Dirichlet and Neumann boundary values of solutions to higher order elliptic equations

Abstract

We show that if $u$ is a solution to a linear elliptic differential equation of order $2m\ge 2$ in the half-space with $t$-independent coefficients, and if $u$ satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of $u$ exist and lie in a Lebesgue space $L^p({\mathbb{R}}^n)$ or Sobolev space ${\dot{W}}_{\pm 1}^p({\mathbb{R}}^n)$. Even in the case where $u$ is a solution to a second order equation, our results are new for certain values of~$p$.

41 pages. arXiv admin note: text overlap with arXiv:1703.06847