Search

Let n ≥ 2 and Ω be a bounded Lipschitz domain in R n. In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Ω. More precisely, for any given p ∈ (2 , ∞) , two necessary and sufficient conditions for W 1 , p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W 1 , q estimates of solutions with q ∈ [ 2 , p ] and some Muckenhoupt weights, are obtained. As applications, for any given p ∈ (1 , ∞) and ω ∈ A p (R n) (the class of Muckenhoupt weights), the authors establish weighted W ω 1 , p estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of boundary bubbling solutions for a supercritical Neumann problem with mixed nonlinearities under a suitable assumption on the mean curvature of the boundary ∂Ω, the behavior of the solutions is as a tower of k bubbles when the exponent goes to Sobolev critical exponent, and the blow-up point is a critical point of the mean curvature. [ABSTRACT FROM AUTHOR]

In this paper, we mainly employed the idea of the previous paper [34] to study the sharp uniform W 1 , p estimates with 1 < p ≤ ∞ for more general elliptic systems with the Neumann boundary condition on a bounded C 1 , η domain, arising in homogenization theory. Based on the skills developed by Z. Shen in [27] and by T. Suslina in [31,32] , we also established the L 2 convergence rates on a bounded C 1 , 1 domain and a Lipschitz domain, respectively. Here we found a “rough” version of the first order correctors (see (1.12) ), which can unify the proof in [27] and [32] . It allows us to skip the corresponding convergence results on R d that are the preconditions in [31,32] . Our results can be regarded as an extension of [23] developed by C. Kenig, F. Lin, Z. Shen, as well as of [32] investigated by T. Suslina. [ABSTRACT FROM AUTHOR]