Homogenization of PDEs on oscillating boundary domains with L1 data and optimal control problems

Abstract

This thesis comprehensively studies the homogenization of partial differential equations (PDEs) and optimal control problems with oscillating coefficients in oscillating domains. After the introduction, the thesis is divided into two parts. In part I, we consider problems in oscillating circular domains with three chapters(chapters 1-3), whereas in part II(chapters 4-5), we deal with domains having oscillations in low dimensions.

The first chapter investigates the homogenization of a second-order elliptic PDE with oscillating coefficients in a circular oscillating boundary domain. By using the polar form of the differential equations and a polar unfolding operator, we consider the general type of oscillations and study the asymptotic behavior of the renormalized solution of the PDE with a source term in $L^1$.

The second chapter examines the homogenization of an elliptic variational form with oscillating coefficients in a circular domain that is highly oscillating itself. The source term is in $L^1$, and we take into account the non-uniform ellipticity that arises due to the highly oscillating boundary, rapidly oscillating coefficient, and the oscillating part made up of highly contrasting materials.

The third chapter focuses on the homogenization of optimal control problems governed by second-order semi-linear elliptic PDEs with matrix coefficients in a circular domain. The cost functionals considered are of general energy type and may have different oscillating matrix coefficients than the constrained PDEs. We prove the existence of well-defined limit problems and derive explicit expressions for the limiting coefficient matrices.

The fourth chapter extends the study to an $n$-dimensional domain with an oscillating boundary that oscillates in $m$ directions, where $1\leq m < n$. This is a relatively unexplored area in the literature, and we show that in this case, the limit problem has derivatives in all non-oscillating directions, or $n-m$. Specifically, we study the homogenization of an elliptic PDE in such a domain with $L^1$ data. Our work expands upon previous research and could have potential applications in various fields.

The fifth chapter investigates the homogenization of optimal control problems governed by second-order semi-linear elliptic PDEs with matrix coefficients in a low-dimensional oscillating domain. The cost functionals considered are of general energy type and may have different oscillating matrix coefficients than the constrained PDEs. We prove the existence of well-defined limit problems and derive explicit expressions for the limiting coefficient matrices. We show that, in this case, the limit problem has derivatives in all non-oscillating directions. Our results show that the limiting cost functional's coefficient matrix is a combination of the original cost functional's and constrained PDE's coefficient matrices.