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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:
Volume 1: Banach, Frechet, Hilbert and Neumann Spaces;
Volume 2: Continuous Functions;
Volume 3: Distributions;
Volume 4: Lebesgue and Sobolev Spaces;
Volume 5: Traces;
Volume 6: Partial Differential Equations.
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, . . . , qd) on an open set Ω of ℝd 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, . . . , ψd) 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 ∂iqi = ∂jqi 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 .
— The fact that if a function satisfies for every , then its support is compact (Theorem 1.22). This is the basis for defining the semi-norms of in Volume 3.
— The concentration theorem for the integral and the construction of an incompressible tubular flow (Theorems 8.18 and 8.17), which are key steps in our construction of the primitives of a field taking values in a Neumann space, as it is explained in the comment Utility of the concentration theorem, p. 186.
Prerequisites. The proofs in the main body of the text only use definitions and results established in Volume 1, whose statements are recalled either in the text or in the Appendix. Detailed proofs are given, including arguments that may seem trivial to experienced readers, and the theorem numbers are systematically referenced.
Comments. Comments with a smaller font size than the main body of the text appeal to external results or results that have not yet been established. The Appendix on Reminders is also written with a smaller font size, since its contents are assumed to be familiar.
Historical notes. Wherever possible, the origin of the concepts and results is given as a footnote2.
Navigation through the book:
— The Table of Contents at the start of the book lists the topics discussed.
— The Table of Notations, p. xv, specifies the meaning of the notation in case there is any doubt.
— The Index, p. 243, provides an alternative access to specific topics.
— All hypotheses are stated directly within the theorems themselves.
— The numbering scheme is shared across every type of statement to make results easier to find by number (for instance, Theorem 2.9 is found between the statements 2.8 and 2.10, which are a definition and a theorem, respectively).
Acknowledgments. Enrique FERNÁNDEZ-CARA suggested to me a large number of improvements to various versions of this work. Jérôme LEMOINE was kind enough to proofread the countless versions of the book and correct just as many mistakes and oversights.
Olivier BESSON, Fulbert MIGNOT, Nicolas DEPAUW, and Didier BRESCH also provided many improvements, in form and in substance.
Pierre DREYFUSS gave me insight into the necessity of simply connected domains for the existence of primitives with Poincaré’s condition, as explained on p. 209 in the comment Is simple connectedness necessary for gluing together local primitives?
Joshua PEPPER spent much time discussing about the best way to adapt this work in English.
Thank you, my friends.
Jacques SIMON
Chapdes-Beaufort
April 2020
Notes
1 1 Students? What might I have answered if one of my MAS students in 1988 had asked for more details about the de Rham duality theorem that I used to obtain the pressure in the Navier-Stokes equations? Perhaps I could say that “Jacques-Louis LIONS, my supervisor, wrote that it follows from the de Rham cohomology theorem, of which I understand neither the statement, nor the proof, nor why it implies the result that we are using.” What a despicably unscientific appeal to authority!This question was the starting point of this work: writing proofs that I can explain to my students for every result that I use. It took me 5 years to find the “elementary” proof of the orthogonality theorem (Theorem 9.2, p. 194) on the existence of the primitives of a field q. I needed a way to obtain fr q • Ai = 0 for every closed path r from the condition fn q • y = 0 for every divergence-free y. It gave me the greatest mathematical satisfaction I have ever experienced to explicitly construct an incompressible tubular flow (see p. 184). Twenty-five years later, I am finally ready to answer any other questions from my (very persistent) students.
2 2 Appeal to the reader. Many important results lack historical notes because I am not familiar with their origins. I hope that my readers will forgive me for these omissions and any injustices they may discover. And I encourage the scholars among you to notify me of any improvements for future editions!
