Boyarsky–Meyers Estimate for Solutions to Zaremba Problem

On the Zaremba problem for the $p$-elliptic equation

“…The problem of estimates for higher integrability of the gradient of the solution of the Zaremba problem is virtually unexplored. In this regard we can mention only [11], where the problem was formulated for a two-dimensional domain and the p-Laplace operator, with a rather particular assumption on the set on which the Dirichlet condition is defined (the paper itself used an approach of [12]), and also the recent papers [13]- [15], in which, for second-order linear elliptic equations, a Bojarski-Meyers estimate for the solution of the Zaremba problem was obtained, for a domain with Lipschitz boundary and a rapid change between Dirichlet and Neumann boundary conditions, with higher integrability exponent independent of the frequency of this change. In the present paper we consider, for the first time, the Zaremba problem for a nonlinear p-Laplace equation with most general conditions on the support of the Dirichlet data.…”

“…Examples of the set F Example 1. Consider a set F of zero (n − 1)-dimensional measure which satisfies condition (2.9) (for a similar example, see [15]). For simplicity we consider a two-dimensional domain.…”