  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.

Analysis of PDEs