# Non-autonomous maximal regularity for divergence-form operators

Let $V \subseteq H$ be Hilbert spaces with dense and continuous embedding. An old problem of J.L. Lions asks for maximal regularity in $H$ of the non-autonomous Cauchy problem $u’(t) + A(t) u(t) = f(t)$, $u(0) = 0$, where each operator $A(t)$ is induced by an elliptic sesquilinear form on $V$. Recent developments have carved out a threshold on the regularity of $A$ as a map $[0,T] \to \mathcal{L}(V, V^*)$: Lions’ question can be answered in the affirmative in case of Hölder-continuity of exponent $\alpha > 1/2$ and there exist counterexamples if only $\alpha < 1/2$. The borderline case, however, was left open, even if all operators are differential operators in divergence-form. In this talk we present a rather simple proof of such a result stemming on some hidden coercivity of the parabolic operator $\partial_t + A$.

This talk is part of the Analysis Seminar series.