J.-L. Lions' Problem Concerning Maximal Regularity of Equations Governed by Non-Autonomous Forms

TL;DR

This paper investigates the conditions under which maximal regularity holds for non-autonomous equations governed by forms, providing a negative answer to longstanding questions and clarifying the minimal regularity requirements.

Contribution

It demonstrates that mere continuity or measurability of the time dependence is insufficient for maximal regularity, and identifies the minimal regularity needed for positive results.

Findings

Maximal regularity does not hold under only continuous or measurable time dependence.

Minimal regularity conditions necessary for maximal regularity are established.

The paper clarifies longstanding open problems from J.-L. Lions' research.

Abstract

An old problem due to J.-L. Lions going back to the 1960s asks whether the abstract Cauchy problem associated to non-autonomous forms has maximal regularity if the time dependence is merely assumed to be continuous or even measurable. We give a negative answer to this question and discuss the minimal regularity needed for positive results.