The two versions of the Dirichlet problem for the heat equation
There are two versions of the Dirichlet problem for the heat equation on an arbitrary open set in Euclidean space. For one of them, there is already a characterization of resolutivity in terms of caloric measure. We prove that there is a similar characterization for the other, that the measure involved is essentially the same caloric measure, and that a boundary function is resolutive with respect to one version of the problem if and only if it is resolutive with respect to the other. We also prove that, for any boundary function, the upper solutions for the two versions coincide.