# The Neumann problem for higher order elliptic equations with symmetric coefficients

## TL;DR

This paper proves the well-posedness of the Neumann boundary value problem for higher order elliptic equations with symmetric, constant-in-vertical-direction coefficients, extending classical results to more complex operators with rough coefficients.

## Contribution

It introduces a higher order Rellich identity and establishes the first well-posedness results for such operators with boundary data in Lebesgue or Sobolev spaces.

## Key findings

- Well-posedness established for higher order elliptic Neumann problems
- Generalization of second order results to higher order operators
- First results for operators with rough variable 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 a higher order operator with rough variable coefficients and boundary data in a Lebesgue or Sobolev space.