# The Kato Square Root Problem on locally uniform domains

## TL;DR

This paper proves the Kato square root estimate for elliptic operators with mixed boundary conditions on complex domains, extending previous results and establishing new regularity properties for fractional Laplacians.

## Contribution

It introduces geometric conditions under which the Kato estimate holds for mixed boundary problems, improving upon prior results even for pure boundary conditions.

## Key findings

- Kato square root estimate established for mixed boundary conditions
- New regularity results for fractional Laplacians with boundary conditions
- Extension of results to unbounded and irregular domains

## Abstract

We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet boundary part is Ahlfors--David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the Neumann boundary part. This improves upon all existing results even in the case of pure Dirichlet or Neumann boundary conditions. We also treat elliptic systems with lower order terms. As a side product we establish new regularity results for the fractional powers of the Laplacian with boundary conditions in our geometric setup.