Continuous Functions. Jacques Simon

      Introduction

      Objective. This book is the second of six volumes in a series dedicated to the mathematical tools for solving partial differential equations derived from physics:

      This second volume is devoted to the partial differentiation of functions and the construction of primitives, which is its inverse mapping, and to their properties, which will be useful for constructing distributions and studying partial differential equations later.

      Target audience. We intended to find simple methods that require a minimal level of knowledge to make these tools accessible to the largest audience possible – PhD candidates, advanced students¹ and engineers – without losing generality and even generalizing some standard results, which may be of interest to some researchers.

Originality. The construction of primitives, the Cauchy integral and the weighting with which they are obtained are performed for a function taking values in a Neumann space, that is, a space in which every Cauchy sequence converges.

      Neumann spaces. The sequential completeness characterizing these spaces is the most general property of E that guarantees that the integral of a continuous function taking values in E will belong to it, see Case where E is not a Neumann space (§ 4.3, p. 92). This property is more general than the more commonly considered property of completeness, that is the convergence of all Cauchy filters; for example, if E is an infinite-dimensional Hilbert space, then E-weak is a Neumann space but is not complete [Vol. 1, Property (4.11), p. 82].

      Moreover, sequential completeness is more straightforward than completeness.

      Semi-norms. We use families of semi-norms, instead of the equivalent notion of locally convex topologies, to be able to define differentiability (p. 73) by comparing the semi-norms of a variation of the variable to the semi-norms of the variation of the value. A section on Familiarization with Semi-normed Spaces can be found on p. xiii. Semi-norms can be manipulated in a similar fashion to normed spaces, except that we are working with several semi-norms instead of a single norm.

      Primitives. We show that any continuous field q = (q₁, . . . , q_d) on an open set Ω of ℝ^d has a primitive f, namely that ∇f = q, if and only if it is orthogonal to the divergence-free test fields, that is, if

for every ψ = (ψ₁, . . . , ψ_d) such that ∇ · ψ = 0. This is the orthogonality theorem (Theorem 9.2).

      When Ω is simply connected, for a primitive f to exist, it is necessary and sufficient for q to have local primitives. This is the local primitive gluing theorem (Theorem 9.4). On any such open set, it is also necessary and sufficient that it verifies Poincaré’s condition ∂_iq_i = ∂_jq_i for every i and j to be satisfied if the field is C¹ (Theorem 9.10), or a weak version of this condition,

for every test function φ, if the field is continuous (Theorem 9.11).

      We explicitly determine all primitives (Theorem 9.17) and construct one that depends continuously on q (Theorem 9.18).

      Integration. We extend the Cauchy integral to uniformly continuous functions taking values in a Neumann space, because this will be an essential tool for constructing primitives.

The properties established here for continuous functions will also be used to extend them to integrable distributions in Volume 4, by continuity or transposition. Indeed, one of the objectives of the Analysis for PDEs series is to extend integration and Sobolev spaces to take values in Neumann spaces. However, it seemed more straightforward to first construct distributions (in Volume 3) using just continuous functions before introducing integrable distributions (in Volume 4), which play the role usually fulfilled by classes of almost everywhere equal integrable functions.

      Weighting. The weighted function

of a function f defined on an open set Ω by the weight μ, a real function with compact support D, is a function defined on the open set

by

This concept will be repeatedly useful. It plays an analogous role to convolution, which is equivalent to it up to a symmetry of μ when

.

      Novelties. Many results are natural extensions of previous results, but the following seemed most noteworthy:

      — The construction of the topology of the space

of continuous functions with compact support using the semi-norms

indexed by

and

(Definition 1.17). This is equivalent to and much simpler than the inductive limit topology of the

.

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