Brelot space

Remarks

Clause 3 implies the convergence axiom of a harmonic space: a locally bounded limit cannot be identically $+\infty$ on any component. Example: $\R^n$ with the classical harmonic functions is a Brelot space. Balls are regular domains, by the Poisson integral, and clause 3 is Harnack's principle. Non-example: $\R \times \R$ with the solutions of the heat equation $\partial_t u = \partial_x^2 u$ fails clause 3. Let $\Phi(x, t) = (4\pi t)^{-1/2} e^{-x^2/(4t)}$ for $t > 0$ and $\Phi(x, t) = 0$ for $t \le 0$, the heat kernel extended by $0$. It solves the heat equation on the domain $U = \R^2 \setminus \set{(0, 0)}$, and $h_n = n \Phi$ increases to $+\infty$ where $t > 0$ and to $0$ where $t \le 0$. The limit is neither a solution on $U$ nor identically $+\infty$.