The Square Root of a Parabolic Operator

Abstract

Let $L(t) = - \mathrm{div} \left( A(x,t) \nabla _x \right)$ for $t \in (0, \tau )$ be a uniformly elliptic operator with boundary conditions on a domain $\Omega$ of $\mathbb {R}^d$ and $\partial = \frac{\partial }{\partial t}$. Define the parabolic operator ${{\mathcal {L}}}= \partial + L$ on $L^2(0, \tau , L^2(\Omega ))$ by $({{\mathcal {L}}}u)(t) := \frac{\partial u(t)}{\partial t} + L(t)u(t)$. We assume a very little of regularity for the boundary of $\Omega$ and we assume that the coefficients A(x, t) are measurable in x and piecewise $C^\alpha$ in t (uniformly in $x \in \Omega$) for some $\alpha > \frac{1}{2}$. We prove the Kato square root property for $\sqrt{{{\mathcal {L}}}}$ and the estimate

$$\begin{aligned}&\Vert \sqrt{{{\mathcal {L}}}}\, u \Vert _{L^2(0,\tau , L^2(\Omega ))} \approx \Vert \nabla _x u \Vert _{L^2(0,\tau , L^2(\Omega ))} + \Vert u \Vert _{H^{\frac{1}{2}}(0,\tau , L^2(\Omega ))}\\&\qquad + \left( \int _0^\tau \Vert u(t) \Vert _{L^2(\Omega )}^2\, \frac{dt}{t} \right) ^{1/2}. \end{aligned}$$

We also prove $L^p$-versions of this result.