The set of mildly regular boundary points has full caloric measure
NEIL A. WATSON · Transactions of the London Mathematical Society · 2023
Abstract We introduce a new kind of regularity in the Dirichlet problem for the heat equation, called mild regularity. The mildly regular boundary points include all types of regular points previously considered. Just as regular boundary points have barriers associated with them, the mildly regular boundary points have mild barriers, which are just as important. We consider whether the mild regularity of a boundary point for an open set is inherited by an open subset or superset. We detail the connection of mild regularity with the Green function. We consider, in particular, subdomains of an arbitrary open set that satisfy a certain connectivity condition which takes account of the special nature of the temporal variable, called zones of influence. Our main theorem states that, for any zone of influence, the set of boundary points that are not mildly regular is caloric measure null.