The Kato square root estimate with Robin boundary conditions

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 $\hat{I}½\cdot A\nabla u+bu=0$ 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.

23 pages