The square root of a parabolic operator

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.