On the behavior of weak convergence under nonlinearities and applications

Abstract

Abstract. 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 ∞ implies strong convergence in L p for all 1 ≤ p < ∞, 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 .

Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

“…The first is that if a sequence of functions w j is bounded in L p (for 1 < p < ∞) and converges almost everywhere, then it converges weakly in L p . The second is that if w j is an almost everywhere and weakly convergent sequence of functions bounded in L p (Ω) for some measure space (Ω, µ) such that µ(Ω) < +∞ and p > 2 then w j converges strongly in L 2 (Ω) (See [15]).…”

Solutions of nonlocal semilinear non-autonomous evolution equations with $L^2-$maximal regularity

“…We will make use of the next lemma, which takes inspiration from [9, Lemma 7.11] (see also [26]). Statement and proof are given in the particular setting of the L 2 space of H−valued functions, though further extensions to a more general setting might be possible.…”

“…t ∈ [0, T ] u(0) = P m λg(w). (26) for some λ ∈ [0, 1] and w ∈ BR,m . By Lemma 11, (F3) and Theorem 2, it is true that u ∈ H 1 ([0, T ], V m ) and that…”

Normalized solutions for the fractional NLS with mass supercritical nonlinearity

“…Proof. We borrow some ideas from [13,Theorem 2.6]. We start exploiting that H s r (R N ) ֒→ L p (R N ) compactly for p ∈ (2, 2 * s ).…”