On the well-posedness of global fully nonlinear first order elliptic systems

1 Introduction

In this paper we consider the problem of existence and uniqueness of global strong solutions u:ℝn→ℝN to the fully nonlinear first order PDE system

where n,N≥2 and F:ℝn×ℝN⁢n→ℝN is a Carathéodory map. The latter means that F⁢(⋅,X) is a measurable map for all X∈ℝN⁢n and F⁢(x,⋅) is a continuous map for almost every x∈ℝn. The gradient D⁢u:ℝn→ℝN⁢n of our solution u=(u1,…,uN)⊤ is viewed as an N×n matrix-valued map D⁢u=(Di⁢uα)i=1,…,nα=1,…,N and the right-hand side f is assumed to be in L2⁢(ℝn,ℝN).

The method we use in this paper to study (1.1) follows that of the recent paper [15] by the second author. Therein, the author introduced and employed a new perturbation method in order to solve (1.1), which is based on the solvability of the respective linearised system and a structural ellipticity hypothesis on F, inspired by the classical work of Campanato in the fully nonlinear second order case ℱ⁢(⋅,D2⁢u)=f (see [3, 4, 5, 6, 7, 8, 9, 21, 20]). Loosely speaking, the ellipticity notion of [15] requires that F is “not too far away” from a linear constant coefficient first order differential operator. In the linear case of constant coefficients, F assumes the form

for some linear map 𝐀:ℝN⁢n→ℝN. We will follow almost the same conventions as in [15], for instance, we will denote the standard bases of ℝn, ℝN and ℝN×n by {ei}, {eα} and {eα⊗ei}, respectively. In the linear case, system (1.1) can be written as

and compactly in vector notation as

The appropriate well-known notion of ellipticity in the linear case is that the nullspace of the linear map 𝐀 contains no rank-one lines. This requirement can be quantified as

which says that all rank-one directions ξ⊗a∈ℝN⁢n are transversal to the nullspace. A prototypical example of such operator 𝐀:ℝ2×2→ℝ2 is given by

and corresponds to Cauchy–Riemann PDEs. In [15] system (1.1) was proved to be well-posed by solving (1.2) via Fourier transform methods and by utilising the following ellipticity notion: (1.1) is an elliptic system (or F is elliptic) when there exists a linear map 𝐀:ℝN⁢n→ℝN which is elliptic in the sense of (1.3) and such that

where

is the “ellipticity constant” of 𝐀. This notion was called “K-Condition” in [15]. The functional space in which well-posedness was obtained is the so-called J. L. Lions space

Here 2* is the conjugate Sobolev exponent, i.e.,

(note that “L2*” means “Lp for p=2*”, not duality) and the natural norm of the space is

In [15] only global strong a.e. solutions on the whole space were considered and for dimensions n≥3 and N≥2, in order to avoid the compatibility difficulties which arise in the case of the Dirichlet problem for first order systems on bounded domains and because the case n=2 has been studied quite extensively.

In this paper we follow the method introduced in [15] and we prove the well-posedness of (1.1) in the space (1.7) for the same dimensions n≥3 and N≥2. This is the content of our Theorem 4.1, whilst we also obtain an a priori quantitative estimate in the form of a “comparison principle” for the distance of two solutions in terms of the distance of the respective right-hand sides of (1.1). The main advance in this paper, which distinguishes it from the results obtained in [15], is that we introduce a new notion of ellipticity for (1.1) which is strictly weaker than (1.5), allowing more general nonlinearities F to be considered. Our new hypothesis of ellipticity is inspired by another recent work of the second author [16] on the second order case. We will refer to our condition as the “AK-Condition” (Definition 3.1). In Examples 3.2 and 3.3 we demonstrate that the new condition is genuinely weaker, and hence our results indeed generalise those of [15]. Further, motivated by [16], we also introduce a related notion which we call pseudo-monotonicity and examine their connection (Lemma 3.4). The idea of the proof of our main result Theorem 4.1 is based, as in [15], on the solvability of the linear system, our ellipticity assumption and on a fixed point argument in the form of Campanato’s near operators, which we recall later for the convenience of the reader (Theorem 2.3).

We conclude the introduction with some comments which contextualise the standing of the topic and connect to previous contributions by other authors. Linear elliptic PDE systems of first order are of paramount importance in several branches of analysis like, for instance, in complex and harmonic analysis. Therefore, they have been extensively studied in several contexts (see, e.g., [2, 1]), including regularity theory of PDEs (see [18, chapter 7] of Morrey’s exposition of the Agmon–Douglis–Nirenberg theory), differential inclusions and compensated compactness theory [12, 19], as well as geometric analysis and the theory of differential forms [10].

However, except for the paper [15], the fully nonlinear system (1.1) is much less studied and understood. By using the Baire category method of the Dacorogna–Marcellini [11] (which is the analytic counterpart of Gromov’s geometric method of convex integration), it can be shown that the Dirichlet problem

has infinitely many strong a.e. solutions in W1,∞⁢(Ω,ℝN), for Ω⊆ℝn, g a Lipschitz map and under certain structural coercivity and compatibility assumptions. However, roughly speaking, the ellipticity and coercivity of F are mutually exclusive properties. In particular, it is well known that the Dirichlet problem (1.8) is not well posed when F is either linear or elliptic.

Further, it is well known that single equations, let alone systems of PDEs, in general do not have classical solutions. In the scalar case N=1, the theory of viscosity solutions of Crandall–Ishii–Lions (we refer to [14] for a pedagogical introduction to the topic) furnishes a very successful setting of generalised solutions in which Hamilton–Jacobi PDEs enjoy strong existence-uniqueness theorems. However, there is no counterpart of this essentially scalar theory for (non-diagonal) systems. The general approach of this paper is inspired by the classical work of Campanato quoted earlier and in a nutshell consists of imposing an appropriate condition that allows to prove well-posedness in the setting of the intermediate theory of strong a.e. solutions. Notwithstanding, very recently the second author in [17] has proposed a new theory of generalised solutions in the context of which he has already obtained existence and uniqueness theorems for second order degenerate elliptic systems. We leave the study of the present problem in the context of “𝒟-solutions” introduced in [17] for future work.

2 Preliminaries

In this section we collect some results taken from our references which are needed for the main results of this paper. The first one below concerns the existence and uniqueness of solutions to the linear first order system with constant coefficient:

with 𝐀:ℝN⁢n→ℝN elliptic in the sense of (1.3), namely when the nullspace of 𝐀 does not contain rank-one lines. By the compactness of the torus, it can be rewritten equivalently as

for some constant ν>0, which can be chosen to be the ellipticity constant of 𝐀 given by (1.6). One can easily see that (2.1) can be rephrased as

where 𝐀⁡a is the N×N matrix given by

It is easy to exhibit examples of tensors 𝐀 satisfying (2.1). A map 𝐀:ℝ2×2→ℝ2 satisfying it is

where κ,λ,μ,ν>0. A higher dimensional example of a map 𝐀:ℝ4×3→ℝ4 is

which corresponds to the electron equation of Dirac in the case where is no external force. For more details we refer to [15].

In the above statement, “sgn”, “cof” and “det” symbolise the sign function on ℝn, the cofactor and the determinant on ℝN×N, respectively. Although the formula (2.2) involves complex quantities, u above is a real vectorial solution. Moreover, the symbol “^” stands for the Fourier transform (with the conventions of [13]) and “ˇ” stands for its inverse.

Next, we recall the strict ellipticity condition of the second author taken from [15] in an alternative form which is more convenient for our analysis. We will relax it in the next section. Let 𝐀:ℝN⁢n→ℝN be a fixed reference linear map satisfying (2.1).

Finally, we recall the next classical result of Campanato, taken from [7], which is needed for the proof of our main result Theorem 4.1:

3 The AK-condition of ellipticity for fully nonlinear first order systems

In this section we introduce and study a new ellipticity condition for the PDE system (1.1), which relaxes the K-Condition Definition 2.2 and still allows to prove existence and uniqueness of strong solutions in the functional space (1.7). Let 𝐀:ℝN⁢n→ℝN be an elliptic reference linear map satisfying (2.1).

Nontrivial fully nonlinear examples of maps F, which are elliptic in the sense of Definition 3.1 above, are easy to find. Consider any fixed map 𝐀:ℝN⁢n→ℝN for which ν⁢(𝐀)>0 and any Carathéodory map L:ℝn×ℝN⁢n→ℝN which is Lipschitz with respect to the second variable and such that

and for some 0<β<1. Let also α be a positive essentially bounded function with 1/α essentially bounded as well. Then the map F:ℝn×ℝN⁢n→ℝN, given by

satisfies Definition 3.1, since

As a consequence, F satisfies the AK-Condition for the same function α⁢(⋅) and for the constants β and γ=(1-β)/2.

The following example shows that, given a reference tensor 𝐀, there exist even linear constant “coefficients” F which are elliptic with respect to 𝐀, in the sense of our AK-Condition Definition 3.1, but which are not elliptic with respect to 𝐀, in the sense of Definition 2.2.

The essential point in the above example that makes Definition 3.1 more general than Definition 2.2 was the introduction of the rescaling function α⁢(⋅). Now we give a more elaborate example which shows that even if we ignore the rescaling function α and normalise it to α⁢(⋅)≡1, Definition 3.1 is still more general than Definition 2.2 with respect to the same fixed reference tensor 𝐀.

We now show that our ellipticity assumption implies a condition of pseudo-monotonicity coupled by a global Lipschitz continuity property. The statement and the proof are modelled after a similar result appearing in [16] which however was in the second order case.

4 Well-posedness of global fully nonlinear first order elliptic systems

In this section we state and prove the main result of this paper which is the following: