On local continuous solvability of equations associated to elliptic and canceling linear differential operators

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Abstract

Consider A(x,D):C(Ω,E)C(Ω,F) an elliptic and canceling linear differential operator of order ν with smooth complex coefficients in ΩRN from a finite dimension complex vector space E to a finite dimension complex vector space F and A(x,D) its adjoint. In this work we characterize the (local) continuous solvability of the partial differential equation A(x,D)v=f (in the distribution sense) for a given distribution f; more precisely we show that any x0Ω is contained in a neighborhood UΩ in which its continuous solvability is characterized by the following condition on f: for every ε>0 and any compact set KU, there exists θ=θ(K,ε)>0 such that the following holds for all smooth function φ supported in K:|f(φ)|θφWν1,1+εA(x,D)φL1, where Wν1,1 stands for the homogenous Sobolev space of all L1 functions whose derivatives of order ν1 belongs to L1(U).

This characterization implies and extends results obtained before for operators associated to elliptic complexes of vector fields (see [1]); we also provide local analogues, for a large range of differential operators, to global results obtained for the classical divergence operator by Bourgain and Brezis in [2] and by De Pauw and Pfeffer in [3].

Résumé

Soit A(x,D):C(Ω,E)C(Ω,F) un opérateur différentiel linéaire elliptique et annulant d'ordre ν à coefficients lisses dans ΩRN, d'un espace vectoriel complexe de dimension finie E dans un espace vectoriel complexe de dimension finie F, et soit A son adjoint. Dans ce travail, nous caractérisons la solvabilité continue (locale) de l'équation aux dérivées partielles A(x,D)v=f (au sens des distributions) pour une distribution donnée f ; plus précisément, nous montrons que chaque point x0Ω est contenu dans un voisinage UΩ dans lequel l'existence d'une solution continue est caractérisée par la propriété suivante sur f : pour tout ε>0 et tout compact KU, il existe une constante θ=θ(K,ε)>0 telle que l'on ait, pour toute fonction indéfiniment dérivable φ nulle hors de K :|f(φ)|θφWν1,1+εA(x,D)φL1,Wν1,1 désigne l'espace de Sobolev homogène des fonctions de L1(U) dont les dérivées d'ordre ν1 appartiennent à L1(U).

Cette caractérisation entraîne et étend des résultats obtenus auparavant pour des opérateurs associés à des complexes de champs de vecteurs (voir [1]) ; nous obtenons aussi des analogues locaux, pour une large classe d'opérateurs différentiels, à des résultats globaux obtenus pour l'équation de la divergence classique par Bourgain et Brezis dans [2] et par De Pauw et Pfeffer dans [3].

Introduction

Consider ΩRN an open set and A(,D) a linear differential operator of order ν with smooth complex coefficients in Ω denoted by:A(x,D)=|α|νaα(x)α:C(Ω;E)C(Ω;F), where E is a complex vector space of finite dimension n and F is a complex vector space of finite dimension nn.

A series of results concerning local L1 estimates for linear differential operators has been studied by J. Hounie and T. Picon in the setting of elliptic systems of complex vector fields, complexes and pseudocomplexes ([4], [5]). The following characterization of local L1 estimates for operators A(x,D) was proved in [6], namely:

Theorem 1.1

Assume, as before, that A(,D) is a linear differential operator of order ν with smooth1 complex coefficients in Ω between the spaces E and F. The following properties are equivalent:

  • (i)

    A(x,D) is elliptic and canceling (see below for a definition of those properties);

  • (ii)

    every point x0Ω is contained in a ball B=B(x0,r)Ω such that the a priori estimateuWν1,N/(N1)CA(x,D)uL1, holds for some C>0 and all smooth functions uC(B;E) having compact support in B.

Here, given kN and 1p, Wk,p(Ω) denotes the homogeneous Sobolev space of complex functions in Lp(Ω) whose weak derivatives of order k belong to Lp(Ω), endowed with the (semi-)norm uWk,p:=|α|=kαuLp.

It turns out that elliptic linear differential operators that satisfy an a priori estimate like (1) can be characterized in terms of properties of their principal symbol aν(x,ξ)=|α|=νaα(x)ξα. Recall that the ellipticity of A(x,D) at x0Ω means that for every ξRN{0} the map aν(x0,ξ):EF is injective.

Definition 1.2

Let x0Ω. A linear partial differential operator A(x,D) of order ν from E to F with principal symbol aν(x,ξ) that satisfies:ξRN{0}aν(x0,ξ)[E]={0} is said to be canceling at x0. If () holds for every x0Ω we say that A(x,D) is canceling.

Examples of canceling operators satisfying () can be founded in [6]; this is the case in particular for operators associated to elliptic system of complex vector fields (see [4], [5]). The canceling property for linear differential operators was originally defined by Van Schaftingen [7] in the setup of homogeneous operators with constant coefficients A(D) and stands out by several applications (and characterizations) in the theory of a priori estimates in L1 norm (see for instance [8] for a brief description).

In this work, we are interested to study the (local) continuous solvability in the weak sense of the equation:A(x,D)v=f, where A(x,D) is an elliptic and canceling linear differential operator. We use the notation A:=At where A denotes the operator obtained from A by conjugating its coefficients and At is its formal transpose — namely this means that, for all smooth functions φ and ψ having compact support in Ω and taking values in E and F respectively, we have:ΩA(x,D)φψ¯=ΩφA(x,D)ψ. We shall say, in the sequel, that equation (2) is continuously solvable in the open set UΩ if there exists vC(U,F) such that, for all φD(U;E), one has:UA(x,D)vφ=Ufφ.

The main result of the present paper is the following. It generalizes previous results on the classical divergence equation (see e.g. Bourgain and Brezis [2] and De Pauw and Pfeffer [3]) as well as a work by the current authors for divergence-type equations (of order 1) associated to elliptic systems of (complex) vector fields (see [1]).

Theorem 1.3

Assume that A(x,D) is an elliptic and canceling linear differential operator. Then every point x0Ω admits an open neighborhood UΩ such that for any fD(U;E), the equation (2) is continuously solvable in U if and only if f is an A-charge in U, meaning that for every ε>0 and every compact set KU, there exists θ=θ(K,ε)>0 such that one has:|f(φ)|θφWν1,1+εA(x,D)φL1, for any φDK(U;E) — where the latter notation denotes the space of smooth function φ in U vanishing outside K.

One simple argument (see Section 5) shows that the above continuity property on f is a necessary condition for the continuous solvability of equation (2) in U. Theorem 1.3 asserts that the continuity property (3) is also sufficient, under the canceling and ellipticity assumptions on the operator. The proof, which will be presented in Section 6, is based on a functional analytic argument inspired from [3] and already improved in [1] for divergence-type equations associated to complexes of vector fields (observe in particular that one recovers [1, Theorem 1.2] when applying Theorem 1.3 to the latter context). However, it should be mentioned here that by allowing in (2) a much larger class of (higher order) differential operators, that method of proof had to be very substantially refined, leading to the use of new tools. Applications of Theorem 1.3 are presented in the Section 7.

We point out that, in the statement of Theorem 1.3, the assumption that the elliptic operator A(x,D) is also canceling — which is characterized by inequality (1) — plays a fundamental role in our proof of Theorem 1.3, as it is the key ingredient in obtaining the generalized (local) L1 Sobolev-Gagliardo-Nirenberg inequality expressed by Proposition 4.10 and derived from (1). However, we should emphasize that this property might not be necessary to obtain a characterization of continuous solutions to the equation (2) formulated along the previous lines. In the context of the Poisson equation with measure data Δu=μ, it follows indeed from a result by Aizenman and Simon [9, Theorem 4.14] (see also Ponce [10, Proposition 18.1], where this question is studied in a luminous fashion) that, given a smooth, bounded open set Ω in Rn and a measure μ in Ω, the Dirichlet problem associated with Δu=μ in Ω has a continuous solution in Ω¯ if and only if, for every ε>0, there exists θ>0 such that for any φC0(Ω¯), one has:|Ωφdμ|θφL1+εΔφL1. This result, however, is proved using very different techniques than the ones developed here; it does not follow from our results since the scalar Laplacean operator is not canceling.

Both authors would like to thank the referee for her/his careful reading of our manuscript, and for all useful suggestions and remarks that were formulated to the authors.

Section snippets

Preliminaries and notations

We always denote by Ω an open set of RN, N2. Unless otherwise specified, all functions are complex valued and the notation Af stands for the Lebesgue integral Af(x)dx. As usual, D(Ω) and D(Ω) are the spaces of complex test functions and distributions, respectively. When KΩ is a compact subset of Ω, we let DK(Ω):=D(Ω)E(K), where E(K) is the space of all distributions with compact support in K. Since the ambient field is C, we identify (formally) each fLloc1(Ω) with the distribution TfD

Canceling and elliptic differential operators

Given A(x,D) as before, the 2ν-order differential operator ΔA:=A(,D)A(,D) may be regarded as an elliptic pseudodifferential operator with symbol in the Hörmander class S2ν(Ω):=S1,02ν(Ω),2 so that there exist properly supported pseudodifferential operators q,q˜OpS2ν(Ω) (parametrixes) and r,r˜OpS(Ω) for

Basic definitions; approximation and compactness

Let Wck,p(Ω;E) be the linear space of all complex functions in Wk,p(Ω;E) whose support is a compact subset of Ω.

The following definition of variation associated to A(x,D) of gWcν1,1(Ω;E) recalls the classical definition of variation when ν=1 and A(x,D)=. It has been formulated for real vector fields by N. Garofalo and D. Nhieu [14] and adapted for complex vector fields in [1].

Definition 4.1

Assume that A(x,D) is a linear differential operator of order ν from E to F having coefficients with derivatives of

A-charges and their extensions to BVA,c

We now get back to the original problem of finding, locally, a continuous solution to (2).

Towards Theorem 1.3

Throughout this section, we assume that the open set UΩ supports inequalities of type (1) and (5); we also assume that one has Λ[D(U;E)]Γ[C(U,F)].

Remark 6.1

It follows from Theorem 1.1, Proposition 3.1 and Remark 5.10 that for any x0Ω, one can find an open neighborhood U of x0 in Ω satisfying all the above assumptions.

Our intention is to prove the following result.

Theorem 6.2

If F:BVA,c(U)F is an A-charge in U, then there exists vC(U,F) for which one has F=Γ(v), i.e. such that one has, for any gBVA,c(U):F(g)=

Application: elliptic complexes of vector fields

Consider n complex vector fields L1,,Ln, n1, with smooth coefficients defined on an open set ΩRN, N2. Naturally, we assume that the vector fields Lj, 1jn do not vanish in Ω; in particular, they may be viewed as non-vanishing sections of the vector bundle CT(Ω) as well as first order differential operators of principal type.

We impose two fundamental properties on those vector fields in our context; namely, we require that:

  • (a)

    L1,,Ln are everywhere linearly independent;

  • (b)

    the system L:={L1,,Ln}

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    This work was partially supported by the “bilateral mobility” program of the Brazilian-French Network in Mathematics (RFBM-GDRI) and the FAPESP grants nos. 2017/17804-6, 2018/15484-7 and 2019/21179-5.

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