Distributions

Chapter 13Existence of Primitives

This chapter is dedicated to obtaining conditions for a distributions field q = (q₁, . . . , q_d) to have a primitive f , i.e. for ∇f = q. The main conditions are the following.

1. — On an arbitrary open set Ω, it suffices that q is orthogonal to divergence-free test fields, namely that 〈q, ψ〉 = 0 for every ψ such that ∇ · ψ = 0. This is the orthogonality theorem for distributions (Theorem 13.5).
2. — When Ω is simply connected, it suffices that q satisfies Poincaré’s condition ∂_iq_j = ∂_jq_i for all i and j. This is Poincaré’s theorem generalized to distributions (Theorem 13.7).

These conditions are necessary and sufficient.

We prove these results as follows.

1. — First of all, we reduce the existence of a primitive on Ω to the existence of a primitive on each of its subsets Ω_1/n = {x : B(x, 1/n) Ω}, i.e. on Ω with a neighborhood of its boundary of width 1/n removed, by a method of gluing. This is the peripheral gluing theorem (Theorem 13.1).
2. — Next, we reduce the existence of a primitive of a distribution field q to the existence of a primitive of a function field, namely q ◊ η_n on Ω_1/n, when q satisfies Poincaré’s condition. This is the theorem on reducing to the function case (Theorem 13.2).
3. — The orthogonality theorem for distributions then easily follows from the orthogonality theorem for functions (Theorem 11.4).
4. — Obtaining Poincaré’s theorem for distributions is more delicate, because the Ω_1/n are not necessarily simply connected, so we cannot ...