$$W^{2,p(\cdot )}$$ W 2 , p ( · ) -regularity for elliptic equations in nondivergence form with BMO coefficients

Abstract

We establish an optimal $$W^{2,p(\cdot )}$$ -estimate to the Dirichlet problem for an elliptic equation in nondivergence form with discontinuous coefficients on a $$C^{1,1}$$ bounded domain for every variable exponent $$p(\cdot )$$ with log-Holder continuity. The matrix of the coefficients is assumed to have a small BMO semi-norm, depending on the exponent, the boundary of the domain, and the matrix itself.