$L^p$-estimates for the square root of elliptic systems with mixed boundary conditions II

TL;DR

This paper establishes $L^p$ estimates for the square roots of complex elliptic systems with mixed boundary conditions, extending the understanding of their functional calculus and boundary behavior in divergence form.

Contribution

It provides new $L^p$ bounds for elliptic systems with mixed boundary conditions, characterizing the range via semigroup and isomorphism properties, and shows the extrapolation range is relatively open.

Findings

$L^p$ estimates are valid on a specific range determined by semigroup properties.

The extrapolation range for $p$ is relatively open in $(1, abla)$.

The boundary conditions influence the $L^p$ bounds and their range.

Abstract

We show $L^p$ estimates for square roots of second order complex elliptic systems $L$ in divergence form on open sets in $\mathbb{R}^d$ subject to mixed boundary conditions. The underlying set is supposed to be locally uniform near the Neumann boundary part, and the Dirichlet boundary part is Ahlfors-David regular. The lower endpoint for the interval where such estimates are available is characterized by $p$-boundedness properties of the semigroup generated by $-L$, and the upper endpoint by extrapolation properties of the Lax-Milgram isomorphism. Also, we show that the extrapolation range is relatively open in $(1,\infty)$.