Nittka's invariance criterion and Hilbert space valued parabolic equations in $L_p$

Abstract

Nittka gave an efficient criterion on a form defined on $L_2(\Omega )$ which implies that the associated semigroup is $L_p$-invariant for some given $p\in (1,\infty )$. We extend this criterion to the Hilbert space valued~$L_2(\Omega ,H)$. As an application we consider elliptic systems of pure second order. Our main result shows that the induced semigroup is $L_p$-contractive for all $p\in [p_{-},p_{+}]$ for some $1<p_{-}<2<p_{+}<\infty$.