The W−1,p Neumann problem for higher order elliptic equations
Ariel Barton · Communications in Partial Differential Equations · 2021
We solve the Neumann problem in the half space R+n+1, for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negative smoothness space Ẇ−1,p, where max(0,12−1n−ε)<1p<12. Our arguments are inspired by an argument of Shen and build on known well posedness results in the case p = 2. We use the same techniques to establish nontangential and square function estimates on layer potentials with inputs in Lp or Ẇ±1,p for a similar range of p, based on known bounds for p near 2; in this case we may relax the requirement of self-adjointess.