Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 71, pp. 1–5.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2023.71

G-CONVERGENCE OF ELLIPTIC OPERATORS IN NON

DIVERGENCE FORM IN Rn

LUIGI D’ONOFRIO

Communicated by Giovanni Molica Bisci

Abstract. The aim of this note is to prove a characterization of the G-limit

of a sequence of elliptic operators in non-divergence form. As we consider any

dimension, for this class of operators, it is not enough to deal with measurable
and bounded coefficients so we need extra regularity assumptions on them.

1. Introduction

The aim of this note is to prove characterization of the G-convergencce of elliptic
operators in non divergence form. Let be Ω ⊂ Rn be a bounded open domain and
let us consider the Dirichlet problem

Lu = h ∈ L2(Ω)

u ∈W 2,2 ∩W 1,2
0 ,

(1.1)

where

Lu =

n∑
i=1

aij(x)
∂2u

∂xi∂xj
= tr(AD2u).

Where with “tr”we denote the trace of a square matrix; A = (aij) is a measurable
n× n matrix-valued defined on Ω satisfying aij = aji and the ellipticity bounds

β|ξ|2 ≤ 〈Aξ, ξ〉 ≤ |ξ!
2

β
(1.2)

for a.e. x ∈ Ω, for all ξ ∈ Rn and 0 < β ≤ 1. We denote by M(β) the set of such
functions A = A(x). The Dirichlet problem (1.1) has unique solution in the plane,
but differently from elliptic operator in divergence form, we need extra assumptions
to guarantee the solvability of (1.1). More precisely

• Talenti in 1966 proved the existence and uniqueness of (1.1) under the
assumption that the coefficient matrix A satisfies the Cordes condition,
that means that the scattering of the eigenvalue of the matrix is small.
We observe that in the plane every 2 × 2 matrix satisfying (1.2) fulfills
automatically the Cordes condition (see [8]).

2020 Mathematics Subject Classification. 35B40.

Key words and phrases. Elliptic operators; G-convergence.
©2023. This work is licensed under a CC BY 4.0 license.
Submitted October 12, 2023. Published October 20, 2023.

1



2 L. D’ONOFRIO EJDE-2023/71

• Miranda in 1963 proved the existence and uniqueness of (1.1) under the
assumption that the coefficient matrix A ∈W 1,n(Ω) (see [5]).
• Chiarenza-Frasca-Longo in 1993 proved the existence and uniqueness of

(1.1) under the assumption that the coefficient matrix A ∈ VMO, where
VMO stands for the space of vanishing mean oscillation functions (see [4]).

In this note we focus our attention to the case in which A ∈ W 1,n(Ω). Now, let
Ak, k = 1, . . ., and A be matrices in M(β) and consider the associated differential
operators Lku = trAkD

2u and Lu = trAD2u. To define G-convergence, let uk and
u the solutions to the Dirichlet problems

Lkuk = f

uk ∈W 2,2 ∩W 1,2
0

and

Lu = f

u ∈W 2,2 ∩W 1,2
0 ,

respectively. We say that {Lk} G-converges to L, and write Lk
G→ L, if for all

f ∈ L2(Ω) we have uk ⇀ u weakly in W 2,2. The Definition of G-convergence is well
posed (see [5]). We recall that the class of non divergence operators whose coefficient
matrix are inM(β) is compact respect to the G-convergence. For properties of G-
convergence of elliptic operator in non divergence form we refer to [9]

2. Main result

Before stating and proving the main result of this note we need to introduce
some auxiliary notion. We have already seen that for u ∈W 2,2(Ω) the operator

L(u) =

n∑
i,j=1

ai,j(x)
∂2u

∂xi∂xj
= tr(AD2u)

is well defined. Now for p ∈ L2 we denote by N the adjoint operator of L,

N(p) =

n∑
i,j=1

ai,j(x)
∂2aijp

∂xi∂xj

defined by the rule ∫
Ω

ϕN(p) =

∫
Ω

pL(ϕ)dx

for all ϕ ∈W 2,2(Ω). By definition N(v) = 0 if and only if∫
Ω

pL(ϕ)dx = 0 .

The reason to study the solutions of the adjoint operator is that they are important
for the solvability of Lu = f and for the properties of the Green’s function for L,
see for example [7].

Now we are in a position to show a reminiscent of the Div-Curl lemma of Murat
and Tartar [6], but we emphasize that our class of operator is of non divergence
form and we are in any dimension.



EJDE-2023/71 G-CONVERGENCE OF ELLIPTIC OPERATORS 3

Lemma 2.1. Let Lk and L be operators with coefficients matrices Ak ∈ W 1,n for
k = 1, . . .. Let A ∈W 1,n be in M(β), let pk ∈ L2(Ω) satisfy

Nk(pk) = 0 , (2.1)

and let uk ∈W 2,2(Ω) be given. If

uk ⇀ u in W 2,2
loc (Ω), and (2.2)

pkAk ⇀ pA in L2
loc(Ω,M), (2.3)

then

tr(pkAkD
2uk)→ tr(pAD2u) (2.4)

in the sense of distributions

Proof. First we note that, for a fixed smooth function ϕ, the commutator

D2(ϕuk)− ϕD2uk = Duk ⊗Dϕ+Dϕ⊗Duk + ukD
2ϕ

only contains uk and its first order derivative, hence in view of compact embeddings
of W 2,2 in W 1,2 and in L2, we have

D2(ϕuk)− ϕD2uk → D2(ϕu)− ϕD2u

strongly in L2. So we can apply (2.3) to have∫
Ω

pkAk[D2(ϕuk)− ϕD2uk]dx→
∫

Ω

pA[D2(ϕu)− ϕD2u] . (2.5)

Since Nk(pk) = 0, for all k, it follows that∫
Ω

pk tr(AkD
2ϕuk) =

∫
Ω

p tr(AD2ϕu) = 0 (2.6)

for all ϕ ∈ C∞
0 . Combining (2.5) and (2.6) we conclude that∫

Ω

ϕ tr(pkAkD
2uk)→

∫
Ω

ϕ tr(pAD2u)

for all ϕ ∈ C∞
0 . �

Theorem 2.2. Let Lk and L be operators with coefficients matrices Ak ∈W 1,n for
k = 1, . . ., and let A ∈W 1,n be in M(β). Assume that pk ∈ L2(Ω) are solutions to
the adjoint operators Nk(pk) = 0. If

pk ⇀ p in L2
loc(Ω), and (2.7)

pkAk ⇀ pA in L2
loc(Ω,M), (2.8)

where p(x) > 0 a.e. in Ω. Then Lk
G→ L.

Proof. Let us fix f ∈ L2(Ω). By Miranda’s Theorem we know that the Dirichlet
problem

Lku = f ∈ L2(Ω)

uk ∈W 2,2 ∩W 1,2
0

(2.9)

has a solution uk. It is well known that the sequence uk is bounded in W 2,2, hence
there exists a subsequence such that

ukh
⇀ u (2.10)



4 L. D’ONOFRIO EJDE-2023/71

weakly in W 2,2. Letting ϕ ∈ C∞
0 (Ω), we have∫

Ω

ϕpkh
fdx =

∫
Ω

ϕ tr(pkh
Akh

D2ukh
)dx, ∀h .

Passing to the limit in the previous expression and using Lemma 2.1, we obtain∫
Ω

ϕpfdx =

∫
Ω

ϕ tr(pAD2u)dx, ∀h,

and hence pf = tr(pAD2u) a.e. in Ω. �

3. Final comments

Theorem 2.2 was proved in the plane with the only ellipticity condition by [3]. As
already explained in the introduction to have the notion of G-convergence in more
than two dimension we need more assumption on the coefficients. In [9] they prove
Theorem 2.2 under the extra-assumption that the Ak, k = 1, . . ., and A, satisfy
the Cordes conditions; that is the operator is almost the Laplace operator as these
coefficients matrix are “near ”the identity matrix. We found these assumption too
restrictive so we prove Theorem 2.2 requiring regularity on the coefficients. Similar
results were proved in the planar case (see [1] and in dimension 3 (see [2] (requiring
that the coefficients matrix be in VMO). However they use a different notion of
G-convergence based on the fact that if the datum in the Dirichlet problem has less
regularity they are able to solve uniquely the Dirichlet problem (1.1) in a suitable
Sobolev space.

Acknowledgments. The author is member of GNAMPA. This research was sup-
ported by PRIN 2022BCFHN2 “Advanced theoretical aspects in PDEs and their
applications”, and by GNAMPA project “Equazioni nonlineari e problemi di tipo
Calabi-Bernstein”.

References

[1] T. Alberico C. Capozzoli, L. D’Onofrio; G-convergence for non-divergence second order op-

erators in the plane, Diff. Int. Equ. A, v. 26 (2013), pages 1127-1138.
[2] T. Alberico C. Capozzoli, L. D’Onofrio, R. Schiattarella; G-convergence for non-divergence

second order operators with VMO coefficients in R3, Discrete and Continuous Dynamical

Systems Series S, v. 12 (2019), pages 129-137.

[3] L. D’Onofrio, L. Greco; A counter-example in G-convergence of non-divergence elliptic op-
erators, Proc. Roy. Math. Society Edinburgh Sect. A, v. 133 (2003), pages 1299-1310.

[4] F. Chiarenza, M. Frasca, P. Longo; W 2,p-solvability of the Dirichlet problem for non-
divergence elliptic equations with VMO- coefficients, Trans. Amer. Math. Society, v. 336

(1993), pages 841-853.

[5] C. Miranda; Sulle equazioni ellettiche del secondo ordine di tipo non variazionale a coeffici-
enti discontinui, Annali di Matematica Pura ed Applicata, v. 63 (1963), pages 355-386.

[6] F. Murat, L. Tartar; H-convergence, Topics in the mathematical modelling of composite

materials, Progress in Nonlinear Differential Equations and Applications, v. 31 (1997), pages
21-43.

[7] P. Syogren; On the adjoint of an elliptic linear differential operator and its potential theory,

Ark. Mat., v. 11 (1973), pages 153-165.
[8] G. Talenti; Sopra una classe di equazioni ellittiche a coefficienti misurabili, Annali di Matem-

atica Pura ed Applicata, v. 69 (1965), pages 285-304 .

[9] V. V. Zhikov, M. M. Sirazhudinov; On G-compactness of nondivergence elliptic operators of
second order, Math USSR Izv., v. 19 (1982), pages 27-40.



EJDE-2023/71 G-CONVERGENCE OF ELLIPTIC OPERATORS 5

Luigi D’Onofrio

Dipartimento di Scienze e Tecnologie, Universita’ degli Studi di Napoli Parthenope,

Italy
Email address: luigi.donofrio@uniparthenope.it


	1. Introduction
	2. Main result
	3. Final comments
	Acknowledgments

	References