Ariel Barton

The L^p Neumann Problem for Higher Order Elliptic Equation

abstract:
We solve the Neumann problem in the half-space $\mathbb{R}^{n+1}_+$ for higher order elliptic differential equations with variable self-adjoint $t$-independent coefficients and with boundary data in $L^p$, where $\max(1,\frac{2n}{n+2}-\varepsilon) < p < 2$.
We also establish nontangential and area integral estimates on layer potentials with inputs in $L^p$ or $\dot W^{\pm1,p}$ for a similar range of $p$, based on the known bounds for $p\geq2$; in this case, we may relax the requirement of self-adjointness.

Mathematics Subject Classification: Primary 35J30, Secondary 35C15

Key words and phrases: Elliptic equation, higher-order differential equation, Neumann problem, layer potentials