On Zaremba problem for second-order linear elliptic equation with drift in case of limit exponent

Abstract

We establish the unique solvability of the Zaremba problem with the homogeneous Dirichlet and Neumann boundary conditions for an inhomogeneous linear second order second order equation in the divergence form with measurable coefficients and lower order terms. The problem is considered in a bounded strictly Lipschitz domain. We suppose that the domain is contained in an $n$--dimensional Euclidean space, where $n\ge2$. If $n>2$, then the lower coefficient belong to the Lebesgue space with the limiting summability exponent from the Sobolev embedding theorem. If $n=2$, then the lower coefficients are summable at each power exceeding two. Apart of the unique solvability, we establish an energy estimate for the solution.