Maximum Principles and Initial Limits

In Theorem 4.5, we proved that any Gauss-Weierstrass integral of a function has certain initial limits m-a. e. on R̲ⁿ. We shall use this result to prove a similar theorem for temperatures on R̲ⁿ × ]0, a[ which are not necessarily the Gauss-Weierstrass integrals of functions. This will imply that Theorem 4.5 can be extended to all Gauss-Weierstrass integrals of measures. In order to do this, we need a maximum principle for more general bounded open sets than was given in Theorem 2.1. We shall, in fact, prove two forms of the maximum principle, both of which are stronger and more generally applicable than is necessary for our application. The first, called the strong maximum principle, asserts that a temperature u on a bounded open set E can have a maximum at a point of E only if u is constant on a particular subset of E. The second, called the weak maximum principle, asserts that, if the values of u(x, t), as (x, t) approaches ∂Е in a particular way, have an upper bound, then the values of u(x, t) for all (x, t) ∈ E have the same upper bound.