Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

Abstract

AbstractWe consider linear elliptic systems whose prototype is $$\begin{aligned} div \, \Lambda \left[ \,\exp (-|x|) - \log |x|\,\right] I \, Du = div \, F + g \text { in}\, B. \end{aligned}$$ d i v Λ exp ( - | x | ) - log | x | I D u = d i v F + g in B . Here B denotes the unit ball of $$\mathbb {R}^n$$ R n , for $$n > 2$$ n > 2 , centered in the origin, I is the identity matrix, F is a matrix in $$W^{1, 2}(B, \mathbb {R}^{n \times n})$$ W 1 , 2 ( B , R n × n ) , g is a vector in $$L^2(B, \mathbb {R}^n)$$ L 2 ( B , R n ) and $$\Lambda $$ Λ is a positive constant. Our result reads that the gradient of the solution $$u \in W_0^{1, 2}(B, \mathbb {R}^n)$$ u ∈ W 0 1 , 2 ( B , R n ) to Dirichlet problem for system (0.1) is weakly differentiable provided the constant $$\Lambda $$ Λ is not large enough.