Abstract. We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps W 1,p (R) × W 1,q (R) → W 1,r (R) with 1 < p, q < ∞ and r ≥ 1, boundedly and continuously. The same result holds on R n when r > 1. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.
In this paper we study the free boundary problem arising as a limit as ε → 0 of the singular perturbation problem div(A(x)∇u) = (x)β ε (u), where A = A(x) is Holder continuous, β ε converges to the Dirac delta δ 0 . By studying some suitable level sets of u ε , uniform geometric properties are obtained and show to hold for the free boundary of the limit function. A detailed analysis of the free boundary condition is also done. At last, using very recent results of Salsa and Ferrari, we prove that if A and are Lipschitz continuous, the free boundary is a C 1,γ surface around H N−1 a.e. point on the free boundary.
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 .
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