Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 103, pp. 1–26.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.103

EXISTENCE AND REGULARITY OF SOLUTIONS FOR ELLIPTIC SYSTEMS

WITH MIXED BOUNDARY CONDITIONS

IMANE BOUSSETOUAN, CHÉRIF AMROUCHE

Communicated by Jesus Ildefonso Diaz

Abstract. In this article we study elliptic systems involving nonstandard mixed boundary
conditions in a bounded domain, possibly multiply connected. We prove the existence and

uniqueness of weak and strong solutions of such problems in the Hilbert setting. Then, we

generalize our results to the Lp-theory using some inf-sup conditions.

1. Introduction

Elliptic problems arise in various fields as in mechanics, chemistry, biology, and other fields.
There has been a considerable amount of works done in the analysis of general and particular
elliptic systems (see for instance [12, 13, 14]). The aim of this work is to study the well-posedness
of linear elliptic systems involving two different kinds of mixed boundary conditions in the Hilbert
framework and then in Lp-theory.

Unless stated otherwise, we assume that Ω is a C 1,1 domain in R3, possibly multiply connected
and its boundary is decomposed as ΓD ∪ ΓN such that ΓD ∩ ΓN = ∅. In what follows, the unit
outer normal to the boundary is denoted by n.

Laplace’s equation has been mostly studied with Dirichlet or Neumann boundary conditions.
In the vector case, we can have the same boundary conditions but they do not lead to an elliptic
system. To recover the ellipticity, various types of physical boundary conditions can be imposed.
Among them, the conditions we assume in this paper.

We consider the Poisson equation

−∆u = f in Ω, (1.1)

with the boundary conditions

u · n = 0, curlu× n = 0 on ΓD, (1.2)

u× n = 0, divu = 0 on ΓN . (1.3)

The condition (1.2) is called “Navier-type boundary condition” [6] or “Hodge boundary condition”
[15] or even “perfect wall” condition [1]. It is equivalent to the Navier boundary condition intro-
duced in [16] in the case of flat boundary and where the friction coefficient vanishes. It appears
in the study of climate modeling, in electromagnetism and in magnetohydrodynamics. It can be
used for ferrofluid flow models, such as those by Rosensweig and Shliomis [17], where the condition
concerning the divergence of internal and external magnetic fields may not be constrained to zero.
In [6], the authors treated the Stokes problem with (1.2) on the boundary and in particular the
system (1.1)-(1.2) with a free-divergence condition, which required more assumptions on the data.
Recently in [3], the authors investigated the Lp-theory of an elliptic system involving the condition
(1.2) on the boundary without a divergence constraint.

2020 Mathematics Subject Classification. 35J05, 35J57, 35D35.
Key words and phrases. Elliptic systems; mixed boundary conditions; Navier-type boundary condition;

weak and strong solutions; Lp-theory.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted September 10, 2024. Published November 3, 2025.

1



2 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

For a better understanding of the conditions (1.2) and (1.3), let us consider any point P on Γ
and choose any neighborhood W of P in Γ, small enough to allow the existence of C 2 curves on
W . The lengths s1, s2 along each family of curves are a possible set of coordinates inW . The unit
tangent vectors to each family of curves are denoted by τ 1 and τ 2. With this notation we have
v = vτ + (v · n)n and vτ =

∑2
k=1 vkτ k, where vk = v · τ k. Consequently, we have the following

formula valuable for any u ∈ D(Ω) (or more generally for any u ∈ W1,p(Ω) with ∆u ∈ Lr(p)(Ω),
see below the definition of the exponent r(p))

divu = divΓ uτ − 2Ku · n+
∂u

∂n
· n on Γ, (1.4)

where K = − 1
2

∑2
j=1

∂n
∂sj

· τ j is the mean curvature of Γ and divΓ is the surface divergence.

Moreover, we have the following relations for any u ∈ D(Ω)

curlu× n = −∇τ (u · n) + Λu+
(∂u
∂n

)
τ

on Γ, (1.5)

where Λ is an operator of order zero, defined by

Λw =

2∑
j=1

( ∂n
∂sj

·wτ

)
τ j .

Taking account of (1.4), the condition (1.3) is equivalent to Fourier-Robin boundary condition
given by

Ku · n+
∂u

∂n
· n = 0.

For the well-posedness of the problem, a quantity that reflects the topological structure of the
domain should be added (see the last relations in Problem (PD) Section 2).

The second elliptic problem we are interested in is given by the equation (1.1) with mixed
normal and tangential components on the boundary, i.e.

u · n = 0,
∂u

∂n
× n = 0 on ΓD, (1.6)

u× n = 0,
∂u

∂n
· n = 0 on ΓN . (1.7)

This kind of conditions occur in the nonlinear model of a nematic liquid crystal studied by Dias
in [11], who proved the existence and uniqueness of a weak solution in the non mixed case. In the
work of Shoenauer [18], the author studied the problem, with (1.6) or (1.7) on the whole boundary.
Taking f ∈ Lp(Ω), he proved the existence and uniqueness of strong solutions in W 2,p(Ω). The
idea of the proof is based on the regularity near the boundary for p = 2. By a bootstrap argument,
the case p > 2 is then deduced. Finally, the case 1 < p < 2 is obtained using a priori estimates.
Brezis et al [8] treated the regularity of the solution in the non mixed case where the source term
f belongs to L1(Ω). Using an additional condition on div f , they proved that the solution belongs

to W1,n/n−1(Ω) and not only to the Sobolev space W
1,n/n−1
∞ (Ω). The functions belonging to the

latter space have gradients in the Lorentz space L
n/n−1
∞ (Ω). Up to our knowledge, the problem

of mixed boundary conditions (1.6)-(1.7) has not been treated in the literature. We assume that
f ∈ Lr(p)(Ω) where

r(p) =


3p
p+3 if p > 3

2

1 + ε if p = 3
2

1 if 1 ≤ p < 3
2 ,

where ε > 0 is arbitrary. Note that W1,r(p)(Ω) ↪→ Lp(Ω) and W1,p′(Ω) ↪→ L[r(p)]′(Ω).
These problems are challenging due to the unusual mixtures of physical boundary conditions

we are considering. Also, the complexity of the geometry of the domain brings more difficulties
to the problems. In fact, the domain is possibly multiply connected and we do not assume that
ΓD and ΓN are connected. We suppose that there exist J connected open surfaces Σj , 1 ≤ j ≤ J ,



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 3

called “cuts”, contained in Γ, such that each surface Σj is an open subset of a smooth manifold

and Σi ∩ Σj = ∅ for i ̸= j. Also, ∂Σ stands for the union of the boundaries of Σj .
Our investigations are separated into three categories: existence, uniqueness and regularity of

the solutions, in the Hilbert framework and then in Lp-theory. We start with the existence and
uniqueness of weak solutions in the Hilbert setting of the first system (1.1)-(1.3). This system is
more general than what has been studied in the literature (see for instance [4, 6, 7]) where the
authors assume that divu = 0 on Ω and add some assumptions on the source term f . The fact
that we propose here a more general setting allows this kind of problems to be applied in some
different areas like in magnetohydrodynamics. Furthermore, the solution of the system without
a divergence constraint requires less conditions on the data. In Proposition 4.12, we prove that
we can recover the condition divu = 0 on Ω. Unlike the case of Dirichlet condition, where
the Poincaré’s inequality plays a fundamental role, we need here some inequalities for vector fields
involving the operators grad, div and curl. To prove the strong regularity results, we use a partition
of unity combined with some regularity properties for the operators div and curl. Let us emphasize
that we will deal separately with the cases where ∂Σ is included in ΓD or in ΓN . The main purpose
of this work is to give a complete Lp-theory for the proposed systems not only for p ≥ 2 but also
for any 1 < p < 2. To do so, we establish an appropriate inf-sup condition for each problem leading
to generalized solutions in W1,p(Ω) for any 1 < p < +∞. We also give some results related to
the case of inhomogeneous boundary conditions. For the second problem (1.1),(1.6) and (1.7),
we prove a particular Poincaré condition in the case of mixed normal and tangential components.
The existence of a strong solution is deduced by using the same arguments as the first system.
For the Lp-theory, we proceed using a strategy based on the first inf-sup condition leading to a
solution in W1,p(Ω) for any 1 < p < +∞ but requiring some assumptions on the geometry of the
domain. We also provide a W2,p(Ω) regularity result.

Let us now briefly outline the structure of this paper. In Section 2, we state some main results.
In Section 3, we introduce the mathematical framework and review some preliminary results. In
Section 4, we focus our attention on the existence, uniqueness and regularity of the solution of
the first elliptic problems dealing with the conditions (1.2) and (1.3) on the boundary. Section 5
is devoted to the study of the second elliptic system with (1.6) and (1.7) in the Hilbert case and
then in Lp-theory.

2. Main results

The first main result of this work deals with the existence and uniqueness of the solution of the
following problem where we assume that ∂Σ ⊂ ΓD,

−∆u = f in Ω,

u× n = a× n, divu = g on ΓN ,

u · n = b, curlu× n = h× n on ΓD,

⟨u · n, 1⟩Γℓ
N
= 0, 1 ≤ ℓ ≤ LN , ⟨u · n, 1⟩Σj

= 0, 1 ≤ j ≤ J,

(2.1)

where f , a, b, g and h are given data. The last relations in (2.1) are zero-fluxes conditions across
the connected components ΓℓN and the cuts Σj respectively and are necessary for the uniqueness
of the solution.

Theorem 2.1. (i) Assume that ∂Σ ⊂ ΓD. Let f ∈ Lr(p)(Ω) with 3/2 ≤ p < +∞, a × n ∈
W1−1/p,p(ΓN ), g ∈ W−1/p,p(ΓN ), b ∈ W 1−1/p,p(ΓD) and h × n ∈ W−1/p,p(ΓD) satisfying the
compatibility condition ∫

Ω

f ·φ dx− ⟨h× n,φ⟩ΓD
+ ⟨g,φ · n⟩ΓN

= 0.

Then Problem (2.1) has a unique solution u ∈ W1,p(Ω) for any 1 < p < +∞ and it satisfies

∥u∥W1,p(Ω) ≤ C
(
∥f∥Lr(p)(Ω) + ∥g∥W−1/p,p(ΓN ) + ∥h× n∥W−1/p,p(ΓD)

+ ∥a× n∥W1−1/p,p(ΓN ) + ∥b∥W 1−1/p,p(ΓD)

)
.



4 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

(ii) If f ∈ L1(Ω), then there exists a unique u ∈ W1,s(Ω) for any s < 3
2 solution of Problem

(2.1) and satisfying

∥u∥W1,s(Ω) ≤ C
(
∥f∥L1(Ω) + ∥g∥W−1/p,p(ΓN ) + ∥h× n∥W−1/p,p(ΓD)

+ ∥a× n∥W1−1/p,p(ΓN ) + ∥b∥W 1−1/p,p(ΓD)

)
.

(iii) Furthermore, if Ω is of class C 2,1, f ∈ Lp(Ω), a× n ∈ W2−1/p,p(ΓN ), g ∈W 1−1/p,p(ΓN ),
b ∈W 2−1/p,p(ΓD), and h× n ∈ W1−1/p,p(ΓD), then u belongs to W2,p(Ω) and

∥u∥W2,p(Ω) ≤ C
(
∥f∥Lp(Ω) + ∥g∥W 1−1/p,p(ΓN ) + ∥h× n∥W1−1/p,p(ΓD)

+ ∥a× n∥W2−1/p,p(ΓN ) + ∥b∥W 2−1/p,p(ΓD)

)
.

The proof of this theorem is given in Section 4 using an inf-sup condition established in Lemma
4.7. We also discuss the case where ∂Σ ⊂ ΓN in the same section.

The second main result concerns the elliptic system (1.1)-(1.7)-(1.6) in which we assume a more
general right-hand side and the corresponding boundary conditions. The proof of this main result
is detailed in Section 5.

−∆u = f + divF in Ω,

u× n = 0, [(∇u+ F)n] · n = g on ΓN ,

u · n = 0, [(∇u+ F)n]× n = h on ΓD.

(2.2)

where f , g and h are given data.

Theorem 2.2. (i) Assume that ∂Σ ⊂ ΓD or ∂Σ ⊂ ΓN . Let f ∈ Lr(p)(Ω), F ∈ Lp(Ω), g ∈
W−1/p,p(ΓN ) and h ∈ W−1/p,p(ΓD) then Problem (2.2) has a unique solution u ∈ W1,p(Ω) and
it satisfies

∥u∥W1,p(Ω) ≤ C
(
∥f∥Lr(p)(Ω) + ∥F|∥Lp(Ω) + ∥g∥W−1/p,p(ΓN ) + ∥h∥W−1/p,p(ΓD)

)
.

(ii) Furthermore, if Ω is of class C 2,1, f ∈ Lp(Ω), g ∈ W 1−1/p,p(ΓN ), and h ∈ W1−1/p,p(ΓD)
with F = 0, then u belongs to W2,p(Ω) and

∥u∥W2,p(Ω) ≤ C
(
∥f∥Lp(Ω) + ∥g∥W 1−1/p,p(ΓN ) + ∥h∥W1−1/p,p(ΓD)

)
.

Note that the assumptions ∂Σ ⊂ ΓD or ∂Σ ⊂ ΓN are related to the chosen method.

3. Preliminaries

In this section, we set some basic notation, we introduce the functional settings and we recall
some tools necessary for our analysis. We follow the convention that C is a constant that may vary
from expression to expression. We denote by X ′ the dual space of the space X and by ⟨·, ·⟩X′,X

the duality product between X and X ′. Vector fields are designated by bold letters and their
corresponding spaces by bold capital characters.

We consider a bounded Lipschitz domain denoted ω. We assume that ω contains a number of
obstacles denoted Ω1,Ω2, . . . ,ΩI , some of them may be non-simply connected. We set

Ω = ω\
(
∪Ii=1 Ωi

)
.

The boundary of Ω is decomposed into two parts ΓD and ΓN , i.e.

Γ = ΓD ∪ ΓN , ΓD ∩ ΓN = ∅.
More precisely,

Γ =
(
∪LD

ℓ=0 Γ
ℓ
D

)
∪
(
∪LN

ℓ=0 Γ
ℓ
N

)
,

where ΓℓD, 0 ≤ ℓ ≤ LD are the connected components of ΓD and similarly ΓℓN , 0 ≤ ℓ ≤ LN are
the connected components of ΓN . We assume that there exists J connected open surfaces Σj ,
1 ≤ j ≤ J , called “cuts” contained in Ω such that



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 5

(i) each surface Σj is an open part of a smooth manifold Mj

(ii) the boundary of Σj is contained in ∂Ω
(iii) the intersection Σi ∩ Σj is empty for i ̸= j
(iv) the open set Ωo = Ω\ ∪Jj=1 Σj is simply-connected.

We denote by ΓjΣ the boundary of each connected open surface Σj and we have

∂Σ = ∪Jj=1Γ
j
Σ

Figure 1. Lipschitz domain Ω

Note that we consider only the cases where ∂Σ ⊂ ΓN or ∂Σ ⊂ ΓD. The case where ∂Σ is
included in ΓD and in ΓN in the same time is not handled in this article.

Referring Figure 1, when ∂Σ ⊂ ΓD (resp ΓN ), this means that ∂ω∪∂Ω5 is included in ΓD (resp
ΓN ).

We denote by [·]j the jump of a function over Σj , i.e. the differences of the traces for any
1 ≤ j ≤ J . For any function q ∈ W1,p(Ωo), ∇q is the gradient of q in the sense of distributions in
D′(Ωo) which belongs to Lp(Ωo) and it can be extended to Lp(Ω). Therefore, to distinguish this

extension from the gradient of q, we denote it by g̃radq.
We introduce the functional framework:

Hp(curl,Ω) = {v ∈ Lp(Ω) : curlv ∈ Lp(Ω)},
Hp(div,Ω) = {v ∈ Lp(Ω) : divv ∈ Lp(Ω)}

and we denote by Xp(Ω) the space

Xp(Ω) = Hp(curl,Ω) ∩Hp(div,Ω)

provided with the norm for 1 < p <∞

∥v∥Xp(Ω) =
(
∥v∥pLp(Ω) + ∥ curlv∥pLp(Ω) + ∥divv∥pLp(Ω)

)1/p

.

The space D(Ω) is dense in both spaces Hp(curl,Ω) and Hp(div,Ω) and the closure of D(Ω) in
Hp(curl,Ω) and Hp(div,Ω) is respectively denoted by Hp

0(curl,Ω) and Hp
0(div,Ω). We also define

the subspaces

Xp
T (Ω) = {v ∈ Xp(Ω) : v · n = 0 on Γ}, Xp

N (Ω) = {v ∈ Xp(Ω) : v × n = 0 on Γ},
Xp

0(Ω) = {v ∈ Xp(Ω) : v × n = 0 on ΓD, v · n = 0 on ΓN},

X̃p
0(Ω) = {v ∈ Xp(Ω) : v × n = 0 on ΓN , v · n = 0 on ΓD}.



6 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

Note that as long as u belongs to Hp(curl,Ω), the tangential boundary value of u is defined in

W− 1
p ,p(Γ) and in the case where u belongs to Hp(div,Ω), the normal boundary value is also

defined in W− 1
p ,p(Γ). Moreover, we have the Green’s formulas

∀φ ∈ W1,p′(Ω), ⟨u× n,φ⟩Γ =

∫
Ω

u · curlφ dx−
∫
Ω

curlu ·φ dx,

where ⟨·, ·⟩Γ denotes the duality product between W− 1
p ,p(Γ) and W

1
p ,p

′
(Γ) and

∀φ ∈W 1,p′(Ω), ⟨u · n, φ⟩Γ =

∫
Ω

u · ∇φdx+

∫
Ω

(divu)φdx,

where ⟨·, ·⟩Γ denotes the duality product between W− 1
p ,p(Γ) and W

1
p ,p

′
(Γ). It is also necessary to

remind the following regularity lemma given in [7].

Lemma 3.1. Let m ∈ N∗ and Ω of class Cm,1. Then the spaces

X m,p(Ω) = {v ∈ Lp(Ω) : divv ∈Wm−1,p(Ω), curlv ∈ Wm−1,p(Ω),v · n ∈Wm− 1
p ,p(Γ)}

and

Y m,p(Ω) =
{
v ∈ Lp(Ω) : divv ∈Wm−1,p(Ω), curlv ∈ Wm−1,p(Ω),v × n ∈ Wm− 1

p ,p(Γ)
}

are continuously embedded in Wm,p(Ω). For each v in Wm,p(Ω), we have the following estimates

∥v∥Wm,p(Ω) ≤ C
(
∥v∥Lp(Ω) + ∥ curlv∥Wm−1,p(Ω) + ∥ divv∥Wm−1,p(Ω) + ∥v · n∥

W
m− 1

p
,p
(Γ)

)
, (3.1)

∥v∥Wm,p(Ω) ≤ C
(
∥v∥Lp(Ω) + ∥ curlv∥Wm−1,p(Ω) + ∥ divv∥Wm−1,p(Ω) + ∥v × n∥

W
m− 1

p
,p
(Γ)

)
.

(3.2)

We define the kernels

Kp
T (Ω) = {v ∈ Xp

T (Ω), divv = 0, curlv = 0 in Ω}
Kp
N (Ω) = {v ∈ Xp

N (Ω),divv = 0, curlv = 0 in Ω},

and the kernels when mixing normal and tangential components,

Kp
0(Ω) = {v ∈ Xp

0(Ω), divv = 0, curlv = 0 in Ω},

K̃p
0(Ω) = {v ∈ X̃p

0(Ω),divv = 0, curlv = 0 in Ω}.

When ∂Σ ⊂ ΓD, the kernel K̃p
0(Ω) is spanned by the functions g̃rad sℓj with 1 ≤ j ≤ J and

1 ≤ ℓ ≤ LN , where each sℓj is the unique solution in H1(Ωo) of the following problem (see [4,
Proposition 3.13])

−∆sℓj = 0 in Ωo,

∂sℓj
∂n

= 0 on ΓD,

sℓj |Γ0
N
= 0 and sℓj |Γm

N
= const, 1 ≤ m ≤ LN ,

[sℓj ]k = const and [
∂sℓj
∂n

]k = 0, 1 ≤ k ≤ J,

⟨
∂sℓj
∂n

, 1⟩Σk
= δjk, 1 ≤ k ≤ J,

⟨
∂sℓj
∂n

, 1⟩Γ0
N
= −1 and ⟨

∂sℓj
∂n

, 1⟩Γm
N
= δℓm, 1 ≤ m ≤ LN .

(3.3)



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 7

When ∂Σ ⊂ ΓN , the space K̃p
0(Ω) is spanned by the functions ∇sℓ with 1 ≤ ℓ ≤ LN where

each sℓ is the unique solution in H1(Ω), of the following problem (see [4])

−∆sℓ = 0 in Ω,

∂sℓ
∂n

= 0 on ΓD,

sℓ|Γ0
N
= 0 and sℓ|Γm

N
= const, 1 ≤ m ≤ LN ,

⟨∂sℓ
∂n

, 1⟩Γ0
N
= −1 and ⟨∂sℓ

∂n
, 1⟩Γm

N
= δℓm, 1 ≤ m ≤ LN .

(3.4)

The kernel Kp
0(Ω) is spanned by the functions g̃radqℓj , 1 ≤ j ≤ J , 1 ≤ ℓ ≤ LD when ∂Σ ⊂ ΓN and

by ∇qℓ, 1 ≤ ℓ ≤ LD when ∂Σ ⊂ ΓD satisfying the same properties as g̃rad sℓj and ∇sℓ when ΓD

and ΓN are swapped. We underline that the kernels Kp
0(Ω) and K̃p

0(Ω) do not depend on p in a
C 1,1 domain.

We recall the inf-sup conditions obtained in [4] in the case where ∂Σ is included either in ΓN
or in ΓD.

Proposition 3.2. (i) If ∂Σ ⊂ ΓN , there exists a constant β1 > 0 depending only on Ω and p,
such that the following inf-sup condition holds

inf
φ∈Ṽp′

0,σ(Ω), φ̸=0

[
sup

ξ∈Ṽp
0,σ(Ω), ξ ̸=0

∣∣ ∫
Ω
curl ξ · curlφ

∣∣
∥ξ∥W1,p(Ω)∥φ∥W1,p′ (Ω)

]
≥ β1. (3.5)

(ii) If ∂Σ ⊂ ΓD, there exists a constant β2 > 0 depending only on Ω and p, such that the
following inf-sup condition holds

inf
φ∈W̃p′

Σ,σ(Ω), φ̸=0

[
sup

ξ∈W̃p
Σ,σ(Ω), ξ ̸=0

∣∣ ∫
Ω
curl ξ · curlφ

∣∣
∥ξ∥W1,p(Ω)∥φ∥W1,p′ (Ω)

]
≥ β2, (3.6)

where

Ṽp
0,σ(Ω) = {v ∈ X̃p

0(Ω), divv = 0 in Ω, ⟨v · n, 1⟩Γℓ
N
= 0, 1 ≤ ℓ ≤ LN},

W̃p
Σ,σ(Ω) = {v ∈ X̃p

0(Ω), divv = 0 in Ω, ⟨v · n, 1⟩Σj
= 0, 1 ≤ j ≤ J, ⟨v · n, 1⟩Γℓ

N
= 0, 0 ≤ ℓ ≤ LN}.

We review some important estimates established in [4]. Note that the space X̃p
0(Ω) ↪→ W1,p(Ω).

Theorem 3.3. Assume that Ω is Lipschitz (resp. C 1,1).

(i) If ∂Σ ⊂ ΓN , then on the space X̃p
0(Ω), the semi-norm

u 7→ ∥divu∥Lp(Ω) + ∥ curlu∥Lp(Ω) +

LN∑
ℓ=1

|⟨u · n, 1⟩Γℓ
N
| (3.7)

is a norm equivalent to the norm ∥ · ∥Xp(Ω) (resp. ∥ · ∥W1,p(Ω)) on X̃p
0(Ω).

(ii) If ∂Σ ⊂ ΓD, the semi-norm

u 7→ ∥divu∥Lp(Ω) + ∥ curlu∥Lp(Ω) +

LN∑
ℓ=1

|⟨u · n, 1⟩Γℓ
N
+

J∑
j=1

| < u · n, 1 >Σj
, (3.8)

is a norm equivalent to the norm ∥ · ∥Xp(Ω) (resp. ∥ · ∥W1,p(Ω)) on X̃p
0(Ω).

The following lemma has been proved for non mixed boundary conditions where Ω is of class C 1,1

(see [2]) and the result remains valid when mixing Dirichlet and Neumann boundary conditions.

Lemma 3.4. Let χ ∈ Lp(Ω) such that ∆χ = 0 in Ω, χ = 0 on ΓN , ∂χ
∂n = 0 on ΓD. Then χ = 0

in Ω.

We also recall the following result [3, Lemma 3.2].



8 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

Lemma 3.5. Suppose that Ω is Lipschitz and let w ∈ Lp(Ω) be such that curlw ∈ Lp(Ω) and
w × n ∈ Lp(Γ). Then divΓ(w × n) ∈W−1/p,p(Γ) and

divΓ(w × n) = curlw · n on Γ. (3.9)

Let us introduce the Banach space

Ep(Ω;∆) = {u ∈ W1,p(Ω) : ∆u ∈ Lr(p)(Ω)},
equipped with the graph norm and the following mappings

ℓ1 : D(Ω) → W−1/p,p(Γ) as ℓ1(u) = curlu× n,

ℓ2 : D(Ω) → W−1/p,p(Γ) as ℓ2(u) = divu.

A vector-valued Laplace operator of a vector field u is equivalently defined as

∆u = ∇(divu)− curl curlu.

Lemma 3.6. (i) The space D(Ω) is dense in the space Ep(Ω;∆).
(ii)The linear mappings ℓ1 and ℓ2 can be extended to linear and continuous mappings from

Ep(Ω;∆) to W−1/p,p(Γ) and from Ep(Ω;∆) to W−1/p,p(Γ) respectively.

(iii) Furthermore, the following Green’s formula holds for any u ∈ Ep(Ω;∆) and φ ∈ W1,p′(Ω)
such that φ× n = 0 on ΓN and φ · n = 0 on ΓD

−
∫
Ω

∆u ·φ =

∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ) + ⟨curlu× n,φ⟩ΓD
− ⟨divu,φ · n⟩ΓN

, (3.10)

where ⟨·, ·⟩ΓD
denotes the duality product between W−1/p,p(ΓD) and W1/p,p′(ΓD) and ⟨·, ·⟩ΓN

the

duality product between W−1/p,p(ΓN ) and W 1/p,p′(ΓN ).

Proof. (i) The proof of the density of D(Ω) into Ep(Ω;∆) is similar to the one given by Grisvard
in [12, Lemma 1.5.3.9, page 59]grisvard.

(ii) Let u ∈ D(Ω) and φ ∈ W1,p′(Ω), then

−
∫
Ω

∆u ·φ =

∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ) + ⟨curlu× n,φ⟩Γ − ⟨divu,φ · n⟩Γ. (3.11)

Let µ ∈ W1/p,p′(Γ). We know that there exists ψ ∈ W1,p′(Ω) such that

divψ = 0 in Ω, ψ = µτ on Γ,

satisfying the estimate

∥ψ∥W1,p′ (Ω) ≤ C∥µτ∥W1/p,p′ (Γ) ≤ C∥µ∥W1/p,p′ (Γ).

We infer that
⟨curlu× n,µ⟩Γ = ⟨curlu× n,µτ ⟩Γ = ⟨curlu× n,ψ⟩Γ.

By replacing in (3.11), we obtain

⟨curlu× n,µ⟩Γ = −
∫
Ω

∆u ·ψ −
∫
Ω

curlu · curlψ,

|⟨curlu× n,µ⟩Γ| ≤ C∥u∥Ep(Ω;∆)∥ψ∥W1,p′ (Ω) ≤ C∥u∥Ep(Ω;∆)∥µ∥W1/p,p′ (Γ).

Consequently,
∥ curlu× n∥W−1/p,p(Γ) ≤ C∥u∥Ep(Ω;∆),

which means that the linear mapping ℓ1 is continuous in D(Ω) for the graph norm of Ep(Ω;∆).
Since the space D(Ω) is dense in Ep(Ω;∆), the mapping ℓ1 can be extended to a linear and
continuous one from Ep(Ω;∆) to W−1/p,p(ΓD).

Now, let µ ∈W 1/p,p′(Γ). We know that there exists ψ ∈W 2,p′(Ω) such that

ψ = 0 on Γ,
∂ψ

∂n
= µ on Γ,

satisfying
∥ψ∥W 2,p′ (Ω) ≤ C∥µ∥W 1/p,p′ (Γ).



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 9

By setting φ = ∇ψ in (3.11), we have

⟨divu, µ⟩Γ =

∫
Ω

∆u · ∇ψ +

∫
Ω

(divu)(∆ψ),

which yields

|⟨divu, µ⟩Γ| ≤ C∥u∥Ep(Ω;∆)∥ψ∥W 2,p′ (Ω) ≤ C∥u∥Ep(Ω;∆)∥µ∥W 1/p,p′ (Γ).

As above, the mapping ℓ2 can be extended to a linear and continuous one from Ep(Ω;∆) to
W−1/p,p(Γ).

(iii) From the density of D(Ω) in Ep(Ω;∆) and the continuity of ℓ1 and ℓ2, we deduce that the

Green’s formula (3.11) is valid for any u ∈ Ep(Ω;∆) and φ ∈ W1,p′(Ω). Taking φ×n = 0 on ΓN
and φ · n = 0 on ΓD in (3.11) leads readily to (3.10). □

4. First elliptic system

The aim of this section is to study the elliptic system where we assume that ∂Σ ⊂ ΓD,

−∆u = f in Ω,

u× n = a× n, divu = g on ΓN ,

u · n = b, curlu× n = h× n on ΓD,

⟨u · n, 1⟩Γℓ
N
= 0, 1 ≤ ℓ ≤ LN , ⟨u · n, 1⟩Σj

= 0, 1 ≤ j ≤ J,

(4.1)

where f , a, b, g and h are given data.
Before solving the general problem (4.1), we start with the case of homogeneous boundary

conditions given by

−∆u = f in Ω,

u× n = 0, divu = 0 on ΓN ,

u · n = 0, curlu× n = 0 on ΓD,

⟨u · n, 1⟩Γℓ
N
= 0, 1 ≤ ℓ ≤ LN , ⟨u · n, 1⟩Σj

= 0, 1 ≤ j ≤ J.

(4.2)

We consider the space

W̃p
Σ(Ω) =

{
v ∈ X̃p

0(Ω) : ⟨v · n, 1⟩Σj = 0, 1 ≤ j ≤ J, ⟨v · n, 1⟩Γℓ
N
= 0, 0 ≤ ℓ ≤ LN

}
.

4.1. Hilbertian case. In the following theorem, we prove the existence and uniqueness of the
weak solution of Problem (4.2) in the Hilbert space. We also discuss its strong solutions.

Theorem 4.1. (i) Assume that ∂Σ ⊂ ΓD. Let f ∈ L6/5(Ω) satisfying the following compatibility

condition for any φ ∈ K̃2
0(Ω): ∫

Ω

f ·φ dx = 0. (4.3)

Then, Problem (4.2) admits a unique solution u ∈ H1(Ω) satisfying

∥u∥H1(Ω) ≤ C∥f∥L6/5(Ω). (4.4)

(ii) Furthermore, if Ω is of class C 2,1 and f ∈ L2(Ω) then u belongs to H2(Ω) and satisfies the
estimate

∥u∥H2(Ω) ≤ C∥f∥L2(Ω). (4.5)

Proof. (i) Since it is rather long, we split the proof into four steps.

Step 1: Compatibility condition. The weak formulation of (4.2) is

Find u ∈ W̃2
Σ(Ω) such that for any φ ∈ W̃2

Σ(Ω),∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ) =

∫
Ω

f ·φ.
(4.6)

As it will be shown is Step 3 that Problem (4.2) and Problem (4.6) are equivalent, so the first

compatibility condition (4.3) appears directly by taking φ ∈ K̃2
0(Ω) in (4.6).



10 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

Step 2: Existence and uniqueness. We define the bilinear form a(·, ·) as

a(u,φ) =

∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ),

which is coercive on W̃2
Σ(Ω) because (3.8). Then, by using Lax Milgram’s theorem we infer that

Problem (4.6) admits a unique solution in W̃2
Σ(Ω). Moreover,

∥u∥2H1(Ω) ≤ C∥f∥L6/5(Ω)∥u∥L6(Ω) ≤ C∥f∥L6/5(Ω)∥u∥H1(Ω),

and the required estimate (4.4) is immediately checked.

Step 3: Equivalence. We show now that Problem (4.6) is equivalent to

Find u ∈ W̃2
Σ(Ω) such that for any φ ∈ X̃2

0(Ω),∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ) =

∫
Ω

f ·φ.
(4.7)

It is clear that any solution of (4.7) also solves (4.6). Conversely, we take φ ∈ X̃2
0(Ω) and we

define φ = φ−∇θ where θ is solution of the problem

∆θ = divφ in Ω, θ = 0 on ΓN ,
∂θ

∂n
= 0 on ΓD. (4.8)

Since ΓD ∩ ΓN = ∅, we can prove that θ belongs to H2(Ω) by an argument of partition of unity
leading to two problems with Dirichlet or Neumann boundary condition in the non mixed case.

Now, we can define the function

φ̃ = φ−
LN∑
ℓ=1

J∑
j=1

( 1

LN
⟨φ · n, 1⟩Σj

+
1

J
⟨φ · n, 1⟩Γℓ

N

)
g̃rad sℓj , (4.9)

It is obvious that φ̃ ∈ L2(Ω), div φ̃ = 0, curl φ̃ = curlφ ∈ L2(Ω). Also φ̃ · n = 0 on ΓD and

φ̃× n = 0 on ΓN , thus φ̃ ∈ X̃2
0(Ω). Moreover, since φ · n = 0 on ΓD and divφ = 0 in Ω, we have

⟨φ · n, 1⟩ΓN
= ⟨φ · n, 1⟩Γ =

∫
Ω

divφ dx = 0. (4.10)

Therefore using (3.3) and (4.9), we infer that for any 1 ≤ m ≤ LN

⟨φ̃ · n, 1⟩Γm
N
= ⟨φ · n, 1⟩Γm

N
−

J∑
j=1

( 1

LN
⟨φ · n, 1⟩Σj

+
1

J
⟨φ · n, 1⟩Γm

N

)
.

So

⟨φ̃ · n, 1⟩Γm
N
= − 1

LN

J∑
j=1

⟨φ · n, 1⟩Σj
. (4.11)

Clearly, using (4.10), from this relation, we obtain after summing

0 =

LN∑
m=1

⟨φ̃ · n, 1⟩Γm
N
= −

J∑
j=1

⟨φ · n, 1⟩Σj
.

Using again (4.11), we obtain ⟨φ̃ · n, 1⟩Γm
N

= 0 for any 1 ≤ m ≤ LN . In the same way for any
1 ≤ j ≤ J , we deduce from (4.10) and (4.9) that

⟨φ̃ · n, 1⟩Σk
= ⟨φ · n, 1⟩Σk

−
LN∑
ℓ=1

( 1

LN
⟨φ · n, 1⟩Σk

+
1

J
⟨φ · n, 1⟩Γℓ

N

)
= ⟨φ · n, 1⟩Σk

− ⟨φ · n, 1⟩Σk
= 0.

As a consequence φ̃ belongs to W̃2
Σ(Ω), which means that any solution of (4.6) also solves (4.7).



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 11

Step 4: Interpretation. Now, we give the interpretation of Problem (4.7). More precisely,
we will prove that Problem (4.6) is equivalent to find u ∈ H1(Ω) solution of Problem (4.2). By
choosing φ ∈ D(Ω), we deduce that

−∆u = f in Ω. (4.12)

Moreover, because u ∈ W̃2
Σ(Ω), it satisfies u× n = 0 on ΓN , u · n = 0 on ΓD, ⟨u · n, 1⟩Γℓ

N
= 0 for

any 1 ≤ ℓ ≤ LN and ⟨u ·n, 1⟩Σj
= 0 for any 1 ≤ j ≤ J . It remains to show that curlu×n = 0 on

ΓD and divu = 0 on ΓN . Multiplying (4.12) by φ ∈ [̃X
2

0(Ω) and using the Green’s formula (3.10)
leads to∫

Ω

f ·φ =

∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ) + ⟨curlu× n,φ⟩ΓD
− ⟨divu,φ · n⟩ΓN

.

Using (4.7), we deduce that

∀φ ∈ X̃2
0(Ω), ⟨curlu× n,φ⟩ΓD

− ⟨divu,φ · n⟩ΓN
= 0

Let µ any element of H1/2(ΓD). So, there exists φ of H1(Ω) such that φ = µτ on ΓD and φ = 0
on ΓN and

⟨curlu× n,µ⟩ΓD
= ⟨curlu× n,µτ ⟩ΓD

= ⟨curlu× n,φ⟩ΓD
= 0.

This implies that curlu× n = 0 on ΓD. Now, let µ be any element of H1/2(ΓN ). So, there exists

ψ in H2(Ω) such that ∂ψ
∂n = µ on ΓN and ψ = 0 on ΓD. Thus

⟨divu, µ⟩ΓN
= ⟨divu, ∂ψ

∂n
⟩ΓN

= ⟨divu,φ · n⟩ΓN
= 0,

which means that divu = 0 on ΓN which is the required property.

(ii) Regularity. Assume that f ∈ L2(Ω) and Ω is of class C 2,1. Arguing with a partition of
unity, let θ be a function defined in C∞

0 (R3), 0 ≤ θ ≤ 1 and such that θ = 1 at the neighborhood
of ΓN and θ = 0 at the neighborhood of ΓD. Setting η = 1 − θ, v = θu, w = ηu and Fθ =
θf − 2∇θ · ∇u− u∆θ, we look for the regularity of v ∈ H1(Ω) which satisfies

−∆v = Fθ in Ω,

v × n = 0, divv = 0 on Γ,
(4.13)

where Fθ ∈ L2(Ω). Observe that divv ∈ L2(Ω) and ∆divv = divFθ ∈ H−1(Ω) with divv = 0 on
Γ. Let χ ∈ H1

0 (Ω) such that ∆χ = divFθ in Ω. The function λ = χ−divv is harmonic and λ = 0
on Γ then λ is zero [2] and thus divv belongs toH1(Ω). We know that curl curlv = −∆v+∇(divv)
which means that curl curlv ∈ L2(Ω) and consequently curlv ∈ X2

T (Ω) ↪→ H1(Ω). We infer that
v belongs to H2(Ω) because v ∈ L2(Ω), curlv ∈ H1(Ω), divv ∈ H1(Ω) and v × n = 0 on Γ
according to (3.2).

Now, we look for the regularity of w ∈ H1(Ω) satisfying

−∆w = Fη in Ω,

w · n = 0, curlw × n = 0 on Γ,
(4.14)

where Fη ∈ L2(Ω). Setting z = curlw so z ∈ L2(Ω) and it satisfies

−∆z = curlFη and div z = 0 in Ω,

z× n = 0 on Γ,

⟨z · n, 1⟩Γi = 0, ∀1 ≤ i ≤ I,

(4.15)

where Γi, 1 ≤ i ≤ I, are the connected components of Γ = ΓD ∪ ΓN . There exists a unique
ψ ∈ H1(Ω) such that −∆ψ = curlFη and divψ = 0 in Ω. Moreover, ψ × n = 0 on Γ and
⟨ψ · n, 1⟩Γi

= 0, for 1 ≤ i ≤ I [7]. The function ϕ = ψ − z satisfies ∆ϕ = 0 in Ω, ϕ× n = 0 on Γ
and ⟨ϕ · n, 1⟩Γi

= 0 for any 1 ≤ i ≤ I thus ϕ = 0 and then z ∈ H1(Ω). Moreover

∇(divw) = ∆w + curl curlw = −Fη + curl z



12 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

which belongs to L2(Ω) and then divw ∈ H1(Ω). We deduce that w belongs to H2(Ω) since
w ∈ L2(Ω), divw ∈ H1(Ω), curlw ∈ H1(Ω) and w ·n = 0 on Γ thanks to (3.1). As a consequence,
u = v +w ∈ H2(Ω) and it satisfies (4.5). □

Remark 4.2. If Ω is only Lipschitz, Problem (4.2) admits a unique solution in X̃2
0(Ω) that does

not belong to H1(Ω) in general but to H1/2(Ω) due to the embedding of X̃2
0(Ω) into H1/2(Ω) (see

[10] for the non mixed case).

Remark 4.3. If Ω is of class C 2,1 and f ∈ L6/5(Ω) then the solution of problem (4.2) belongs to
W2,6/5(Ω). The proof follows the same lines of the one of Theorem 4.1 point (ii).

Remark 4.4. As in the step 3 of the proof of Theorem 4.1, we can check that if a × n = 0
and b = 0 in (2.1), the problem: Find u ∈ H1(Ω) solution of (2.1), is equivalent to the weak
formulation (4.6), in which, we replace the linear form φ 7→

∫
Ω
f ·φ by

φ 7→
∫
Ω

f ·φ+ ⟨h× n,φ⟩ΓD
− ⟨g,φ · n⟩ΓN

In the next, we will see the equivalent version of Theorem 4.1 in the case where ∂Σ ⊂ ΓN .

Theorem 4.5. (i) Assume that ∂Σ ⊂ ΓN and f ∈ L6/5(Ω) satisfying (4.3). Then the problem

−∆u = f in Ω,

u× n = 0, divu = 0 on ΓN ,

u · n = 0, curlu× n = 0 on ΓD,

⟨u · n, 1⟩Γℓ
N
= 0, 1 ≤ ℓ ≤ LN ,

(4.16)

admits a unique solution u ∈ H1(Ω) satisfying estimate (4.4).
(ii) Moreover, if Ω is of class C 2,1 and f ∈ L2(Ω) then u belongs to H2(Ω) and satisfies the

estimate (4.5).

Sketch of the proof. The space we are working here is

Ṽp
0(Ω) =

{
v ∈ X̃p

0(Ω), ⟨v · n, 1⟩Γℓ
N
= 0, 0 ≤ ℓ ≤ LN

}
.

The weak formulation of (P0
N ) is

Find u ∈ Ṽ2
0(Ω) such that for any φ ∈ Ṽ2

0(Ω),∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ) =

∫
Ω

f ·φ.
(4.17)

To extend (4.17) to any test function φ ∈ X̃2
0(Ω), we set

φ̃ = φ−∇θ −
LN∑
ℓ=1

⟨(φ−∇θ) · n, 1⟩Γℓ
N
∇sℓ,

where θ is defined in (4.8). It is clear that φ̃ ∈ Ṽ2
0(Ω). The remainder of the questions of the

existence, uniqueness and regularity can be solved with the same arguments as in Theorem 4.1. □

Remark 4.6. If Ω is only Lipschitz, Problem (4.16) admits a unique solution u in X̃2
0(Ω) that is

not necessary in H1(Ω).

In the next section, we will analyze generalized and strong solutions in Lp-theory in the case
where ∂Σ ⊂ ΓD and then ∂Σ ⊂ ΓN . We will also treat the case of non-homogeneous boundary
conditions.



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 13

4.2. Lp-theory. Unlike the inf-sup conditions of Proposition 3.2 which involve divergence-free
vector fields, we extend in the following the condition (3.6) to a more general case where the
divergence is not specified, this result is useful in the forthcoming analysis.

Lemma 4.7. Assume that ∂Σ ⊂ ΓD and 1 < p < +∞. Then there exists a constant β > 0
depending only on Ω and p such that the following inf-sup condition holds

inf
φ∈W̃p′

Σ (Ω),φ ̸=0

sup
u∈W̃p

Σ(Ω). u̸=0

∣∣ ∫
Ω
curlu · curlφ+ (divu)(divφ)

∣∣
∥u∥W1,p(Ω)∥φ∥W1,p′ (Ω)

≥ β. (4.18)

Proof. Let g ∈ Lp(Ω) and λ ∈ Lp(Ω). Then there exists a unique θ ∈W 1,p(Ω) such that θ = 0 on
ΓD, (∇θ − g) · n = 0 on ΓN satisfying ∆θ = div g in Ω and

∥θ∥W 1,p(Ω) ≤ C∥g∥Lp(Ω).

We set z = g −∇θ and

z̃ = z−
LD∑
ℓ=1

⟨z · n, 1⟩Γℓ
D
∇qℓ.

Thus z̃ ∈ Lp(Ω), div z̃ = 0 in Ω, z̃ · n = 0 on ΓN and ⟨z̃ · n, 1⟩Γm
D
= 0, for any 1 ≤ m ≤ LD.

From [4, Theorem 4.18] where ΓD and ΓN are swapped, there exists a vector potential ψ ∈
W̃p

Σ,σ(Ω) such that z̃ = curlψ and it satisfies the estimate

∥ψ∥W1,p(Ω) ≤ C∥z̃∥Lp(Ω) ≤ C∥g∥Lp(Ω).

Now, let χ ∈W 2,p(Ω) be the unique solution of

∆χ = λ in Ω,
∂χ

∂n
= 0 on ΓD, χ = 0 on ΓN

and satisfying

∥χ∥W 2,p(Ω) ≤ C∥λ∥Lp(Ω).

Now, we set v = ψ +∇χ and

u = v −
LN∑
ℓ=1

J∑
j=1

( 1

LN
⟨u · n, 1⟩Σj +

1

J
⟨u · n, 1⟩Γℓ

N

)
g̃rad sℓj .

Then u ∈ W1,p(Ω) and

curlu = curlψ = z̃, divu = λ in Ω

u · n = 0 on ΓD, u× n = 0 on ΓN

⟨u · n, 1⟩Γm
N
= 0, 1 ≤ m ≤ LN , ⟨u · n, 1⟩Σj = 0, 1 ≤ j ≤ J,

with the estimate

∥u∥W1,p(Ω) ≤ C(∥g∥Lp(Ω) + ∥λ∥Lp(Ω)). (4.19)

Let φ ∈ W̃p′

Σ (Ω), so the following estimate holds

∥φ∥W1,p′ (Ω) ≤ C
(
∥ curlφ∥Lp′ (Ω) + ∥ divφ∥Lp′ (Ω)

)
= C sup

g∈Lp(Ω), g ̸=0, λ∈Lp(Ω), λ ̸=0

∣∣ ∫
Ω
curlφ · g + (divφ)λ dx

∣∣
∥g∥Lp(Ω) + ∥λ∥Lp(Ω)

.

But ∫
Ω

curlφ · g dx =

∫
Ω

curlφ ·
(
z̃+∇θ +

LD∑
ℓ=1

⟨z · n, 1⟩Γℓ
D
∇ qℓ

)
dx.

Note that ∫
Ω

curlφ · ∇θ dx =

∫
Ω

curlφ · ∇qℓ dx = 0.



14 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

So, from (4.19) we obtain∣∣ ∫
Ω
curlφ · g + (divφ)λ dx

∣∣
∥g∥Lp(Ω) + ∥λ∥Lp(Ω)

=

∣∣ ∫
Ω
curlφ · curlu+ (divφ)(divu) dx

∣∣
∥g∥Lp(Ω) + ∥λ∥Lp(Ω)

≤ 1

C

∣∣ ∫
Ω
curlφ · curlu+ (divφ)(divu) dx

∣∣
∥u∥W1,p(Ω)

.

Consequently, the required condition (4.18) is obtained. □

In a similar way as above, we can establish the following condition when ∂Σ ⊂ ΓN .

Lemma 4.8. Assume that ∂Σ ⊂ ΓN and 1 < p < +∞. Then there exists a constant β > 0
depending only on Ω and p such that the following inf-sup condition holds

inf
φ∈Ṽp′

Σ (Ω),φ ̸=0

sup
u∈Ṽp

Σ(Ω), u̸=0

∣∣ ∫
Ω
curlu · curlφ+ (divu)(divφ)

∣∣
∥u∥W1,p(Ω)∥φ∥W1,p′ (Ω)

≥ β. (4.20)

In the next theorem, we investigate the Lp-theory of the solutions of Problem (4.2) where the
role of the inf-sup condition (4.18) is illustrated. We recall the definition of the exponent

r(p) =


3p
p+3 if p > 3/2

1 + ε if p = 3/2

1 if 1 ≤ p < 3/2,

(4.21)

where ε > 0 is arbitrary.

Theorem 4.9. Assume that ∂Σ ⊂ ΓD.
(i) Let f ∈ Lr(p)(Ω) with 3/2 ≤ p < +∞ satisfying the compatibility condition (4.3). Then,

there exists a unique solution u ∈ W1,p(Ω) to Problem (4.2). Moreover, u satisfies the following
estimate

∥u∥W1,p(Ω) ≤ C∥f∥Lr(p)(Ω). (4.22)

(ii) If f ∈ L1(Ω), then there exists unique u ∈ W1,s(Ω) for any s < 3
2 , solution of Problem

(4.2) and satisfying

∥u∥W1,s(Ω) ≤ C∥f∥L1(Ω). (4.23)

(iii) Furthermore, if Ω is of class C 2,1 and f ∈ Lp(Ω) with 1 < p < +∞ then u belongs to
W2,p(Ω) and satisfies the estimate

∥u∥W2,p(Ω) ≤ C∥f∥Lp(Ω). (4.24)

Proof. Step 1: Uniqueness. Suppose that f = 0 in Ω and let u ∈ W1,p(Ω) solution of Problem
(4.2). Applying the divergence operator, we obtain ∆(divu) = 0 in Ω. But divu = 0 on ΓN and
curlu×n = 0 on ΓD in the sense of W−1/p,p(ΓD) which implies that divΓD

(curlu×n) = 0 in the
sense of W−1−1/p,p(ΓD). Moreover, curl curlu · n = divΓD

(curlu× n) = 0 on ΓD. Since ∆u = 0
in Ω, we have

curl curlu = ∇(divu) in Ω.

So, ∂(divu)
∂n = 0 on ΓD this yields divu = 0 in Ω according to Lemma 3.4.

Now, we set z = curlu in (4.2), then curl z = ∇(divu) = 0 and div z = 0 in Ω, z × n = 0 on

ΓD, z · n = 0 on ΓN , which means that z ∈ Kp
0(Ω) so we can write z =

∑LD

ℓ=1⟨z · n, 1⟩Γℓ
D
∇qℓ but

since ⟨z · n, 1⟩Γℓ
D
= 0 for any 1 ≤ ℓ ≤ LD, we infer that z = 0. Finally, we have

curlu = 0, divu = 0 in Ω, u× n = 0 on ΓN , u · n = 0 on ΓD.

So, u ∈ K̃p
0(Ω) and since the fluxes of u over Σj and ΓℓN are equal to zero, we deduce that u = 0

in Ω and this completes the uniqueness proof.



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 15

Step 2: Existence. We consider the problem

Find u ∈ W̃p
Σ(Ω) such that for any φ ∈ W̃p′

Σ (Ω),∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ) =

∫
Ω

f ·φ,
(4.25)

We recall that the inf-sup condition (4.18) holds for any 1 < p < +∞. Then, it suffices to show

that the right-hand side of (4.25) is a linear and continuous form on W̃p
Σ(Ω) for any 1 < p < +∞.

(i) When p ≥ 3
2 , since f ∈ Lr(p)(Ω) so the right-hand side is a linear and continuous form

on W̃p
Σ(Ω). By the inf-sup condition (4.18), Problem (4.25) has a unique solution in W̃p

Σ(Ω) ↪→
W1,p(Ω). As in the proof of Theorem 4.1, we show that (4.25) is equivalent to

Find u ∈ W̃p
Σ(Ω) such that for any φ ∈ X̃p′

0 (Ω),∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ) =

∫
Ω

f ·φ,
(4.26)

and that (4.26) is equivalent to: find u ∈ W̃p
Σ(Ω) solution of Problem (4.2) by using the same

arguments. Observe that W̃p
Σ(Ω) ↪→ W1,p(Ω).

(ii) When 1 ≤ p < 3
2 , from the definition of the exponent r(p) given in (4.21), we have f ∈ L1(Ω)

and then the right-hand side of (4.25) is a linear and continuous form on W̃p
Σ(Ω) since W̃

p′

Σ (Ω) ↪→
W1,p′(Ω) ↪→ L∞(Ω). Using the same argument than above, we obtain a solution u ∈ ∩W1,s(Ω)
for any s < 3

2 , satisfying the estimate (4.23).

(iii) Regularity. Assume that Ω is of class C 2,1 and f ∈ Lp(Ω), we prove that the solution of
(4.2) belongs to W2,p(Ω) with the same approach used in the last step of the proof of Theorem
4.1. □

Remark 4.10. If Ω is of class C 2,1 and f ∈ Lr(p)(Ω) then the solution of problem (4.2) belongs
to W2,r(p)(Ω).

Remark 4.11. In some previous works [6, 7], similar problems have been studied in the case where
ΓN = ∅ with divu = 0 in Ω and the solution is obtained when div f = 0 in Ω and f · n = 0 on Γ
are imposed. Having in mind these results, the question we may ask is under which assumptions,
the solution of (4.2) satisfies divu = 0 in Ω?

Proposition 4.12. Let f ∈ Lr(p)(Ω) satisfying the compatibility condition (4.3), if in addition
div f = 0 in Ω and f · n = 0 on ΓD, then the unique solution u given in Theorem 4.9 satisfies
divu = 0 in Ω.

Proof. Note that the solution u satisfies ∆(divu) = 0 in Ω because ∆(divu) = div f in Ω. Since
divu ∈ Lp(Ω) thus ∂

∂n divu ∈W−1−1/p,p(Γ).

We need to prove that ∂
∂n divu = 0 on Γ, in particular that ∂

∂n divu = 0 on ΓD. Using the

Green formula (3.10), we have for any χ ∈W 2,p′(Ω) with ∂χ
∂n = 0 on ΓD and χ = 0 on ΓN

−
∫
Ω

∆u · ∇χ =

∫
Ω

(divu)(∆χ)− ⟨curlu× n,∇χ⟩ΓD
+ ⟨divu, ∂χ

∂n
⟩ΓN

.

But divu = 0 on ΓN and curlu× n = 0 on ΓD, so we have

⟨curlu× n,∇χ⟩ΓD
= ⟨curlu× n,∇τχ⟩ΓD

= −⟨divΓ(curlu× n), χ⟩ΓD
.

As −∆u · n = f · n = 0 on ΓD and χ = 0 on ΓN , we have

−
∫
Ω

∆u · ∇χ =

∫
Ω

χ(div∆u)− ⟨∆u · n, χ⟩Γ = 0

Now, using the following Green formula gives

0 =

∫
Ω

(divu)(∆χ) =

∫
Ω

χ∆(divu) + ⟨ ∂
∂n

divu, χ⟩ΓD
= ⟨ ∂

∂n
divu, χ⟩ΓD

.

We deduce that ∂
∂n divu = 0 on ΓD and we already know that divu = 0 on ΓN and ∆(divu) = 0

then divu = 0 in Ω according to Lemma 3.4. □



16 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

In a similar way as in Theorem 4.9, we can establish the existence, uniqueness and regularity
of the solution of Problem (P0

N ) in Lp-theory.

Theorem 4.13. Assume that ∂Σ ⊂ ΓN .
(i) Let f ∈ Lr(p)(Ω) with 3/2 ≤ p < +∞, satisfying the compatibility condition (4.3) for any

φ ∈ K̃p
0(Ω). Then, there exists a unique solution u ∈ W1,p(Ω) of Problem (4.16). Moreover, u

satisfies

∥u∥W1,p(Ω) ≤ C∥f∥Lr(p)(Ω).

(ii) If f ∈ L1(Ω). Then, there exists a unique solution u ∈ W1,s(Ω) for any s < 3/2 of Problem
(4.16). Moreover, u satisfies

∥u∥W1,s(Ω) ≤ C∥f∥L1(Ω).

(iii) Furthermore, if Ω is of class C 2,1 and f ∈ Lp(Ω) then u belongs to W2,p(Ω) and satisfies

∥u∥W2,p(Ω) ≤ C∥f∥Lp(Ω).

In Theorem 4.9, we have studied the problem with homogeneous boundary conditions. We go
back now to the original problem including non-homogeneous boundary conditions (2.1) in the
case where ∂Σ ⊂ ΓD.

Theorem 4.14. (i) Assume that ∂Σ ⊂ ΓD. Let f ∈ Lr(p)(Ω) with 3/2 ≤ p < +∞, a × n ∈
W1−1/p,p(ΓN ), g ∈ W−1/p,p(ΓN ), b ∈ W 1−1/p,p(ΓD) and h × n ∈ W−1/p,p(ΓD) satisfying the
following compatibility condition∫

Ω

f ·φ dx− ⟨h× n,φ⟩ΓD
+ ⟨g,φ · n⟩ΓN

= 0. (4.27)

Then Problem (2.1) has a unique solution u ∈ W1,p(Ω) for any 1 < p < +∞ and it satisfies

∥u∥W1,p(Ω) ≤ C
(
∥f∥Lr(p)(Ω) + ∥g∥W−1/p,p(ΓN ) + ∥h× n∥W−1/p,p(ΓD)

+ ∥a× n∥W1−1/p,p(ΓN ) + ∥b∥W 1−1/p,p(ΓD)

)
.

(4.28)

(ii) If f ∈ L1(Ω), then there exists a unique u ∈ W1,s(Ω) for any s < 3
2 solution of Problem

(2.1) and satisfying

∥u∥W1,s(Ω) ≤ C
(
∥f∥L1(Ω) + ∥g∥W−1/p,p(ΓN ) + ∥h× n∥W−1/p,p(ΓD)

+ ∥a× n∥W1−1/p,p(ΓN ) + ∥b∥W 1−1/p,p(ΓD)

)
.

(iii) Furthermore, if Ω is of class C 2,1, f ∈ Lp(Ω) with 1 < p < +∞, a × n ∈ W2−1/p,p(ΓN ),
g ∈W 1−1/p,p(ΓN ), b ∈W 2−1/p,p(ΓD) and h× n ∈ W1−1/p,p(ΓD) then u belongs to W2,p(Ω) and
the following estimate holds

∥u∥W2,p(Ω) ≤ C
(
∥f∥Lp(Ω) + ∥g∥W 1−1/p,p(ΓN ) + ∥h× n∥W1−1/p,p(ΓD)

+ ∥a× n∥W2−1/p,p(ΓN ) + ∥b∥W 2−1/p,p(ΓD)

)
.

(4.29)

Proof. (i) Step 1: Uniqueness. By taking f = 0, h = 0, a = 0, b = 0 and g = 0 in Problem
(2.1), we proceed as in the proof of Theorem 4.9 to deduce that u is unique.

Step 2: Compatibility condition. According to (3.11), every solution of (2.1) also solves

Find u ∈ W1,p(Ω) such that for any φ ∈ W̃p′

Σ (Ω),∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ) =

∫
Ω

f ·φ−⟨h× n,φ⟩ΓD
+ ⟨g,φ · n⟩ΓN

.
(4.30)

By taking φ ∈ K̃p′

0 (Ω), the compatibility condition (4.27) is obtained.



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 17

Step 3: Equivalence. We prove that (4.30) is equivalent to

Find u ∈ W1,p(Ω) such that for any φ ∈ X̃p′

0 (Ω),∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ) =

∫
Ω

f ·φ−⟨h× n,φ⟩ΓD
+ ⟨g,φ · n⟩ΓN

,
(4.31)

in the same way as in the proof of Theorem 4.1.

Step 4: Existence and estimate. Let χ ∈W 2,p(Ω) such that

∆χ = 0 in Ω,
∂χ

∂n
= b on ΓD, χ = 0 on ΓN ,

with the estimate

∥χ∥W 2,p(Ω) ≤ C∥b∥W 1−1/p,p(ΓD).

We set

ξ = u−∇χ−
J∑
j=1

LN∑
ℓ=1

( 1

J
⟨(u−∇χ) · n, 1⟩Γℓ

N
+

1

LN
⟨(u−∇χ) · n, 1⟩Σj

)
g̃rad sℓj , (4.32)

where the properties of g̃rad sℓj are given in (3.3). We are now reduced to solve the following
problem

−∆ξ = f in Ω,

ξ × n = a× n, div ξ = g on ΓN ,

ξ · n = 0, curl ξ × n = h× n on ΓD,

⟨ξ · n, 1⟩Γℓ
N
= 0, 1 ≤ ℓ ≤ LN , ⟨ξ · n, 1⟩Σj = 0, 1 ≤ j ≤ J.

Since aτ ∈ W1−1/p,p(ΓN ), there exists a unique (ξ0, π0) ∈ W1,p(Ω)× Lp(Ω) solution of

∆ξ0 +∇π0 = 0, div ξ0 = 0, in Ω,

ξ0 = aτ on ΓN , ξ0 = 0 on ΓD.

Note that since ξ0 ∈ W1,p(Ω) and ∆ξ0 ∈ [Hp′

0 (div,Ω)]′, so curl ξ0×n makes sense in W−1/p,p(Γ)
according to Corollary 4.2 [6]. Setting z̃ = ξ − ξ0 so z̃ satisfies

−∆z̃ = f +∆ξ0 in Ω,

z̃× n = 0, div z̃ = g on ΓN ,

z̃ · n = 0, curl z̃× n = h× n− curl ξ0 × n on ΓD,

⟨z̃ · n, 1⟩Γℓ
N
= 0, 1 ≤ ℓ ≤ LN , ⟨z̃ · n, 1⟩Σj

= −⟨ξ0 · n, 1⟩Σj
, 1 ≤ j ≤ J.

We finally set

z = z̃−
J∑
j=1

LN∑
ℓ=1

( 1

J
⟨z̃ · n, 1⟩Γℓ

N
+

1

LN
⟨z̃ · n, 1⟩Σj

)
g̃rad sℓj ,

we obtain
−∆z = f +∆ξ0 in Ω,

z× n = 0, div z = g on ΓN ,

z · n = 0, curl(z+ ξ0)× n = h× n on ΓD,

⟨z · n, 1⟩Γℓ
N
= 0, 1 ≤ ℓ ≤ LN , ⟨z · n, 1⟩Σj = 0, 1 ≤ j ≤ J.

(4.33)

This problem is equivalent to

Find z ∈ W̃p
Σ(Ω) such that for any φ ∈ W̃p′

Σ (Ω),∫
Ω

curl z · curlφ+

∫
Ω

(div z)(divφ) =

∫
Ω

f ·φ−
∫
ΓD

curl ξ0 · curlφ− ⟨h× n,φ⟩ΓD
+ ⟨g,φ · n⟩ΓN

.

(4.34)



18 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

Note that the right-hand side is a linear and continuous form on W̃p
Σ(Ω) for any p ≥ 3

2 , by use

of the inf-sup condition (4.18), this problem admits a unique solution z ∈ W̃p
Σ(Ω) ↪→ W1,p(Ω)

satisfying

∥z∥W1,p(Ω) ≤ C
(
∥f∥Lr(p)(Ω) + ∥g∥W−1/p,p(ΓN ) + ∥h× n∥W−1/p,p(ΓD) + ∥ curl ξ0∥Lp(Ω)

)
≤ C

(
∥f∥Lr(p)(Ω) + ∥g∥W−1/p,p(ΓN ) + ∥h× n∥W−1/p,p(ΓD) + ∥a× n∥W−1/p,p(ΓN )

)
.

From the construction of z, we have

∥ξ∥W1,p(Ω) ≤ C
(
∥a× n∥W1−1/p,p(ΓN ) + ∥z∥W1,p(Ω)

)
.

By (4.32), we have

∥u∥W1,p(Ω) ≤ ∥ξ∥W1,p(Ω) + ∥∇χ∥W1,p(Ω) ≤ ∥ξ∥W1,p(Ω) + ∥b∥W 1−1/p,p(ΓD).

The estimate (4.29) is then deduced.

Step 5: Interpretation. To give the interpretation of (4.31), otherwise, to prove that (4.31) is
equivalent to find u ∈ W1,p(Ω) solution of Problem (2.1), we use Remark 4.4 and we follow the
same lines of the proof of Theorem 4.1.

(ii) When 1 ≤ p < 3
2 , we prove that there exists a unique solution u ∈ ∩W1,s(Ω) for any s < 3

2
using the same arguments than in the proof of Theorem 4.9 point ii).

(iii) Regularity. Assume that f ∈ Lp(Ω) and Ω is of class C 2,1 as in Theorem 4.1. We can prove
the W2,p(Ω) regularity by means of a partition of unity such that v = θz and w = ηz. Indeed,
∆(divv) = divFθ ∈ W−1,p(Ω), divv = g on Γ where g ∈ W 1−1/p,p(Γ) so divv ∈ W 1,p(Ω).

Moreover, v ∈ W1,p(Ω) thus curlv ∈ X̃p
T (Ω) ↪→ W1,p(Ω). As, v ∈ Lp(Ω) and v × n = 0 on Γ

hence, v ∈ W2,p(Ω) according to (3.2).
Setting λ = curlw, so −∆λ = curlFη, divλ = 0 in Ω and λ × n = h × n ∈ W1−1/p,p(Γ)

then λ ∈ W1,p(Ω) referring to [7]. Thus, ∇(divw) = −Fη + curlλ ∈ Lp(Ω). So w ∈ Lp(Ω),
curlw ∈ W1,p(Ω) and divw ∈W 1,p(Ω) with w ·n = 0 on Γ, we deduce that w belongs to W2,p(Ω)
thanks to (3.1). Therefore, z ∈ W2,p(Ω) and consequently u belongs to W2,p(Ω) satisfying
(4.29). □

Also, Problem (4.16) can be considered in the case of non-homogeneous boundary conditions
and can be solved with the same tools as above.

Theorem 4.15. Assume that ∂Σ ⊂ ΓN .
(i) Let f ∈ Lr(p)(Ω) with 3/2 ≤ p < +∞, a × n ∈ W1−1/p,p(ΓN ), g ∈ W−1/p,p(ΓN ), b ∈

W 1−1/p,p(ΓD) and h × n ∈ W−1/p,p(ΓD) satisfying the compatibility condition (4.27). Then the
problem

−∆u = f in Ω,

u× n = a× n, divu = g on ΓN ,

u · n = b, curlu× n = h× n on ΓD,

⟨u · n, 1⟩Γℓ
N
= 0, 1 ≤ ℓ ≤ LN ,

(4.35)

has a unique solution u ∈ W1,p(Ω) satisfying (4.28).
(ii) If f ∈ L1(Ω) then there exists a unique u ∈ W1,s(Ω) solution of Problem (4.35)
(iii) Furthermore, if Ω is of class C 2,1, f ∈ Lp(Ω), a× n ∈ W2−1/p,p(ΓN ), g ∈W 1−1/p,p(ΓN ),

b ∈W 2−1/p,p(ΓD) and h×n ∈ W1−1/p,p(ΓD) then u belongs to W2,p(Ω) and the estimate (4.29)
holds.

We consider now a more general right-hand side than Problem (2.1). This choice is motivated
by the generalized Neumann problem which allows to have a right-hand side in negative spaces



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 19

and this explains the boundary condition on ΓD.

−∆u = f + curlF in Ω,

u× n = 0, divu = g on ΓN ,

u · n = 0, (curlu− F)× n = h× n on ΓD,

⟨u · n, 1⟩Γℓ
N
= 0, 1 ≤ ℓ ≤ LN , ⟨u · n, 1⟩Σj = 0, 1 ≤ j ≤ J.

(4.36)

Theorem 4.16. Assume that ∂Σ ⊂ ΓD.
(i) Let f ∈ Lr(p)(Ω), F ∈ Lp(Ω) with 1 < p < +∞, g ∈W−1/p,p(ΓN ) and h×n ∈ W−1/p,p(ΓD)

satisfying the following compatibility condition for any φ ∈ K̃p
0(Ω):∫

Ω

f ·φ dx− ⟨h× n,φ⟩ΓD
+ ⟨g,φ · n⟩ΓN

= 0. (4.37)

Then, Problem (4.36) has a unique solution u ∈ W1,p(Ω) and it satisfies

∥u∥W1,p(Ω) ≤ C
(
∥f∥Lr(p)(Ω) + ∥F∥Lp(Ω) + ∥g∥W−1/p,p(ΓN ) + ∥h× n∥W−1/p,p(ΓD)

)
(4.38)

(ii) Furthermore, if Ω is of class C 2,1, f ∈ Lp(Ω), g ∈W 1−1/p,p(ΓN ) and h×n ∈ W1−1/p,p(ΓD)
with F = 0 then u belongs to W2,p(Ω) and the following estimate holds

∥u∥W2,p(Ω) ≤ C
(
∥f∥Lp(Ω) + ∥g∥W 1−1/p,p(ΓN ) + ∥h× n∥W1−1/p,p(ΓD)

)
. (4.39)

Proof. (i) The weak formulation of Problem (4.36) is

Find u ∈ W̃p
Σ(Ω) such that for any φ ∈ W̃

p

Σ(Ω),∫
Ω

curlu · curlφ+

∫
Ω

(divu)(divφ)

=

∫
Ω

f ·φ+

∫
Ω

F · curlφ+ ⟨h× n,φ⟩ΓD
+ ⟨g,φ · n⟩ΓN

.

(4.40)

By use of the inf-sup condition (4.18), then Problem (4.36) admits a unique solution u ∈ W1,p(Ω)
satisfying the estimate (4.38). The rest of the proof is similar to that of Theorem 4.14.

(ii) With F = 0, we can prove that u belongs to W2,p(Ω) using the same arguments of Problem
(2.1) in Theorem 4.14. □

Remark 4.17. In item (ii) of the the above theorem, we have assumed that f ∈ Lp(Ω) and
F = 0 to get a solution in W2,p(Ω). In the case where F ̸= 0, it suffices to take f ∈ Lp(Ω) and
curlF ∈ Lp(Ω) to have the same regularity result.

Remark 4.18. We can also consider the same generalized right-hand side for Problem (4.35) and
we obtain the same results as for (4.36).

5. Second elliptic problem

We recall the elliptic problem

−∆u = f + divF in Ω,

u× n = a× n, [(∇u+ F)n] · n = g on ΓN ,

u · n = b, [(∇u+ F)n]× n = h on ΓD,

(5.1)

where f , a, b, g and h are given data. We start as the for the first problem by the following special
case:

−∆u = f + divF in Ω,

u× n = 0, [(∇u+ F)n] · n = g on ΓN ,

u · n = 0, [(∇u+ F)n]× n = h on ΓD.

(5.2)

We introduce the space

Vp
n,τ (Ω) =

{
v ∈ W1,p(Ω), v · n = 0 on ΓD, v × n = 0 on ΓN

}
.



20 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

Remark 5.1. Let Ω be a Lipschitz domain and v ∈ Ep(∆;Ω), then ∂v
∂n ∈ W−1/p,p(Γ) and we

have for any φ ∈ W1,p′(Ω) the Green formula∫
Ω

φ ·∆v +

∫
Ω

∇v : ∇φ = ⟨∂v
∂n

,φ⟩Γ, (5.3)

Furthermore, if Ω is of class C 1,1, then
(
∂u
∂n

)
τ
makes sense inW−1/p,p(Γ) and for any φ ∈ W1,p′(Ω)

such that φ · n = 0 on Γ, ∫
Ω

φ ·∆v +

∫
Ω

∇v : ∇φ = ⟨
(∂v
∂n

)
τ
,φ⟩Γ. (5.4)

In the next proposition, we prove that the Poincaré inequality holds for all u ∈ Vp
n,τ (Ω).

Proposition 5.2. For any v ∈ Vp
n,τ (Ω), we have

∥v∥Lp(Ω) ≤ C∥∇v∥Lp(Ω) (5.5)

where C > 0 is a constant that only depends on Ω.

Proof. Reasoning by absurdity and assuming that (5.5) is not valid. Then, we can find a sequence
{vk}k ∈ Vp

n,τ (Ω) such that

∥∇vk∥Lp(Ω) <
1

k
∥vk∥Lp(Ω). (5.6)

Setting then uk = vk

∥vk∥Lp(Ω)
, we deduce that (uk) is bounded in W1,p(Ω) with ∥uk∥Lp(Ω) = 1.

Moreover

∇uk → 0 strongly in Lp(Ω),

otherwise ∇u = 0 in D′(Ω). Since Ω is supposed to be a connected set, we have u = a where a is a
constant in R3. Because of the weak convergence of uk to a in W1,p(Ω) and that 0 = uk ·n⇀ a ·n
in W 1−1/p,p(ΓD), also 0 = uk × n⇀ a× n in W1−1/p,p(ΓN ) thus a = 0 on Γ. Therefore

uk → 0 in Lp(Ω),

which contradicts ∥uk∥Lp(Ω) = 1. □

According to Lemma 3.6, we recall that the space D(Ω) is dense in Ep(∆;Ω) and the mapping
v 7→ ∂v

∂n is continuous from Ep(∆;Ω) into W−1/p,p(Γ).

Lemma 5.3. For each v ∈ Ep(∆;Ω), the following relations hold

divv = divΓ vτ − 2Kv · n+
∂v

∂n
· n on Γ, (5.7)

curlv × n = −∇τ (v · n) + Λv +
(∂v
∂n

)
τ

on Γ, (5.8)

Proof. Using the above density, there exists a sequence (vk)k ∈ D(Ω) such that vk → v in
Ep(∆;Ω). Thus

divvk = divΓ vkτ − 2Kvk · n+
∂vk
∂n

· n on Γ. (5.9)

Since vk → v in W1,p(Ω) and ∆vk → ∆v in Lr(p)(Ω) then divvk → divv in W−1/p,p(Γ).
Moreover vkτ → vτ in W1−1/p,p(Γ) thus

divΓ vkτ → divΓ vτ in W−1/p,p(Γ).

Since n ∈W 1,∞(Ω), we have

2Kvk · n → 2Kv · n in W 1−1/p,p(Γ),

Now, since ∂vk

∂n → ∂v
∂n in W−1/p,p(Γ) so

∂vk
∂n

· n → ∂v

∂n
· n in W−1/p,p(Γ).

Therefore, the relation (5.7) holds. Then we follow the same lines as when proving (5.8). □



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 21

Definition 5.4. Given f ∈ Lr(p)(Ω), F ∈ Lp(Ω), g ∈ W−1/p,p(ΓN ) and h ∈ W−1/p,p(ΓD) such
that h · n = 0 on ΓN then a function u ∈ Vp

n,τ (Ω) is called weak solution of Problem (5.2) if it

satisfies for any φ ∈ Vp′

n,τ (Ω) that∫
Ω

∇u : ∇φ =

∫
Ω

f ·φ−
∫
Ω

F : ∇φ+ ⟨h,φτ ⟩ΓD
+ ⟨g,φ · n⟩ΓN

. (5.10)

Next, we prove that (5.10) and Problem (5.2) are equivalent.

Proposition 5.5. Let f ∈ Lr(p)(Ω), F ∈ Lp(Ω), g ∈ W−1/p,p(ΓN ) and h ∈ W−1/p,p(ΓD) such
that h · n = 0 on ΓN then the following statements are equivalent:

(i) u ∈ Vp
n,τ (Ω) is solution of (5.2) in the sense of (5.10).

(ii) There exists (u, π) ∈ W1,p(Ω)× Lp0(Ω) satisfying

−∆u = f + divF, divu = 0 in Ω,

u× n = 0 on ΓN ,

[(∇u+ F)n] · n = g in W−1/p,p(ΓN ),

u · n = 0 on ΓD,

[(∇u+ F)n]τ = h in W−1/p,p(ΓD).

(5.11)

Proof. If u ∈ W1,p(Ω) is solution of (5.11), thanks to Green’s formula (5.3), it is easily checked
that u is a weak solution of (5.2) in the sense of (5.10).

Conversely, we can let u ∈ Vp
n,τ (Ω) be a weak solution of (5.2). Taking the test function

φ ∈ D(Ω) in (5.10) leads to

⟨−∆u,φ⟩D′(Ω)×D(Ω) =

∫
Ω

∇u : ∇φ =

∫
Ω

f ·φ−
∫
Ω

F : ∇φ.

So
−∆u = f + divF in Ω. (5.12)

As u ∈ Vp
n,τ (Ω), we have u × n = 0 on ΓN and u · n = 0 on ΓD so it remains to show that

[(∇u+ F)n] · n = g on ΓN and [(∇u+ F)n]τ = h on ΓD. Taking the dual product of (5.12) with

φ ∈ Vp′

n,τ (Ω) and from (5.3), we infer that for any φ ∈ Vp′

n,τ (Ω),

⟨[(∇u+ F)n]τ ,φτ ⟩ΓD
+ ⟨[(∇u+ F)n] · n,φ · n⟩ΓN

= ⟨h,φτ ⟩ΓD
+ ⟨g,φ · n⟩ΓN

.

Let µ ∈ W1/p,p′(ΓD), so there exists φ ∈ W1,p′(Ω) with divφ = 0 in Ω and such that φ = µτ

on ΓD and φ = 0 on ΓN . We can then write

⟨[(∇u+ F)n]τ − h,µ⟩ΓD
= ⟨[(∇u+ F)n]τ − h,µτ ⟩ΓD

= ⟨[(∇u+ F)n]τ − h,φ⟩ΓD
= 0.

Thus [(∇u+ F)n]τ = h in W−1/p,p(ΓD). Now, let µ be any element of W 1/p,p′(ΓN ). Then there

exists ψ ∈W 2,p′(Ω) such that ∆ψ = 0 in Ω, ∂ψ∂n = µ on ΓN and ψ = 0 on ΓD. By setting ∇ψ = φ
we have

⟨[(∇u+ F)n] · n− g, µ⟩ΓN
= ⟨[(∇u+ F)n] · n− g,

∂ψ

∂n
⟩ΓN

= ⟨[(∇u+ F)n] · n− g,φ · n⟩ΓN
= 0,

which implies that [(∇u+ F)n] · n = g ∈W−1/p,p(ΓN ). □

5.1. Hilbertian case. In the following, we prove the existence and uniqueness of weak and strong
solutions associated to Problem (5.2) in L2-theory.

Theorem 5.6. (i) Assume that f ∈ L6/5(Ω), F ∈ L2(Ω), g ∈ H−1/2(ΓN ) and h ∈ H−1/2(ΓD)
such that h · n = 0 on ΓN . Then, Problem (5.2) admits a unique solution u ∈ V2

n,τ (Ω) satisfying

∥u∥H1(Ω) ≤ C
(
∥f∥L6/5(Ω) + ∥F∥L2(Ω) + ∥g∥H−1/2(ΓN ) + ∥h∥H−1/2(ΓD)

)
. (5.13)

(ii) Furthermore, if Ω is of class C 2,1, f ∈ L2(Ω), g ∈ H1/2(ΓN ) and h ∈ H1/2(ΓD) then the
solution of Problem (5.2), with F = 0 belongs to H2(Ω) and satisfies the estimate

∥u∥H2(Ω) ≤ C
(
∥f∥L2(Ω) + ∥g∥H1/2(ΓN ) + ∥h∥H1/2(ΓD)

)
. (5.14)



22 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

Proof. (i) The bilinear form

a(u,φ) =

∫
Ω

∇u : ∇φ (5.15)

is continuous and coercive on V2
n,τ (Ω) due to Poincaré’s inequality (5.5). Moreover, the second-

hand side of (5.10) is a linear and continuous form on V2
n,τ (Ω), one may then apply the Lax

Milgram theorem in order to get the existence and uniqueness of a solution u ∈ V2
n,τ (Ω) to (5.10)

satisfying (5.13).
(ii) Regularity. Assume that f ∈ L2(Ω) and Ω is of class C 2,1 so there exists a unique u ∈ H1(Ω)

solution of (5.2). The idea of the proof is basically similar to that of Theorem 4.1, with a partition
of unity where v = θu and w = ηu, we look for the regularity of v ∈ H1(Ω) satisfying

−∆v = Fθ in Ω,

v × n = 0,
∂v

∂n
· n = θg on Γ,

(5.16)

where Fθ = θf − 2∇θ · ∇u− u∆θ ∈ L2(Ω). From (5.7) we have

divv = −2Kv · n+ θg on Γ.

Since K belongs to W 1,∞(Γ) because of the C 2,1 regularity of Ω, divv belongs to H1/2(Γ).
Moreover, we have ∆divv = divFθ ∈ H−1(Ω) then divv belongs to H1(Ω). Let χ ∈ H1(Ω) such
that ∆χ = divFθ in Ω and χ = −2Kv ·n+ θg on Γ. The function λ = χ− divv is harmonic and
λ = 0 on Γ then λ is zero [2] and thus divv belongs to H1(Ω). We know that curlv ∈ L2(Ω) and
curl curlv = −∆v+∇(divv) which means that curl curlv ∈ L2(Ω) and div curlv = 0. According
to Lemma 3.5, we have curlv · n = divΓ(v × n) = 0 on Γ since v × n = 0 on Γ. Consequently
curlv ∈ X2

T (Ω) ↪→ H1(Ω). Because v ∈ L2(Ω), curlv ∈ H1(Ω), divv ∈ H1(Ω) and v × n = 0 on
Γ, we infer that v belongs to H2(Ω) from (3.2).

Now, we look for the regularity of w ∈ H1(Ω) satisfying

−∆w = Fη in Ω,

w · n = 0,
∂w

∂n
× n = ηh on Γ.

(5.17)

Setting z = curlw so div z = 0 in Ω and −∆z = curlFη in Ω. Since w · n = 0 and ∂w
∂n × n = ηh

on Γ, we deduce from (5.8) that

z× n = Λw + ηh on Γ,

thus z× n ∈ H1/2(Γ) so z satisfies

−∆z = curlFη and div z = 0 in Ω,

z× n = Λw + ηh on Γ,

⟨z · n, 1⟩Γi
= 0, ∀1 ≤ i ≤ I,

(5.18)

as in the proof of Theorem 4.1 point (ii), we can show that z ∈ H1(Ω). Since ∇(divw) =
− curl curlw+∆w = − curl z−Fη ∈ L2(Ω); therefore, divw ∈ H1(Ω). Moreover, w ∈ L2(Ω) and
w · n = 0 on Γ, we infer that w belongs to H2(Ω) by (3.1). As a consequence, u ∈ H2(Ω). □

5.2. Lp-theory. The task now is to introduce a strategy to get a solution in W1,p(Ω) for any
1 < p < +∞, based on some inf-sup conditions and on where ∂Σ lies.

Theorem 5.7. Assume that ∂Σ ⊂ ΓD or ∂Σ ⊂ ΓN . Then for any ℓ ∈ [Vp′

n,τ (Ω)]
′ the problem:

Find u ∈ Vp
n,τ (Ω) such that for any φ ∈ Vp′

n,τ (Ω) such that

a(u,φ) = ⟨ℓ,φ⟩,

where the bilinear form a(·, ·) is defined in (5.15), has a unique solution.



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 23

Proof. (1) We assume that ∂Σ ⊂ ΓD:

Step 1: Case p ≥ 2. Since [Vp′

n,τ (Ω)]
′ ↪→ [V2

n,τ (Ω)]
′ by the Lax Milgram theorem, there exists a

unique solution u ∈ V2
n,τ (Ω) such that

a(u,φ) = ⟨ℓ,φ⟩, ∀φ ∈ V2
n,τ (Ω). (5.19)

Recall that we proved an inf-sup condition in Lemma 4.7 associated with the form

b(u,φ) =

∫
Ω

curlu · curlφ+ (divu)(divφ) dx.

Let v ∈ H1(Ω) with ∆v ∈ L6/5(Ω), so we have the following Green formula for any φ ∈ V2
n,τ (Ω),

−⟨∆v,φ⟩Ω =

∫
Ω

∇v : ∇φ dx− ⟨∂v
∂n

,φ⟩Γ

but we know that v = vτ + (v · n)n thus

∂v

∂n
= (

∂v

∂n
)τ + (

∂v

∂n
· n)n.

So we obtain

−⟨∆v,φ⟩Ω =

∫
Ω

∇v : ∇φ dx− ⟨
(∂v
∂n

)
τ
,φτ ⟩ΓD

− ⟨∂v
∂n

· n,φ · n⟩ΓN
.

On the other hand, we have the Green formula for any φ ∈ V2
n,τ (Ω),

−⟨∆v,φ⟩Ω =

∫
Ω

curlv · curlφ+ (divv)(divφ)− ⟨curlv × n,φ⟩ΓD
− ⟨divv,φ · n⟩ΓN

.

Then, we can write∫
Ω

∇v : ∇φ dx =

∫
Ω

curlv · curlφ+ (divv)(divφ)− ⟨curlv × n,φ⟩ΓD

− ⟨divv,φ · n⟩ΓN
+ ⟨

(∂v
∂n

)
τ
,φτ ⟩ΓD

+ ⟨∂v
∂n

· n,φ · n⟩ΓN
.

(5.20)

From the relations (1.4) and (1.5), we deduce that

divv − ∂v

∂n
· n = −2Kv · n+ divΓ vτ , curlv × n−

(∂v
∂n

)
τ
= Λv −∇τ (v · n).

Replacing this in (5.20) gives that v ∈ D(Ω) satisfies∫
Ω

∇v : ∇φ dx =

∫
Ω

curlv · curlφ+ (divv)(divφ)− ⟨Λv,φ⟩ΓD
+ 2⟨Kv · n,φ · n⟩ΓN

. (5.21)

From the density of D(Ω) into H1(Ω), the formula (5.21) holds for any v ∈ V2
n,τ (Ω) and φ ∈

V2
n,τ (Ω). Consequently, (5.19) becomes∫

Ω

curlu · curlφ+ (divu)(divφ) = ⟨ℓ,φ⟩+ ⟨Λu,φ⟩ΓD
− 2⟨Ku · n,φ · n⟩ΓN

.

We discuss the regularity of the solution u case by case.
(i) For any p ≤ 6: Since Ω is of class C 1,1, we haveK ∈ L∞(Γ) and u·n ∈ H1/2(ΓN ) ↪→ L4(ΓN ).

Thus Ku ·n ∈ L4(ΓN ) ↪→W−1/6,6(ΓN ) and similarly Λu ∈ L4(ΓD) ↪→ W−1/6,6(ΓD). Since p ≤ 6
then Ku · n ∈W−1/p,p(ΓN ) and Λu ∈ W−1/p,p(ΓD) so that the mapping

⟨L,φ⟩ = ⟨ℓ,φ⟩+ ⟨Λu,φ⟩ΓD
− 2⟨Ku · n,φ · n⟩ΓN

, ∀φ ∈ Vp
n,τ (Ω) (5.22)

defines an element in the dual space of Vp′

n,τ (Ω). Then, according to the inf-sup condition (4.18)
and Babùska-Brezzi theorem, there exists a unique w ∈ Vp

n,τ (Ω) satisfying∫
Ω

curlw · curlφ+ (divw)(divφ) = ⟨L,φ⟩
[W̃p′

Σ (Ω)]′×W̃p′
Σ (Ω)

, ∀φ ∈ W̃p′

Σ (Ω). (5.23)



24 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

Note that g̃rad sℓj ∈ V2
n,τ (Ω) ↪→ Vp′

n,τ (Ω) and we have

⟨L, g̃rad sℓj⟩[W̃p′
Σ (Ω)]′×W̃p′

Σ (Ω)
= ⟨ℓ, g̃rad sℓj⟩+ ⟨Λu, g̃rad sℓj⟩ΓD

− 2⟨Ku · n, g̃rad sℓj · n⟩ΓN

=

∫
Ω

curlw · curl g̃rad sℓj +
∫
Ω

(divw)(div g̃rad sℓj) = 0.

To extend (5.23) to any test function φ in Vp′

n,τ (Ω), we set

φ̃ = φ−
LN∑
ℓ=1

J∑
j=1

( 1

LN
⟨φ · n, 1⟩Σj +

1

J
⟨φ · n, 1⟩Γℓ

N

)
g̃radsℓj

which yields ⟨L,φ⟩ = ⟨L, φ̃⟩ and φ̃ ∈ W̃p′

Σ (Ω). Thus, we obtain∫
Ω

curlw · curlφ+(divw)(divφ) =

∫
Ω

curlw · curl φ̃ dx+
∫
Ω

(divw)(div φ̃) dx = ⟨L, φ̃⟩ = ⟨L,φ⟩.

We deduce that w ∈ Vp
n,τ (Ω) satisfies∫

Ω

curlw · curlφ+ (divw)(divφ) = ⟨L,φ⟩, ∀φ ∈ Vp′

n,τ (Ω). (5.24)

Since V2
n,τ (Ω) ↪→ Vp′

n,τ (Ω), we infer for any φ ∈ V2
n,τ (Ω) that∫

Ω

curlw · curlφ+ (divw)(divφ) =

∫
Ω

curlu · curlφ+ (divu)(divφ),

taking φ = w − u leads to curlw = curlu and divw = divu in Ω. Finally, since u ∈ L6(Ω) ↪→
Lp(Ω), curlu ∈ Lp(Ω) and divu ∈ Lp(Ω). Moreover u ·n = 0 on ΓD and u×n = 0 on ΓN , hence
u ∈ W1,p(Ω) according to [4, Corollary 3.5].

(ii) For any p > 6: we have u ∈ W1,6(Ω) thus u ·n and uτ belong respectively toW 1−1/6,6(ΓN )
and W1−1/6,6(ΓD). But W

1−1/6,6(Γ) ↪→ W−1/p,p(Γ), then the mapping L defined in (5.22) is an

element in the dual space of Vp′

n,τ (Ω). Thus, there exists a unique w ∈ Vp
n,τ (Ω) such that (5.24)

holds for any φ ∈ Vp′

n,τ (Ω). Then, we infer similarly that u belongs to W1,p(Ω).

Step 2: Case 1 < p < 2. We consider the operator A ∈ L(Vp
n,τ (Ω), (V

p′

n,τ (Ω))
′) associated with

the bilinear form

a(u,φ) =

∫
Ω

∇u : ∇φ dx,

such that < Au,φ >= a(u,φ). From the first step and by virtue of the Babùska-Brezzi theorem,
we deduce that the operator A is an isomorphism for p > 2 and its dual A′ is an isomorphism
from Vp′

n,τ (Ω) into (Vp
n,τ (Ω))

′ for p′ < 2. This means that the operator A is an isomorphism for
any p < 2. Consequently the following inf-sup condition holds for any 1 < p <∞,

inf
φ∈Vp′

n,τ (Ω)

sup
u∈Vp

n,τ (Ω)

∣∣ ∫
Ω
∇u : ∇φ

∣∣
∥∇u∥Lp(Ω)∥∇φ∥Lp′ (Ω)

> 0.

This completes the proof for this first case.

(2) We now assume that ∂Σ ⊂ ΓN . The proof is exactly the same as above except the case
where p ≤ 6, in which we recall the inf-sup condition (4.20) and Babùska-Brezzi theorem, to prove
the existence and uniqueness of w ∈ Vp

n,τ (Ω) satisfying∫
Ω

curlw · curlφ+ (divw)(divφ) = ⟨L,φ⟩
[Ṽp′

0 (Ω)]′×Ṽp′
0 (Ω)

, ∀φ ∈ Ṽp′

0 (Ω). (5.25)

Note that ∇sℓ ∈ V2
n,τ (Ω) ↪→ Vp′

n,τ (Ω) and we have

⟨L,∇sℓ⟩[Ṽp′
0 (Ω)]′×Ṽp′

0 (Ω)
= ⟨ℓ,∇sℓ⟩+ ⟨Λu,∇sℓ⟩ΓD

− 2⟨Ku · n,∇sℓ · n⟩ΓN

=

∫
Ω

curlw · curl∇sℓ +
∫
Ω

(divw)(div∇sℓ) = 0.



EJDE-2025/103 EXISTENCE AND REGULARITY FOR ELLIPTIC SYSTEMS 25

To extend (5.25) to any test function φ in Vp′

n,τ (Ω), we set

φ̃ = φ−
LN∑
ℓ=1

⟨φ · n, 1⟩Γℓ
N
∇sℓ

which yields ⟨L,φ⟩ = ⟨L, φ̃⟩ and φ̃ ∈ Ṽp′

0 (Ω). The remainder of the proof is exactly the same as
above. □

Applying the Babùska-Brezzi theorem, we obtain the following result.

Proposition 5.8. Assume that ∂Σ ⊂ ΓD or ∂Σ ⊂ ΓN then there exists a constant γ > 0 depending
only on Ω and p such that the following inf-sup condition holds

inf
φ∈Vp′

n,τ (Ω),φ ̸=0

[
sup

u∈Vp
n,τ (Ω), u̸=0

∣∣ ∫
Ω
∇u : ∇φ

∣∣
∥u∥W1,p(Ω)∥φ∥W1,p′ (Ω)

]
≥ γ. (5.26)

Now, we are in position to show that the solution of Problem (5.2) belongs to W1,p(Ω).

Theorem 5.9. (i) Assume that ∂Σ ⊂ ΓD or ∂Σ ⊂ ΓN . Let f ∈ Lr(p)(Ω), F ∈ Lp(Ω), g ∈
W−1/p,p(ΓN ) and h ∈ W−1/p,p(ΓD) then Problem (2.2) has a unique solution u ∈ W1,p(Ω) and
it satisfies

∥u∥W1,p(Ω) ≤ C
(
∥f∥Lr(p)(Ω) + ∥F|∥Lp(Ω) + ∥g∥W−1/p,p(ΓN ) + ∥h∥W−1/p,p(ΓD)

)
.

(ii) Furthermore, if Ω is of class C 2,1, f ∈ Lp(Ω), g ∈ W 1−1/p,p(ΓN ) and h ∈ W1−1/p,p(ΓD)
with F = 0, then u belongs to W2,p(Ω) and the following estimate holds

∥u∥W2,p(Ω) ≤ C
(
∥f∥Lp(Ω) + ∥g∥W 1−1/p,p(ΓN ) + ∥h∥W1−1/p,p(ΓD)

)
.

Proof. (i) The first result is straightforward due to Proposition 5.8.
(ii) Assume that f ∈ Lp(Ω) ↪→ Lr(p)(Ω), there exists a unique solution u ∈ W1,p(Ω) solution of

(2.2). We recall the systems (5.16) and (5.17) in Theorem 5.6 as their solutions v and w remain
in W1,p(Ω). We have

divv = Kv · n+ θg ∈W 1−1/p,p(Γ)

since K ∈ W1,∞(Γ). Moreover, −∆(divv) = divFθ ∈ W−1,p(Ω). It follows that divv belongs
to W 1,p(Ω). As −∆v = curl curlv − ∇(divu) then curlv ∈ Xp

T (Ω) ↪→ W1,p(Ω). Consequently,
v ∈ W2,p(Ω).

Similarly, taking z = curlw in Problem (5.17) leads to −∆z = curlFη ∈ W−1,p(Ω) and

z × n = Λw + ηh ∈ W1−1/p,p(Ω). Following the regularity step in the proof of Theorem 5.6,
we obtain that curlw ∈ W1,p(Ω) and ∇(divw) ∈ Lp(Ω) leading to divw ∈ W 1,p(Ω). Thus
w ∈ W2,p(Ω), we finally infer that u belongs to W2,p(Ω). □

Remark 5.10. Note that the condition ∂Σ ⊂ ΓD or ∂Σ ⊂ ΓN is not natural due to the boundary
conditions of Problem (5.2). However, it gives a generalized solution in W1,p(Ω) for any 1 < p <
+∞.

Remark 5.11. Without any restriction on ∂Σ, we can prove that the solution of Problem (5.2)
belongs to W1,p(Ω) for any p ≥ 2 using a partition of unity and with F = 0.

Remark 5.12. In the case where F = 0 and f ∈ Lr(p)(Ω) for any p ≥ 3
2 , the solution of Problem

(5.2) belongs to W1,p(Ω) and when f ∈ L1(Ω), the solution belongs to W1,s(Ω) for any s < 3/2.

In the sequel, we generalize the existence, uniqueness and regularity of the solutions of Problem
(5.2) to the case of non-homogeneous boundary conditions introduced above and call it (Q).

Theorem 5.13. (i) Assume that ∂Σ ⊂ ΓD or ∂Σ ⊂ ΓN . Let f ∈ Lr(p)(Ω), F ∈ Lp(Ω), a × n ∈
W1−1/p,p(ΓN ), g ∈ W−1/p,p(ΓN ), b ∈ W 1−1/p,p(ΓD) and h ∈ W−1/p,p(ΓD) then Problem (5.1)
has a unique solution u ∈ W1,p(Ω) and it satisfies

∥u∥W1,p(Ω) ≤ C
(
∥f∥Lr(p)(Ω) + ∥F∥Lp(Ω) + ∥g∥W−1/p,p(ΓN )



26 I. BOUSSETOUAN, C. AMROUCHE EJDE-2025/103

+ ∥h∥W−1/p,p(ΓD) + ∥a× n∥W1−1/p,p(ΓN ) + ∥b∥W 1−1/p,p(ΓD)

)
.

(ii) Furthermore, if Ω is of class C 2,1, f ∈ Lp(Ω), a× n ∈ W2−1/p,p(ΓN ), g ∈ W 1−1/p,p(ΓN ),
b ∈ W 2−1/p,p(ΓD) and h × n ∈ W1−1/p,p(ΓD) with F = 0, then u belongs to W2,p(Ω) and the
following estimate holds

∥u∥W2,p(Ω) ≤ C
(
∥f∥Lp(Ω) + ∥g∥W 1−1/p,p(ΓN ) + ∥h∥W1−1/p,p(ΓD)

+ ∥a× n∥W2−1/p,p(ΓN ) + ∥b∥W 2−1/p,p(ΓD)

)
.

We skip the proof in this paper as it can be done with the same arguments as for Problem (Q).

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Imane Boussetouan
National Higher School of Technology and Engineering, Department CPST, Annaba 23005, Algeria

Email address: i.boussetouan@ensti-annaba.dz

Chérif Amrouche

LMAP, UMR CNRS 5142, Batiment IPRA, Avenue de l’Université BP 1155, Pau Cedex,64013, France

Email address: cherif.amrouche@univ-pau.fr


	1. Introduction
	2. Main results
	3. Preliminaries
	4. First elliptic system
	4.1. Hilbertian case
	4.2. Lp-theory

	5. Second elliptic problem
	5.1. Hilbertian case
	5.2. Lp-theory

	References