Establishes well-posedness of the dot W^{-1,p} Neumann problem for higher-order elliptic operators with t-independent self-adjoint coefficients in the half-space for max(0, 1/2 - 1/n - eps) < 1/p < 1/2.
Dirichlet and Neumann boundary values of solutions to higher order elliptic equations
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abstract
We show that if $u$ is a solution to a linear elliptic differential equation of order $2m\geq 2$ 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 $\dot W^p_{\pm 1}(\mathbb{R}^n)$. Even in the case where $u$ is a solution to a second order equation, our results are new for certain values of~$p$.
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2019 1verdicts
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The $\dot W^{-1,p}$ Neumann problem for higher order elliptic equations
Establishes well-posedness of the dot W^{-1,p} Neumann problem for higher-order elliptic operators with t-independent self-adjoint coefficients in the half-space for max(0, 1/2 - 1/n - eps) < 1/p < 1/2.