The Neumann problem for higher order elliptic equations with symmetric coefficients

Abstract

In this paper we establish well posedness of the Neumann problem with boundary data in $$L^2$$ or the Sobolev space $$\dot{W}^2_{-1}$$ , in the half space, for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint. This generalizes the well known well posedness result of the second order case and is based on a higher order and one sided version of the classic Rellich identity, and is the first known well posedness result for an elliptic divergence form higher order operator with rough variable coefficients and boundary data in a Lebesgue or Sobolev space.