Homogenization and Shape Differentiation of Quasilinear Elliptic Equations

TL;DR

This thesis investigates homogenization, shape differentiation, and symmetrization techniques for quasilinear elliptic equations, extending classical results to less smooth nonlinearities and complex geometries, with applications to eigenvalue problems.

Contribution

It extends homogenization and shape differentiation theories to less smooth nonlinearities and general shapes, including critical cases, and explores eigenvalue basis optimality for elliptic operators.

Findings

Extended homogenization results to less smooth nonlinearities.

Developed shape differentiation methods for non-smooth nonlinearities.

Proved the uniqueness of the eigenvalue basis for optimal function approximation.

Abstract

This thesis is divided into five chapters. The aim is the study of the effectiveness of a chemical as defined by R. Aris for semilinear elliptic equations. The first chapter focuses on homogenization on quasilinear diffusion-reaction problems in domains with small particles. The classical results are extend to less smooth nonlinearities, and more general shapes, specially new for the critical case. The second chapter deals with Steiner symmetrisation of semilinear elliptic and parabolic equations. The third chapter deals with shape differentiation, with smooth and non smooth nonlinearities. The fourth chapter deals with linear elliptic equations with a potential, $-\Delta u + \nabla u \cdot b + V u$, where the potential, $V$, "blows up" near the boundary. This kind of equations appear as a result of the shape differentiation process, in the non-smooth case. The fifth chapter develops the second part of the thesis, and includes results obtained during 2017 visit to Prof. Brezis. We showed that the basis of eigenvalues of $-\Delta$ with Dirichlet boundary conditions is the unique basis to approximate functions in $H_0^1$ in $L^2$ in an optimal way.