Regularity for non-autonomous parabolic equations with right-hand side singular measures involved

Abstract

This article provides a theory for non-autonomous parabolic equations the right hand side of which includes singular measures - depending on the time parameter - on the spatial domain. In two space dimensions all bounded Radon measures are admissable as such. In higher dimensions the focus is on measures whose support is concentrated on l-sets in the sense of Jonsson and Wallin. It is shown that they may interpreted as elements from a Sobolev space W. So the right hand side is considered as an element from a W-valued Lebesgue space on the time interval. Having this at hand, previous results on maximal (non-autonomous) maximal parabolic regularity apply and show that the solution lies in the corresponding space of maximal parabolic regularity. In contrast to other work in this field we only require absolute minimal smothness for the data of the problem: the domain, the coefficients - and mixed boundary conditions are allowed. Under minimally stronger assumptions we even show the Hölder property in space and time. Overall, this work contains an interplay of geometric measure theory with advanced parabolic theory which delivers as much parabolic regularity for the solution as one can maximally expect.