Some properties of (C,E,P)-algebras : Overgneration and 0-order estimates.

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Some properties of (C,E,P)-algebras : Overgneration and 0-order estimates.

Antoine Delcroix

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Antoine Delcroix. Some properties of (C,E,P)-algebras : Overgneration and 0-order estimates.. 2008.

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Some properties of (C,E,P)-algebras : Overgneration and 0-order estimates.

Antoine Delcroix ^∗ October 4, 2008

Abstract

We give a new definition of the so-called overgenerated rings, which are the usual tool used to define the asymptotic structure of a (C,E,P)-algebra, written as a factor space M(A,E,P)/N(I_A,E,P). With this new definition and in the particular case of E = C^∞, we show that a moderate element i.e. in M(A,E,P) is negligible if and only if it satisfies the C⁰-order estimate for the ideal N_(IA,E,P).

Mathematics Subject Classification (2000): 35A20, 35A25, 35D05, 46F30, 46T30 Keyword: Properties of non linear generalized functions, (C,E,P)-algebra.

1 Introduction

More than ten years ago, J.-A. Marti has introduced the structure of (C,E,P)-algebras [8] which is a refinement of the Colombeau simplified algebras of new generalized functions [2, 6, 10].

These (C,E,P)-algebras are constructed on a base (pre)sheaf E, which is usually a (pre)sheaf of algebras equipped with a topology P and on an asymptotic structure given by a ring C of generalized constants. Except in a few cases (for example [1]), the sheaf E is chosen to be the sheaf of smooth functions. Roughly speaking a presheaf of (C,E,P)-algebras is a presheaf of factor algebrasM_(A,E,P)/N_(IA,E,P), whereM_(A,E,P)(respN_(IA,E,P)) is the (pre)sheaf of algebras of moderate elements (resp the (pre)sheaf of ideals of negligible elements) over the factor ring C = A/IA. (The moderateness and the negligibility are defined by the asymptotic structure given by the ringC.)

Although these (C,E,P)-type algebras have proved their efficiency to give a meaning and to solve singular differential problems, the investigations on their intrinsic properties have not yet been developed, expect first attempts concerning topology in [4] and asymptotic analysis in [3].

In this paper, after recalling for sake of self contentedness the basic notions on (C,E,P)-type algebras (Section 2), we present a new definition of the so-called overgenerated rings in Section 3. This concept of overgeneration is one essential part of the theory for it allows to adapt the algebraic structure to the singularities of the problems. Taking advantage of this new definition and analogously to the theorem 1.2.3. of [6] for Colombeau simplified algebras, we show in Section 4 that a moderate element (i.e. inM_(A,E,P)) is negligible (i.e. inN_(IA,E,P)) if and only if it satisfies the C⁰-order estimate of the ideal, for the case ofE = C^∞ endowed with its classical topology. This property allows radical simplification of many proofs of existence and uniqueness for differential problems.

∗Equipe AANL, Laboratoire AOC – Campus de Fouillole, Universit´e des Antilles et de la Guyane, 97159 Pointe-`a-Pitre, Guadeloupe (France), E-mail: [email protected]

2 The presheaf of (C,E,P)-type algebras

We begin by recalling the algebraical construction of (C,E,P)-algebras [8, 9] as improved in [3].

Let:

(1) Λ be a directed set with partial order relation;

(2) A be a solid subring of the ring K^Λ (K=R or C): whenever (|sλ|)λ ≤ (|rλ|)λ for some ((sλ)λ,(rλ)λ)∈K^Λ×A, that is,|sλ| ≤ |rλ|for all λ, it follows that (sλ)λ ∈A;

(3)IAbe a solid ideal of A;

(4)E be a sheaf ofK-topological algebras over a topological space X.

Suppose that for any open set Ω inX, the topology of the algebraE(Ω) is defined by a family P(Ω) of seminorms such that:

(5) Whenever Ω₁, Ω₂ are two open subsets ofX with Ω₂⊂Ω₁and ρ¹₂ is the restriction operator E(Ω1)→ E(Ω2), then, for each p₂ ∈ P(Ω2), the seminormp₁=p₂◦ρ¹₂ extends p₂ to P(Ω1);

(6) Whenever Θ = (Ω_h)_h∈H is a family of open sets in X with Ω = ∪h∈HΩ_h, then, for each p∈ P(Ω), there exist a finite subfamily (Ωi)_1≤i≤nof Θ and corresponding seminormspi ∈ P(Ωi), 1≤i≤n, such that

∀u∈ E(Ω), p(u)≤p₁(u|_Ω1) +. . .+pn(u|_Ωn).

Define C =A/IA and |B|={(|rλ|)_λ, (rλ)_λ ∈B} (B =A or IA). From (2), it follows that

|A|is a subset ofA and that A₊={(bλ)λ∈A,∀λ∈Λ, bλ ≥0}=|A|. The same holds for IA. Furthermore, (2) implies also that Ais a K-algebra [3]. With these notations, set

M(Ω) =M_(A,E,P)(Ω) =n

(uλ)λ ∈[E(Ω)]^Λ| ∀p∈ P(Ω), ((p(uλ))_λ ∈ |A|o , N(Ω) =N_(IA,E,P)(Ω) =n

(u_λ)_λ ∈[E(Ω)]^Λ| ∀p∈ P(Ω), (p(u_λ))_λ ∈ |IA|o .

Proposition-Definition 1 [3, 8]

(i) M_(A,E,P) (resp. N_(IA,E,P)) is a sheaf of K-subalgebras (resp. of ideals) of the sheafE^Λ (resp.

of M_(A,E,P)).

(ii) The factorM_(A,E,P)/N_(IA,E,P) is a presheaf of algebras over the factor ring C=A/IA, with localization principle, called presheaf of (C,E,P)-algebras.

Remark that, with (2), the constant sheafM_(A,K,|.|)/N_(IA,K,|.|) is exactly equal toC=A/I_A. Notation 1 We denote by [(uλ)_λ]_A= [uλ]_A or [uλ], when no confusion may arise, the class of (uλ)_λ∈Λ in A(Ω).

Remark 1 We suppose in addition that {(aλ)_λ ∈A |lim_Λa_λ= 0} 6=∅ and that I_A satisfies (7) IA⊂ {(aλ)_λ∈A |lim_Λaλ = 0},

Then there exists a canonical sheaf embedding of E into A through the morphism of algebra σ_Ω:E(Ω)→ A(Ω), f 7→[(f)_λ].

Indeed, if [(f)_λ] = 0, we have: ∀p ∈ P(Ω), (p(f))_λ ∈ |IA|. From (7), it follows that ∀p ∈ P(Ω), p(f) = 0. Thus f = 0.

Remark 2 For the above algebraic considerations of this section (and specially Proposition- Definition 1), we don’t need Λ to be a directed set. However, the previous remark shows the importance of this assumption in order to get non trivial extensions.

3 Overgenerated rings

In almost all works using (C,E,P)-type algebras (see, for examples, [5, 8, 9]), the ring A and the ideal IAare constructed as polynomially overgenerated rings and satisfy the assumption (7) of Remark 1. For (aλ)_λ,(bλ)_λ ∈R^Λ, we shall use the following notation

aλ ≪bλ⇔ ∃λ0 ∈Λ, ∀λλ0 :aλ≤bλ. We first give an improved definition of the overgeneration.

Proposition-Definition 2 (Polynomially overgenerated rings)ConsiderB₀a family of nets in (R^∗

+)^Λ and B the subset of elements in(R^∗

+)^Λ obtained as rational functions with coefficients in R^∗

+ of elements inB0 as variables. Set A_B =

(a_λ)_λ ∈K^Λ | ∃(b_λ)_λ ∈ B:|aλ| ≪b_λ .

The set A_B is a solid subring of K^Λ, called the ring (polynomially) overgenerated byB0 (or by B).

Usually, the set B0 is finite and given by the problem itself. (See [5, 9].) The term polyno- mially refers to the fact that the growth of elements ofA_B is at most polynomial with respect to the elements ofB0. This polynomial overgeneration is sufficient for the non linearities considered in the quoted references, but, for example, does not permit to obtain (C,E,P)-algebras stable by exponential.

Remark 3 With this definitionBis stable by inverse. In many practical cases and, for example, in the case of Colombeau simplified algebras, which are a particular case of (C,E,P)-algebras, B is exactly the set of invertible elements of the ring of generalized constants.

As a “canonical” ideal of A_B, one usually choose

(8) I_B =

(aλ)_λ ∈K^Λ| ∀(bλ)_λ∈ B :|aλ| ≪bλ .

A routine checking shows that I_B is a solid ideal of A_B. We shall always assume the existence of (rλ)_λ ∈ B such that lim_Λrλ = 0, in order to have (7) and, thus, the canonical embedding of E(Ω) into A(Ω). (This assumption is satisfied in all practical applications.) We denote by CB =AB/IB the corresponding ring of generalized numbers.

4 An Austrian Lemma in (C,E,P)-algebras

We take hereE = C^∞withX=R^d,P R^d

being the usual family of seminorms (PK,l)_K,l defined by

P_K,l(u) = sup

|α|≤l

PK,α(u) with PK,α(u) = sup

x∈K

|∂^αu(x)|, K ⊂⊂Ω, l= 0 or l= 1.

and ∂^α = ∂^α1+...+αd

∂z^α₁¹...∂z_d^αd forz= (z₁, . . . , z_d)∈Ω,l∈N,α= (α₁, ..., α_d) ∈N^d. We consider a ring of generalized constants C = A_B/I_B overgenerated as stated in Proposition-Definition 2. The idealI_B is defined by (8) and the set of indices Λ is assumed to be left filtering. Recall that

M(R^d) =M_(AB,C^∞_,P)(R^d) ={(uλ)_λ∈C^∞(Ω)^Λ:∀p∈ P(R^d), (p(u_λ))_λ∈ |AB|}, N(R^d) =N_(IB,C^∞_,P)(R^d) ={(uλ)_λ ∈C^∞(Ω)^Λ:∀p∈ P(R^d), (p(uλ))_λ ∈ |I_B|}.

Proposition 3 Assume that there exists(aλ)_λ ∈ Bwithlim_Λaλ = 0.Then(uλ)_λ∈ M_(AB,C^∞_,P)(R^d) is in N_(IB,C^∞_,P)(R^d) if, and only if,

∀K ⊂⊂R², P_K,0(u_λ)∈ |IB|.

Remark 4 We recall that the set B is stable by inverse, which could be assumed for all the (C,E,P)-algebras considered up to now in the literature. Notice also that one has the existence of (aλ)_λ ∈ B such that lim_Λaλ = 0 in all practical cases.

Proof. Take K ⊂⊂ Ω. We have to prove that ∀l ∈ N, PK,l(uλ) ∈ |I_B|. By induction, it suffices to prove thatP_K,0(u_λ)∈ |IB|impliesP_K,1(u_λ)∈ |IB|. In fact, this amounts to show that P_K,0(u_λ)∈ |IB| implies P_K,0((∂/∂x_i)u_λ)∈ |IB| fori∈ {1, . . . , d}. Set δ = min(1,dist(K, ∂Ω)) and L=K+B(0, δ/2). We have K⊂⊂L⊂⊂Ω. Since (uλ)_λ ∈ M(R^d), there exists (βλ)_λ ∈ B such that

∃λ0∈Λ,∀λλ₀, P_L,2(u_λ)≤β_λ.

We may assume that lim_Λβλ = +∞. Indeed, for any (βλ)_λ ∈ B, we set β^′_λ =a⁻¹_λ +βλ where (a_λ)_λ ∈ B is such that lim_Λa_λ = 0. Thus, lim_Λmaxβ_λ^′ = +∞. Take any (cλ)_λ ∈ B and define b_λ = a_λc_λ/(a_λ+c_λ). Clearly we have b_λ ∈ |AB|, b_λ ≤c_λ and b_λ ≤ a_λ. Thus lim_Λb_λ = 0. Let (ei)_1≤i≤dbe the canonical base ofR^d. There exists λ₁ such that, for allx∈K,x+bλβ_λ⁻¹ei ∈L when λλ₁, since lim_Λβ_λ⁻¹ = 0. By the Taylor theorem we have, forx∈K,

u_λ x+b_λβ_λ⁻¹e_i

=u_λ(x) +b_λβ_λ⁻¹ ∂

∂x_iu_λ(x) + 1

2 b_λβ_λ⁻¹2 ∂²

∂x²_iu_λ x+θb_λβ⁻¹_λ e_i with 0≤θ≤1. It follows that

∂

∂xi

u_λ(x) =b⁻¹_λ β_λ u_λ x+b_λβ_λ⁻¹ei

−u_λ(x)

−1

2 b_λβ_λ⁻¹ ∂²

∂x²_iu_λ x+θb_λβ_λ⁻¹ei .

Thus

∂

∂xi

uλ(x)

≤2b⁻¹_λ βλP_L,0(uλ) +1

2bλβ_λ⁻¹P_L,2(uλ)≤2b⁻¹_λ βλP_L,0(uλ) + 1 2bλ

for λλ₂ with λ₂ λ_j, 0≤j ≤1. As P_K,0(u_λ) ∈ |IB|, we have P_L,0(u_λ) ≤(1/4)b²_λβ⁻¹_λ ∈ B forλλ₃ for someλ₃. Thus

∂

∂xi

u_λ(x)

≤b_λ forλλ₄ withλ₄λ_j, 3≤j≤4.

Finally, P_K,0((∂/∂xi)uλ)∈ |I_B|as expected.

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