The Kato square root estimate with Robin boundary conditions

TL;DR

This paper establishes the Kato square root estimate for elliptic operators with Robin boundary conditions on bounded domains, addressing a longstanding challenge in the analysis of such boundary value problems.

Contribution

It proves the Kato estimate for divergence form elliptic operators with Robin boundary conditions, a case previously unresolved due to the lack of a first-order approach.

Findings

Proves the Kato square root estimate for Robin boundary conditions.

Handles possibly unbounded boundary conductivity functions.

Extends the class of elliptic operators satisfying the Kato estimate.

Abstract

We prove the Kato square root estimate for second-order divergence form elliptic operators $-div(A\nabla)$ on a bounded, locally uniform domain $D \subseteq \mathbb{R}^n$, for accretive coefficients $A \in L^\infty(D; \mathbb{C}^n)$, under the Robin boundary condition $\nu \cdot A\nabla u + bu = 0$ for a (possibly unbounded) boundary conductivity $b$. In contrast to essentially all previous estimates of Kato square root operators, no first-order approach seems possible for the Robin boundary conditions.