Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

We consider linear elliptic systems whose prototype is divΛ[exp(−|x|)−log|x|]IDu=divF+g inB. (0.1) Here B denotes the unit ball of Rn , for n>2 , centered in the origin, I is the identity matrix, F is a matrix in W1,2(B,Rn×n) , g is a vector in L2(B,Rn) and Λ is a positive constant. Our result reads that the gradient of the solution u∈W1,20(B,Rn) to Dirichlet problem for system (0.1) is weakly differentiable provided the constant Λ is not large enough.