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 ∞ L^∞ implies strong convergence in L p L^p for all 1 ≤ p > ∞ 1≤ p > ∞ , weak convergence in L 1 L^1 vs. strong convergence in L 1 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 W1,p .