Mean Oscillation Gradient Estimates for Elliptic Systems in Divergence Form with VMO Coefficients

Abstract

We consider gradient estimates for H1 solutions of linear elliptic systems in divergence form partial _{alpha }(A_{ij}^{alpha beta } partial _{beta } u^{j}) = 0. It is known that the Dini continuity of coefficient matrix A = (A_{ij}^{alpha beta }) is essential for the differentiability of solutions. We prove the following results:(a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the L2 mean oscillation ωA,2 of A satisfiesXA,2:=lim supr→0r∫r2ωA,2(t)t2expC∗∫tRωA,2(s)sdsdt<∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ X_{A,2} := \\limsup\\limits_{r\\rightarrow 0} r {{\\int \\limits }_{r}^{2}} \\frac {\\omega _{A,2}(t)}{t^{2}} \\exp \\left (C_{*} {{\\int \\limits }_{t}^{R}} \\frac {\\omega _{A,2}(s)}{s} ds\\right ) dt < \\infty , $\\end{document} where C∗ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.(b) If XA,2 = 0, then ∇u ∈ VMO.(c) Finally, examples satisfying XA,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when nabla u in L^{infty } cap VMO.