Introduction

Objective. This book is the third of seven volumes dedicated to solving partial differential equations in physics:

Volume 1: Banach, Frechet, Hilbert and Neumann Spaces

Volume 2: Continuous Functions

Volume 3: Distributions

Volume 4: Integration

Volume 5: Sobolev Spaces

Volume 6: Traces

Volume 7: Partial Differential equations

This third volume aims to construct the space of distributions with real or vectorial values and to provide the main properties that are useful in studying partial differential equations.

Intended audience. We1 have looked for simple methods that require a minimal level of knowledge to make this tool accessible to as wide an audience as possible - doctoral students, university students, engineers - without loosing generality and even generalizing certain results, which may be of interest to some researchers.

This has led us to choose an unconventional approach that prioritizes semi-norms and sequential properties, whether related to completeness, compactness or continuity.

Utility of distributions. The main advantage of distributions is that they provide derivatives of all continuous or integrable functions, even those which are not differentiable, and thus broaden the scope of application of differential calculus. This is especially useful for solving partial differential equations.

To this end, a family of objects, the distributions, is defined, with the following properties.

• - Any continuous function is a distribution.

• - Any distribution has partial derivatives, which are distributions.

• - For a differentiable function, we find the conventional derivatives.

• - Any limit of distributions is a distribution.

• - Any Cauchy sequence of distributions has a limit.

These properties may be roughly summarized by saying that the space ´ of distributions is the completion with respect to derivation of the space of continuous functions. This construction, due to Laurent SCHWARTZ, [69] and [72], is completed here for distributions on an open subset O of d with values in a Neumann space E, i.e. a sequentially complete separable semi-normed space. This includes values in a Banach or Fréchet space.

Originality. The quest for simple methods2 giving general properties led us to proceed as follows.

• - Directly consider vectorial values, i.e. constructing ´(O; E) without any prior study of real distributions.

• - Assume that E is sequentially complete, i.e. a Neumann space.

• - Use semi-norms to construct the topologies of E, (O), ´(O; E), etc.

• - Equip ´(O; E) with the simple topology.

• - Introduce weighting to generalize the convolution to open domains.

• - Explicitly construct the primitives.

• - Separate the variables using a "basic" method.

• - Only use integration for continuous functions.

Let us take a closer look at these points that lie off the beaten track.

Vector values. We consider distributions with values in a general Neumann space E even though the partial differential equations in physics generally have real values. This is useful in evolution equations to separate the time t from the variable of space x. A distribution over t, x with real values is then identified with a distribution over t with values in a space E of distributions on x, for example, with an element of ´((0, T); E) where E = ´(O), which is itself a Neumann space. This identification is made possible by the fundamental kernel theorem, p. 312.

A list of the most useful Neumann spaces is given on page 43.

For stationary equations, the real distributions (that is, the case where E = ) are sufficient. We will directly work on the case where E is a Neumann space in order to avoid repetitions, the generalization often consisting of replacing with E and the absolute value | | with a semi-norm of E in the statements and proofs, when using appropriate methods.

Particular features in the case of vector values. The main differences as compared to distributions with real values are as follows, for a general space E.

• - The space ´(O; E) is not reflexive and its topology of pointwise convergence on (O) does not coincide with its weak topology.

• - The bounded subsets of ´(O; E) are not relatively compact.

• - The distributions over O are not of a finite order over its compact parts: they cannot always be expressed as finite order derivatives of continuous functions.

• - Variables may be separated by constructing a bijection from ´(O1 × O2; E) onto ´(O1; ´(O2; E)) (even for a real distribution, i.e. for E = , this brings in vector values, in this case in ´(O2)).

Sequential completeness. We assume that E is a Neumann space, i.e. that all its Cauchy series converge, since this is an essential condition for continuous functions to be distributions. That is, for (O; E) ? ´(O; E), see section 3.4, The case where E is not a Neumann space, p. 53.

This property is simpler than the completeness, i.e. the convergence of all the Cauchy filters, and is especially more general: for example, if H is a Hilbert space with infinite dimensions, H-weak is sequentially complete but is not complete [Vol. 1, Property (4.11), p. 63].

It is also simpler and more general than quasi-completeness, i.e. the completeness of bounded subsets, used by Laurent SCHWARTZ [72, p. 2, 50 and 52].

Semi-norms. We use families of semi-norms rather than locally convex topologies, which are equivalent, in order to be able to define Lp(O; E) in Volume 4. Indeed, it is possible to raise a semi-norm to a power p, but not a convex neighborhood!

The handling of semi-normed spaces is simple, although it is less familiar than that of topological spaces: it follows the handling of normed spaces, the main difference being that there are several semi-norms or norms instead of a single norm. For example, we bring in the topology of (O) through the family of semi-norms indexed by p ? + (O), which is much simpler than its (equivalent) construction as the inductive limit of the K(O).

Simply topology. We equip the space ´(O; E) with the family of semi-norms indexed by f ? (O) and ? ? E (set indexing the semi-norms of E), i.e. with the topology of simple convergence on (O), as it is well-suited to our study . . . and is simple. This simplicity is achieved without restricting ourselves to a pseudo-topology as is done in several texts.

In addition, this topology has the same convergent sequences and the same bounded sets as the topology of uniform convergence on the bounded subsets of (O) used by Laurent SCHWARTZ. The reasons for our choices are detailed on p. 45.

Open domain and weighting. We consider distributions defined on an open subset O of d. As these do not necessarily have an extension to all of d, we introduce an operation, we call it weighting, which plays a role for O that is similar to the role played by convolution for d and which we constantly use.

The weighted distribution f µ of a distribution f, defined on an open set O, by a weight µ, which is a real distribution on d with a compact support D, is a distribution defined on the open set OD ?= {x ? d : x + D ? O}. When f and µ are functions, it is given by When O = d, the convolution is recovered up to a symmetry on µ, and all its properties are recovered up to a possible sign.

Primitives. We show that a field of distributions q = (q1, . . . , qd) has a primitive f, that is ?f = q, if and only if it satisfies <q, ?> = 0E for all the test fields ? = (?1, . . . , ?d) such that ? . ? = 0. It is the orthogonality theorem. We explicitly determine all the primitives and among these determine one which depends continuously on q.

We also demonstrate that when O is simply connected it is necessary and sufficient that ?iqj = ?jqi for all i and j. It is the Poincaré's generalized theorem.

Separation of variables. We show that the separation of variables is bijective from ´(O1 O2; E) onto ´(O1; ´(O2; E)) by means of inequalities. These are certainly laborious to establish, but they avoid the difficult topological properties used by Laurent SCHWARTZ in...