Brelot space

Summary

A Brelot space is a connected, locally connected space with a sheaf of harmonic functions, as in a harmonic space. In it, an increasing sequence of harmonic functions on a domain tends either to a harmonic function or to $+\infty$ everywhere. Its axioms, due to Marcel Brelot, model elliptic equations such as the Laplace equation. In a Brelot space a Harnack inequality holds, as for classical harmonic functions. The heat equation gives a harmonic space that is not a Brelot space.