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

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 $(1,\infty )$.

arXiv commentfinal version, to appear in JDE

References in corpus (4)

• Lp-estimates for the square root of elliptic systems with mixed boundary conditions
• Interpolation theory for Sobolev functions with partially vanishing trace on irregular open sets
• Boundary value problems and Hardy spaces for elliptic systems with block structure
• The Kato Square Root Problem on locally uniform domains

Cited by in corpus (1)

• Weighted non-autonomous $L^q(L^p)$ maximal regularity for complex systems