Boyarsky–Meyers Estimate for Solutions to Zaremba Problem

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 $$L_{2+\delta }$$ , $$\delta >0$$ , 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.