On the behavior of weak convergence under nonlinearities and applications

This paper provides a sufficient condition to guarantee the stability of weak limits under nonlinear operators acting on vector-valued Lebesgue spaces. This nonlinear framework places the weak convergence in perspective. Such an approach allows short and insightful proofs of important results in Functional Analysis such as: weak convergence in $L^\infty$ implies strong convergence in $L^p$ for all $1\le p < \infty$, weak convergence in $L^1$ vs. strong convergence in $L^1$ and the Brezis-Lieb theorem. The final goal is to use this framework as a strategy to grapple with a nonlinear weak spectral problem on $W^{1,p}$.