The square root of a parabolic operator

TL;DR

This paper establishes the Kato square root property for a class of non-autonomous parabolic operators with minimal boundary regularity, providing key estimates and Lp-versions relevant to elliptic and parabolic PDE analysis.

Contribution

It proves the Kato square root property for non-autonomous parabolic operators with irregular boundaries and measurable coefficients, extending previous results to less regular settings.

Findings

Proved the Kato square root property for the operator L.

Established estimates relating the square root of L to gradient and boundary terms.

Extended results to Lp spaces for broader applicability.

Abstract

Let L(t) = --div (A(x, t)$\nabla$ x) for t $\in$ (0, $\tau$) be a uniformly elliptic operator with boundary conditions on a domain $\Omega$ of R d and $\partial$ = $\partial$ $\partial$t. Define the parabolic operator L = $\partial$ + L on L 2 (0, $\tau$, L 2 ($\Omega$)) by (Lu)(t) := $\partial$u(t) $\partial$t + L(t)u(t). We assume a very little of regularity for the boundary of $\Omega$ and assume that the coefficients A(x, t) are measurable in x and piecewise C $\alpha$ in t for some $\alpha$ > 1 2. We prove the Kato square root property for $\sqrt$ L and the estimate $\sqrt$ L u L 2 (0,$\tau$,L 2 ($\Omega$)) $\approx$ $\nabla$ x u L 2 (0,$\tau$,L 2 ($\Omega$)) + u H 1 2 (0,$\tau$,L 2 ($\Omega$)) + $\tau$ 0 u(t) 2 L 2 ($\Omega$) dt t 1/2. We also prove L p-versions of this result. Keywords: elliptic and parabolic operators, the Kato square root property, maximal regularity, the holomorphic functional calculus, non-autonomous evolution equations.