Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

Abstract

We consider linear elliptic systems whose prototype is 0.1divΛexp(-|x|)-log|x|IDu=divF+ginB.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} div \\, \\Lambda \\left[ \\,\\exp (-|x|) - \\log |x|\\,\\right] I \\, Du = div \\, F + g \\text { in}\\, B. \\end{aligned}$$\\end{document}Here B denotes the unit ball of mathbb {R}^n, for n > 2, centered in the origin, I is the identity matrix, F is a matrix in W^{1, 2}(B, mathbb {R}^{n times n}), g is a vector in L^2(B, mathbb {R}^n) and Lambda is a positive constant. Our result reads that the gradient of the solution u in W_0^{1, 2}(B, mathbb {R}^n) to Dirichlet problem for system (0.1) is weakly differentiable provided the constant Lambda is not large enough.