM: Control, Optimisation, and Calculus of Variations 27 (2021): S4.), where the oscillations are on the
curved interface with general cost functional. Due to time constraints, we may not discuss other chapters
of the thesis.
In the first part of my talk, I will briefly discuss the periodic unfolding method and its construction as
it is the main tool in our analysis.
The second part of the talk will be homogenizing optimal control problems subject to the considered PDEs.
The interesting result is the difference in the limit behavior of the optimal control problem, which depends
on the control’s action, whether it is on the conductive part or insulating part. In both cases, we derive
the two-scale limit controls problems which are quite similar as far as analysis is concerned. But, if the
controls are acting on the conductive region, a complete-scale separation is available, whereas a complete
separation is not visible in the insulating case due to the intrinsic nature of the problem. In this case,
to obtain the limit optimal control problem in the macro scale, two cross-sectional cell problems are
introduced. We do obtain the homogenized equation for the state, but the two-scale separation of the cost
functional remains as an open question.