The variational solution to the Zaremba problem for a divergent linear second order elliptic equation with measurable coefficients is considered. The problem is set in a local Lipschitz graph domain. An estimate in L_2+δ, δ > 0, for the gradient of a solution, is proved. An example of the problem with the Dirichlet data supported by a fractal set of zero (n - 1)-dimensional measure and non-zero p-capacity, p > 1 is constructed.