On local continuous solvability of equations associated to elliptic and canceling linear differential operators☆
Introduction
Consider an open set and a linear differential operator of order ν with smooth complex coefficients in Ω denoted by: where E is a complex vector space of finite dimension n and F is a complex vector space of finite dimension .
A series of results concerning local 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 estimates for operators was proved in [6], namely:
Theorem 1.1 Assume, as before, that 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: is elliptic and canceling (see below for a definition of those properties); every point is contained in a ball such that the a priori estimate holds for some and all smooth functions having compact support in B.
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 . Recall that the ellipticity of at means that for every the map is injective.
Definition 1.2 Let . A linear partial differential operator of order ν from E to F with principal symbol that satisfies: is said to be canceling at . If holds for every we say that 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 and stands out by several applications (and characterizations) in the theory of a priori estimates in 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: where is an elliptic and canceling linear differential operator. We use the notation where denotes the operator obtained from A by conjugating its coefficients and 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: We shall say, in the sequel, that equation (2) is continuously solvable in the open set if there exists such that, for all , one has:
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 is an elliptic and canceling linear differential operator. Then every point admits an open neighborhood such that for any , the equation (2) is continuously solvable in U if and only if f is an -charge in U, meaning that for every and every compact set , there exists such that one has: for any — 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 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) 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 , 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 and a measure μ in Ω, the Dirichlet problem associated with in Ω has a continuous solution in if and only if, for every , there exists such that for any , one has: 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 , . Unless otherwise specified, all functions are complex valued and the notation stands for the Lebesgue integral . As usual, and are the spaces of complex test functions and distributions, respectively. When is a compact subset of Ω, we let , where is the space of all distributions with compact support in K. Since the ambient field is , we identify (formally) each with the distribution
Canceling and elliptic differential operators
Given as before, the 2ν-order differential operator may be regarded as an elliptic pseudodifferential operator with symbol in the Hörmander class ,2 so that there exist properly supported pseudodifferential operators (parametrixes) and for
Basic definitions; approximation and compactness
Let be the linear space of all complex functions in whose support is a compact subset of Ω.
The following definition of variation associated to of recalls the classical definition of variation when and . 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 is a linear differential operator of order ν from E to F having coefficients with derivatives of
-charges and their extensions to
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 supports inequalities of type (1) and (5); we also assume that one has . Remark 6.1 It follows from Theorem 1.1, Proposition 3.1 and Remark 5.10 that for any , one can find an open neighborhood U of in Ω satisfying all the above assumptions.
Our intention is to prove the following result.
Theorem 6.2 If is an -charge in U, then there exists for which one has , i.e. such that one has, for any :
Application: elliptic complexes of vector fields
Consider n complex vector fields , , with smooth coefficients defined on an open set , . Naturally, we assume that the vector fields , do not vanish in Ω; in particular, they may be viewed as non-vanishing sections of the vector bundle 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)
are everywhere linearly independent;
- (b)
the system
References (22)
- et al.
Continuous solutions for divergence-type equations associated to elliptic systems of complex vector fields
J. Funct. Anal.
(2018) Functional Analysis
(1995)- et al.
On the equation and application to control of phases
J. Am. Math. Soc.
(2003) - et al.
Distributions for which has a continuous solution
Commun. Pure Appl. Math.
(2008) - et al.
Local Gagliardo-Nirenberg estimates for elliptic systems of vector fields
Math. Res. Lett.
(2011) - et al.
Local estimates for elliptic systems of complex vector fields
Proc. Am. Math. Soc.
(2015) - et al.
Sobolev estimates for (pseudo)-differential operators and applications
Math. Nachr.
(2016) Limiting Sobolev inequalities for vector fields and canceling linear differential operators
J. Eur. Math. Soc.
(2013)Limiting Bourgain-Brezis estimates for systems of linear differential equations: theme and variations
J. Fixed Point Theory Appl.
(2014)- et al.
Brownian motion and Harnack inequality for Schrödinger operators
Commun. Pure Appl. Math.
(1982)
Elliptic PDEs, Measures and Capacities. From the Poisson Equations to Nonlinear Thomas-Fermi Problems
Cited by (0)
- ☆
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.