Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi)convex domains
Abstract
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 realvalued, 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.
 Publication:

Journal of Differential Equations
 Pub Date:
 March 2020
 DOI:
 10.1016/j.jde.2019.09.036
 arXiv:
 arXiv:2003.07609
 Bibcode:
 2020JDE...268.2510Y
 Keywords:

 primary;
 35J25;
 secondary;
 35J15;
 42B35;
 42B37;
 Mathematics  Analysis of PDEs;
 Mathematics  Functional Analysis;
 Primary 35J25;
 Secondary 35J15;
 42B35;
 42B37
 EPrint:
 54 pages