The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$

TL;DR

This paper proves that a specific square root operator acts as a topological isomorphism between certain Sobolev and Lebesgue spaces under mixed boundary conditions, extending known results from p=2 to p in (1,2).

Contribution

It establishes the isomorphism property of the square root operator for p in (1,2) based on the assumption that it holds for p=2, under general geometric boundary conditions.

Findings

Operator provides a topological isomorphism for p in (1,2).

Results depend on boundary regularity conditions.

Extends known p=2 case to a range of p values.

Abstract

We show that, under general conditions, the operator $\bigl (-\nabla \cdot \mu \nabla +1\bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(\Omega)$ and $L^p(\Omega)$, for $p \in {]1,2[}$ if one presupposes that this isomorphism holds true for $p=2$. The domain $\Omega$ is assumed to be bounded, the Dirichlet part $D$ of the boundary has to satisfy the well-known Ahlfors-David condition, whilst for the points from $\overline {\partial \Omega \setminus D}$ the existence of bi-Lipschitzian boundary charts is required.