Arab J Math (2014) 3:325–340DOI 10.1007/s40065-014-0103-8

Belkacem Chaouchi · Fatiha Boutaous

An abstract approach for the study of an elliptic problemin a nonsmooth cylinder

Received: 13 December 2013 / Accepted: 11 May 2014 / Published online: 4 July 2014© The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract The aim of this work is the resolution of a non-autonomous abstract differential equation of elliptictype set on unbounded domain. The study is performed in the framework of Hölder spaces. An example for aconcrete elliptic problem in nonsmooth cylindrical domains will illustrate the theory.

Mathematics Subject Classification 34G10 · 34K10 · 12H2O

1 Introduction

In this work, we deal with non-autonomous problems of the form

u′′(t) + A(t)u(t) − λu(t) = f (t), t ≥ 0, λ > 0, (1)

subject to the following boundary conditions

u(0) = 0, u(+∞) = 0, (2)

where:

(i) f ∈ BUC2σ ([0, +∞[; E), σ ∈]0, 1/2[, denoting the space of bounded and 2σ−Hölder continuousfunctions f : [0,+∞[→ E, endowed with the norm

‖ f ‖BUC2σ ([0,+∞[;E) = supt≥0

‖ f (t)‖E + supt �=τ

‖ f (t) − f (τ )‖E

|t − τ |2σ ,

where E is a complex Banach space.

B. Chaouchi (B)Lab. de l’Energie et des Systémes Intelligents, Khemis Miliana University, 44225 Khemis Miliana, AlgeriaE-mail: chaouchicukm@gmail.com

F. BoutaousDépartement de Mathématiques, Faculté des Sciences, Sâad Dahlab University, B.P. 270, Blida, AlgeriaE-mail: boutaous.fatiha2009@hotmail.com

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326 Arab J Math (2014) 3:325–340

(ii) (A(t))t≥0 is a family of closed linear operators satisfying certain assumptions to be specified later on.

The aim of the present paper is twofold:

1. Give a complete study of Problems (1) and (2). We will then establish existence, uniqueness of the strictsolution, that means a function u such that

⎧⎨

⎩

u ∈ BUC2([0, +∞[, E),for every t ≥ 0 : u(t) ∈ D(A(t))and (A(t) − λ)u(t) ∈ BUC([0,+∞[; E),

and satisfying (1) and (2). Just, we recall here that for k ∈ N , BUCk([0, +∞[; E) is the space of vector-valued functions with uniformly continuous and bounded derivatives up to order k in [0,+∞[.

2. Exploit and apply the above results to establish some Hölder continuous regularity results for a concreteboundary value problem set on a singular domain.

It should be noted that the solvability of boundary value problems for differential operator equations on boundeddomains has been widely studied. For an overview on these kind of problems and some historical references,see [11,15,16]. Several methods have been developed for the solution of this kind of problems. Among thesemethods, we cite as an example the semigroup techniques and the well-known sum’s operators theory, see [6].In this work, our strategy is based essentially on the use of the Dunford’s integrals as in [3] and the methodsapplied in [1] and [12]. Note that besides being complementary to [5], the present paper contains importantnew results. In fact, we present some new Hölder continuous regularity results for an elliptic equation set onnonsmooth cylindrical domains. These results can be hardly obtained using the classical standard techniquessuch as the classical variational methods or the potential theory. This paper is organized as follows. In Sect. 2,we build the natural representation of the solution of (1) and (2) using the Dunford operational calculus. Weprove also some results, which allow us to justify the optimal smoothness of the previous representation. InSect. 3, we give a concrete example to which our abstract results can be apply.

2 Optimal results for Problems (1) and (2)

2.1 Assumptions and representation of the solution

For simplicity of notation, we set

Q(t) = A(t) − λ, λ > 0.

Throughout this work, we assume that the the family of linear closed operators (Q(t), D(Q(t)))t≥0 enjoysthe following properties:

1. ∀t ≥ 0, ∃ δ0 ∈]π2 , π[such that

ρ(Q(t)) ⊃∑

δ0

= {z ∈ C : reiδ, r ∈ [0,+∞[, δ ∈] − δ0, +δ0[}, (3)

here, ρ(Q(t)) is the resolvent set of Q(t).2. ∃C1 > 0 : ∀z ∈ ∑

δ0, ∀t ≥ 0

‖(Q(t) − z I )−1‖L(E) ≤ C1

|z| + 1. (4)

3. For all z ∈ ∑δ0

, the mapping t → (Q(t)− z I )−1 defined on R+, is of class C2. Furthermore, we supposealso that:There exist C2 > 0, σ ∈]0, 1/2[ such that for all z ∈ ∑

δ0and all t, τ ≥ 0,

{‖ ∂∂t (Q(t) − z I )−1‖L(E) ≤ C2|z|+1 ,

‖ ∂2

∂t2(Q(t) − z I )−1‖L(E) ≤ C2|z|+1 ,

(5)

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Arab J Math (2014) 3:325–340 327

and∥∥∥∥

∂

∂t(Q(t) − z I )−1 − ∂

∂τ(Q(τ ) − z I )−1

∥∥∥∥

L(E)

≤ C2|t − τ |2σ|z| + 1

,

∥∥∥∥

∂2

∂t2(Q(t) − z I )−1 − ∂2

∂τ 2(Q(τ ) − z I )−1

∥∥∥∥

L(E)

≤ C2|t − τ |2σ|z| + 1

. (6)

Remark 2.1 The above hypotheses are known in the literature as the Da Prato–Grisvard hypotheses, see [6]and [12]. Just, we note that:

1. All the constants given above are independent of t .2. Hypotheses (3) and (4) express the ellipticity of (1). Moreover, all the previous assumptions remain true if

we replace z by z + √λ.

Throughout the rest of this paper, C stands for a generic constant and σ ∈]0, 1/2[,We know that in the case when

Q(t) = Q,

is a constant operator satisfying the hypothesis (3) and (4), the representation of the solution u of (1) and (2)is given by

u(t) = − 1

2iπ

∫

γ

+∞∫

0

k√−z(t, s)(Q − z)−1 f (s)ds, (7)

where

k√−z(t, s) =

⎧⎪⎨

⎪⎩

e−√−zt sinh√−zs√−z

0 ≤ s ≤ t,

e−√−zs sinh√−zt√−z

s ≥ t.(8)

Here, the curve γ is the boundary of the sector∑

δ0oriented from ∞e+iδ0 to ∞e−iδ0 and

√−z is theanalytic determination defined by �√−z > 0, see [3].

In our situation, our representation formula can be heuristically derived by the following argument:Taking the constant case into account, we look for a solution of Problems (1) and (2) in the following form

u(t) = − 1

2iπ

∫

γ

+∞∫

0

k√−z(t, s)(Q(t) − z)−1 f ∗(s)ds. (9)

We are then concerned with the determination of the unknown function f ∗ in order that (9) is a strict solutionof Problems (1) and (2).

2.2 Study of the regularity of the formal solution (9)

First, one has

Proposition 2.2 Assume that f ∗ ∈ BUC2σ ([0, +∞[; E). Then,

1. For all t ≥ 0, one has

u(t) ∈ D(A(t)).

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328 Arab J Math (2014) 3:325–340

2. For all t ≥ 0, the function f ∗ (introduced in 9) satisfies the following equation

f ∗(t) − Op( f ∗)(t) = f (t), (10)

where

Op( f ∗)(t) = + 1

2iπ

∫

γ

⎛

⎝

t∫

0

e−√−zt sinh√−zs√−z

∂2

∂t2(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

+ 1

2iπ

∫

γ

⎛

⎝

+∞∫

t

e−√−zs sinh√−zt√−z

∂2

∂t2(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

− 1

iπ

∫

γ

⎛

⎝

t∫

0

e−√−zt sinh√−zs

∂

∂t(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

+ 1

iπ

∫

γ

⎛

⎝

+∞∫

t

e−√−zs cosh√−zt

∂

∂t(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz.

Proof Statement 1 follows from Proposition 3.1 in [3].Concerning Statement 2, one has

u′(t) = − 1

2iπ

∫

γ

⎛

⎝

t∫

0

e−√−zt sinh√−zs√−z

∂

∂t(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

− 1

2iπ

∫

γ

⎛

⎝

+∞∫

t

e−√−zs sinh√−zt√−z

∂

∂t(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

− 1

2iπ

∫

γ

e−√−zt sinh√−zt√−z

(Q(t) − z)−1 f ∗(t)dz

+ 1

2iπ

∫

γ

⎛

⎝

t∫

0

e−√−zt sinh√−zs(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

+ 1

2iπ

∫

γ

e−√−zt sinh√−zt√−z

(Q(t) − z)−1 f ∗(t)dz

− 1

2iπ

∫

γ

⎛

⎝

+∞∫

t

e−√−zs cosh√−zt (Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz.

To calculate u′′(t), we follow the same reasoning as in [12]. We set

u′ε(t)

= − 1

2iπ

∫

γ

⎛

⎝

t−ε∫

0

k√−z(t, s)∂

∂t(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

− 1

2iπ

∫

γ

⎛

⎝

+∞∫

t+ε

k√−z(t, s)∂

∂t(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

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Arab J Math (2014) 3:325–340 329

+ 1

2iπ

∫

γ

⎛

⎝

t−ε∫

0

e−√−zt sinh√−zs(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

− 1

2iπ

∫

γ

⎛

⎝

+∞∫

t+ε

e−√−zs cosh√−zt (Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz.

First, it is clear that

limε→0

u′ε(t) = u′(t).

On the other hand, one has

u′′ε (t)

= R(ε)( f ∗)(t) + I(ε)( f ∗)(t) − Op(ε)( f ∗)(t),

where

R(ε)( f ∗)(t)

= − 1

2iπ

∫

γ

e−√−zt sinh√−z(t − ε)√−z

∂

∂t(Q(t) − z I )−1 f ∗(t − ε)dz

+ 1

2iπ

∫

γ

e−√−z(t+ε) sinh√−zt√−z

∂

∂t(Q(t) − z I )−1 f ∗(t + ε)dz,

and

I(ε)( f ∗)(t)

= + 1

2iπ

∫

γ

e−√−zt sinh√−z(t − ε)(Q(t) − z I )−1 f ∗(t − ε)dz

+ 1

2iπ

∫

γ

e−√−z(t+ε) cosh√−zt (Q(t) − z I )−1 f ∗(t + ε)dz

+ 1

2iπ

∫

γ

(−√−z)

⎛

⎝

t−ε∫

0

e−√−zt sinh√−zs(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

+ 1

2iπ

∫

γ

(−√−z)

⎛

⎝

+∞∫

t+ε

e−√−zs sinh√−zt (Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz,

and

Op(ε)( f ∗)(t)

= + 1

2iπ

∫

γ

⎛

⎝

t−ε∫

0

e−√−zt sinh√−zs√−z

∂2

∂t2(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

+ 1

2iπ

∫

γ

⎛

⎝

+∞∫

t+ε

e−√−zs sinh√−zt√−z

∂2

∂t2(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

− 1

iπ

∫

γ

⎛

⎝

t−ε∫

0

e−√−zt sinh√−zs

∂

∂t(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

123

330 Arab J Math (2014) 3:325–340

+ 1

iπ

∫

γ

⎛

⎝

+∞∫

t+ε

e−√−zs cosh√−zt

∂

∂t(Q(t) − z I )−1 f ∗(s)ds

⎞

⎠ dz.

For R(ε)( f ∗)(t), observe that∥∥∥∥∥

e−√−zt sinh√−z(t − ε)√−z

∂

∂t(Q(t) − z I )−1 f ∗(t − ε)

∥∥∥∥∥

L(E)

≤ e−�√−zte�√−z(t−ε)

|z|1/2|z| ‖ f ∗‖BUC2σ ([0,+∞[,E)

and∥∥∥∥∥

e−√−z(t+ε) sinh√−zt√−z

∂

∂t(Q(t) − z I )−1 f ∗(t + ε)

∥∥∥∥∥

L(E)

≤ e−�√−z(t+ε)e−�√−zt

|z|1/2|z| ‖ f ∗‖BUC2σ ([0,+∞[,E),

by the dominated convergence theorem, we get

limε→0

R(ε)( f ∗)(.) = 0.

For the quantity Op(ε)( f ∗)(t), we see that

limε→0

Op(ε)( f ∗)(.) = Op( f ∗)(.).

It remains to treat the quantity I(ε)( f ∗)(t), using the identity

Q(t)(Q(t) − z I )−1 = I + z(Q(t) − z I )−1,

we get

I(ε)( f ∗)(t) = I ′(ε)( f ∗)(t) + I ′′

(ε)( f ∗)(t),

where

I ′(ε)( f ∗)(t)

= + 1

2iπ

∫

γ

(e−√−zt sinh√−z(t − ε)

Q(t)(Q(t) − z I )−1

zf ∗(t − ε)dz

+ 1

2iπ

∫

γ

e−√−z(t+ε) cosh√−zt

Q(t)(Q(t) − z I )−1

zf ∗(t + ε)dz

+ 1

2iπ

∫

γ

⎛

⎝

t−ε∫

0

e−√−zt sinh√−zs√−z

(Q(t)(Q(t) − z I )−1) f ∗(s)ds

⎞

⎠ dz

+ 1

2iπ

∫

γ

⎛

⎝

+∞∫

t+ε

e−√−zs sinh√−zt√−z

(Q(t)(Q(t) − z I )−1) f ∗(s)ds

⎞

⎠ dz,

123

Arab J Math (2014) 3:325–340 331

and

I ′′(ε)

(f ∗) (t)

= − 1

2iπ

∫

γ1

t−ε∫

0

e−√−zt sinh√−zs√−z

f ∗(s)dsdz

− 1

2iπ

∫

γ1

+∞∫

t+ε

e−√−zs sinh√−zt√−z

f ∗(s)dsdz

− 1

2iπ

∫

γ1

e−√−zt sinh√−z (t − ε)

zf ∗(t − ε)dz

− 1

2iπ

∫

γ1

e−√−z(t+ε) cosh√−zt

zf ∗(t + ε)dz.

Now for the quantity I ′(ε) ( f ∗) (t), one has

I ′(ε)

(f ∗) (t)

= + 1

2iπ

∫

γ1

e−√−zt sinh√−z (t − ε)

Q (t) (Q (t) − z I )−1

zf ∗(t − ε)dz

− 1

2iπ

∫

γ1

e−√−z(t+ε) sinh√−zt

Q (t) (Q (t) − z I )−1

zf ∗(t)dz

+ 1

2iπ

∫

γ1

e−√−z(t+ε) cosh√−zt

Q (t) (Q (t) − z I )−1

zf ∗(t + ε)dz

− 1

2iπ

∫

γ1

e−√−zt cosh√−z (t − ε)

Q (t) (Q (t) − z I )−1

zf ∗(t)dz

+ 1

2iπ

∫

γ1

⎛

⎝

t−ε∫

0

e−√−zt sinh√−zs√−z

Q (t) (Q (t) − z I )−1 (f ∗(s) − f ∗(t)

)ds

⎞

⎠ dz

+ 1

2iπ

∫

γ1

⎛

⎝

+∞∫

t+ε

e−√−zs sinh√−zt√−z

Q (t) (Q (t) − z I )−1 (f ∗(s) − f ∗(t)

)ds

⎞

⎠ dz

+ 1

2iπ

∫

γ1

e−√−zt Q (t) (Q (t) − z I )−1

zf ∗(t)dz.

Consequently,

limε→0

I ′(ε)

(f ∗) (t)

= − 1

2iπ

∫

γ1

⎛

⎝

+∞∫

0

k√−z(t, s)Q (t) (Q (t) − z)−1 (f ∗(s) − f ∗(t)

)ds

⎞

⎠ dz

+ 1

2iπ

∫

γ1

e−√−zt Q (t) (Q (t) − z I )−1

zf ∗(t)dz.

123

332 Arab J Math (2014) 3:325–340

Concerning the term I ′′(ε) ( f ∗) (t), we write

I ′′(ε)

(f ∗) (t)

= − 1

2iπ

∫

γ1

⎛

⎝

t−ε∫

0

e−√−zt sinh√−zs√−z

(f ∗(s) − f ∗(t − ε)

)ds

⎞

⎠ dz

− 1

2iπ

∫

γ1

e−√−zt

√−z

t−ε∫

0

sinh√−zsds f ∗(t − ε)dz

− 1

2iπ

∫

γ1

⎛

⎝

+∞∫

t+ε

e−√−zs sinh√−zt√−z

(f ∗(s) − f ∗(t + ε)

)ds

⎞

⎠ dz

− 1

2iπ

∫

γ1

sinh√−zt√−z

⎛

⎝

+∞∫

t+ε

e−√−zs)ds

⎞

⎠ f ∗(t + ε)dz

− 1

2iπ

∫

γ1

e−√−zt sinh√−z (t − ε)

zf ∗(t − ε)dz

− 1

2iπ

∫

γ1

e−√−z(t+ε) cosh√−zt

zf ∗(t + ε)dz.

A direct computation gives

limε→0

I ′′(ε)

(f ∗) (.) = 0.

On the other hand, one has

Q (t) u(t)

= − 1

2iπ

∫

γ1

+∞∫

0

k√−z(t, s)Q (t) (Q (t) − z I )−1 (F∗(s) − f ∗(t)

)dsdz

− 1

2iπ

∫

γ1

e−√−zt

zQ (t) (Q (t) − z I )−1 f ∗(t)dz

+ 1

2iπ

∫

γ1

1

zQ (t) (Q (t) − z I )−1 f ∗(t)dz.

Summing up, we deduce that, for all t ≥ 0 :f (t)

= limε→0

u′′ε (t) + Q (t) u(t)

= u′′ (t) + Q (t) u(t)

= f ∗(t) − Op(

f ∗) (t) . ��

Now, we need the following important result concerning f ∗.Proposition 2.3 There exists λ∗ > 0, such that, for all λ ≥ λ∗, the operator

I − Op : BUC([0, +∞[ ; E) → BUC([0,+∞[ ; E)f ∗(.) → f = f ∗(.) − Op ( f ∗) (.) ,

is an isomorphism.

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Arab J Math (2014) 3:325–340 333

Proof It suffices to adapt the techniques used in [4, Proposition 5.2, p. 27]. ��To study the regularity of the formal solution, we need the following result

Proposition 2.4 Assume that f ∗ ∈ BUC2σ ([0,+∞[ ; E). Then, the vector-valued function t → Op ( f ∗) (t)belongs to the space BUC2σ ([0, +∞[ ; E).

Proof Let t > τ ≥ 0, thus

Op(

f ∗) (t) − Op(

f ∗) (τ ) = (I ) + (I I ),

where

(I) = 1

2iπ

∫

γ

⎛

⎝

t∫

0

e−√−zt sinh√−zs√−z

∂2

∂t2(Q (t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

− 1

2iπ

∫

γ

⎛

⎝

τ∫

0

e−√−zτ sinh√−zs√−z

∂2

∂τ 2(Q (τ ) − z I )−1 f ∗(s)ds

⎞

⎠ dz

+ 1

2iπ

∫

γ

⎛

⎝

+∞∫

t

e−√−zs sinh√−zt√−z

∂2

∂t2(Q (t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

− 1

2iπ

∫

γ

⎛

⎝

+∞∫

τ

e−√−zs sinh√−zτ√−z

∂2

∂τ 2(Q (τ ) − z I )−1 f ∗(s)ds

⎞

⎠ dz,

and

(II) = − 1

iπ

∫

γ

⎛

⎝

t∫

0

e−√−zt sinh√−zs

∂

∂t(Q (t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

+ 1

iπ

∫

γ

⎛

⎝

τ∫

0

e−√−zτ sinh√−zs

∂

∂τ(Q (τ ) − z I )−1 f ∗(s)ds

⎞

⎠ dz

+ 1

iπ

∫

γ

⎛

⎝

+∞∫

t

e−√−zs cosh√−zt

∂

∂t(Q (t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

− 1

iπ

∫

γ

⎛

⎝

+∞∫

τ

e−√−zs cosh√−zτ

∂

∂τ(Q (τ ) − z I )−1 f ∗(s)ds

⎞

⎠ dz.

(I) and (II) can be treated similarly. So, we restrict ourselves to treat the first quantity. One has

(I ) = (I1) + (I2),

where

(I1) = + 1

2iπ

∫

γ

τ∫

0

e−√−zt sinh√−zs√−z

∂2

∂t2(Q (t) − z I )−1 f ∗(s)dsdz

+ 1

2iπ

∫

γ

t∫

τ

e−√−zt sinh√−zs√−z

∂2

∂t2(Q (t) − z I )−1 f ∗(s)dsdz

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334 Arab J Math (2014) 3:325–340

+ 1

2iπ

∫

γ

τ∫

0

e−√−zτ sinh√−zs√−z

∂2

∂t2(Q (t) − z I )−1 f ∗(s)dsdz

− 1

2iπ

∫

γ

τ∫

0

e−√−zτ sinh√−zs√−z

∂2

∂t2(Q (t) − z I )−1 f ∗(s)dsdz

− 1

2iπ

∫

γ

τ∫

0

e−√−zτ sinh√−zs√−z

∂2

∂τ 2(Q (τ ) − z I )−1 f ∗(s)dsdz,

and

(I2) + 1

2iπ

∫

γ

⎛

⎝

+∞∫

t

e−√−zs sinh√−zt√−z

∂2

∂t2(Q (t) − z I )−1 f ∗(s)ds

⎞

⎠ dz

− 1

2iπ

∫

γ

⎛

⎝

+∞∫

t

e−√−zs sinh√−zτ√−z

∂2

∂τ 2(Q (τ ) − z I )−1 f ∗(s)ds

⎞

⎠ dz,

We can write I1 as

(I1) = (J1) + (J2) + (J3).

where

(J1) = 1

2iπ

∫

γ1

⎛

⎝

τ∫

0

(e−√−zt − e−√−zτ

)sinh

√−zs√−z

∂2

∂t2(Q (t) − z I )−1 f ∗(s)ds

⎞

⎠ dz,

(J2) = 1

2iπ

∫

γ1

⎛

⎝

τ∫

0

e−√−zτ sinh√−zs√−z

(∂2

∂t2(Q (t) − z I )−1 − ∂2

∂τ 2(Q (τ ) − z I )−1

)

f ∗(s)ds

⎞

⎠ dz,

(J3) = + 1

2iπ

∫

γ1

⎛

⎝

t∫

τ

e−√−zt sinh√−zs√−z

∂2

∂t2(Q (t) − z I )−1 f ∗(s)ds

⎞

⎠ dz,

thanks to the differentiability properties of the resolvent, that is, (5 )and (6), we conclude that

‖(I1)‖ = O |t − τ |2σ .

The same arguments applied for I2 give

‖(I2)‖ = O |t − τ |2σ . ��

Summing up, we are in position to give our main maximal regularity results concerning Problems (1) and(2).

Proposition 2.5 Let f ∈ BUC2σ ([0,+∞[ ; E). Then, there exists λ∗ > 0, such that , for all λ ≥ λ∗, Problems(1) and (2) have a unique strict solution

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Arab J Math (2014) 3:325–340 335

u(t) = − 1

2iπ

∫

γ

+∞∫

0

k√−z(t, s) (Q (t) − z)−1 {(I − Op)−1 ( f )

}(s) ds. (11)

Moreover, one has

Q(.)u(.), u′′(.) ∈ BUC2σ ([0,+∞[ ; E).

Proof See Proposition 3.1 in [3]. ��The following lemma is needed to prove the optimal regularity of the strict solution (11) when f ∗ is taken inL∞(]0,+∞[ ; DQ(.) (σ, +∞)) ∩ BUC([0,+∞[ , E) where

DQ(.)(σ, +∞) ={

ϕ ∈ E : supr>0

∥∥rσ Q (.) (Q (.) − r I )−1ϕ

∥∥

E < +∞}

,

for more details about these spaces, see [7] and [13].

Lemma 2.6 Let f ∗ ∈ L∞(]0,+∞[ ; DQ(.) (σ,+∞)) ∩ BUC([0,+∞[ , E). Then, for all t ≥ 0,

u (t) ∈ D(A (t)).

Proof We are interesting with the convergence of the integral

1

2iπ

∫

γ1

+∞∫

0

k√−z(t, s)Q (t) (Q (t) − z I )−1 f ∗(s)dsdz,

where f ∗ ∈ L∞(]0,+∞[ ; DQ(.) (σ,+∞)).One has

∥∥∥∥∥∥

+∞∫

0

k√−z(t, s)Q (t) (Q (t) − z I )−1 f ∗(s)ds

∥∥∥∥∥∥

E

≤ (L1) + (L2),

with

‖(L1)‖ ≤ Ce−(�√−z)t∣∣√−z

∣∣ |z|σ

1

�√−zsinh(�√−zt),

and

‖(L2)‖ ≤ Ccosh(�√−zt)e−(�√−z)t

|z|1/2 |z|σ �√−z,

so∥∥∥∥∥∥

+∞∫

0

k√−z(t, s)Q (t) (Q (t) − z I )−1 f ∗(s)ds

∥∥∥∥∥∥

E

≤ C

|z|1+σ. ��

Therefore, we can deduce the following result.

Proposition 2.7 Let f ∈ L∞(]0,+∞[ ; DQ(.) (σ,+∞)) ∩ BUC ([0,+∞[ ; E). Then, there exists λ∗ > 0,such that , for all λ ≥ λ∗, Problems (1) and (2) have a unique strict solution given by (11). Moreover,

Q(.)u(.), u′′(.) ∈ L∞(]0,+∞[ ; DQ(.) (σ,+∞)) ∩ BUC ([0,+∞[ ; E) .

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336 Arab J Math (2014) 3:325–340

3 On the study of a concrete elliptic problem set on nonsmooth cylindrical domain

3.1 Position of the problem

Now, we will apply the abstract regularity results obtained in the previous section for the study of a concreteelliptic problem. We consider the following problem:

Let Π be an open set of R3 defined by

Π = {(x1, x2, x3) ∈ R3 : (x1, x2) ∈ Ω, 0 < x3 < b

},

where b is a finite positive number and Ω is the planar cusp domain defined by

Ω = {(x1, x2) ∈ R2 : 0 < x1 < a, −ψ (x1) < x2 < ψ (x1)

}.

Here

1. ψ (x1) = (x1)α , (1 < α ≤ 2),2. a is a finite positive number small enough.

In Π , we consider the boundary value problem

Δu − λ(

1ψ(x1)

)2u = h , λ > 0,

u|∂Π = 0.(12)

The right-hand side h belongs to the Hölder space C2σ (Π) and satisfies the following condition

h = 0 on ∂Γ(a). (13)

where ∂Γ(a) denotes the the boundary of the lateral surface

Γ(a) = {(a, x2, x3) ∈ R3 : −ψ (a) < x2 < ψ (a) , 0 < x3 < b

}.

It should be noted that Problem (12) is a particular case of some elliptic equations frequently encounteredin engineering application. In fact, applications of such equation are abundant in fluid dynamics and themodelization of weather prediction.

It is well known that the solvability of elliptic problems posed in singular domains was intensively investi-gated by numerous authors via several techniques, see, for example [8,10]. Most of these studies have focusedon the study of existence, uniqueness and the behavior of solutions near the singular parts of the boundary.In [5], an abstract approach was used to establish some Hölder continuous regularity results for the Dirichletproblem for Laplace equation posed in planar cusp domain. These authors have used the abstract differentialequation theory which seems more adapted for this kind of problems. In our situation, the study of the concreteproblem (12) will be reduced to the study of an abstract differential equation of elliptic type with variableoperator coefficients., that is, Problems (1) and (2).

3.2 Change of variables

As in [5], we use the following change of variables

T : Π → Π∞(x1, x2, x3) → (ξ, η, υ) :=

(1

α−1 (x1)1−α , x2ψ(x1)

, x3)

,

with

Π∞ = ]ξ0, +∞[ × D, ξ0 = 1

α − 1(a)1−α > 0 and D = ]−1, 1[ × ]0, b[ .

which means that the cuspidal edge (0, 0, x3) , (0 < x3 < b) is transformed in

D∞ = {+∞} × D.

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Arab J Math (2014) 3:325–340 337

Note that for any x1 ∈]0, a[, x3 ∈ ]0, b[⎧⎪⎨

⎪⎩

limx→0+ T (x1, ψ (x1) , x3) = lim

x→0+

(1

α−1 (x1)1−α , 1, υ)

= (+∞, 1, υ)

limx→0+ T (x1, −ψ (x1) , x3) = lim

x→0+

(1

α−1 (x1)1−α , −1, υ)

= (+∞,−1, υ) .

Now, define the following change of functions⎧⎪⎨

⎪⎩

v (ξ, η, υ) := u (x1, x2, x3) = u(exp

(ln[(α−1)ξ ]

1−α

), η exp

(α ln[(α−1)ξ ]

1−α

), υ

),

g (ξ, η, υ) := h (x1, x2, x3) = h(exp

(ln[(α−1)ξ ]

1−α

), η exp

(α ln[(α−1)ξ ]

1−α

), υ

).

Remark 3.1 Observe that if h is continuous in (0, 0, x3) , one has

limξ→+∞ g(ξ, η, υ)

= limξ→+∞ h

(

exp

(ln [(α − 1)ξ ]

1 − α

)

, η exp

(α ln [(α − 1)ξ ]

1 − α

)

, υ

)

= h(0, 0, x3).

From the above changes of variables, one has

Δu

= θ2ξ2β

(

Δ(ξ,η)v + θ−2ξ−2β∂2υv + α2θ−2/β(

η

ξ

)2

∂2η + 2αθ−1/β η

ξ∂2ξη

+αθ−1/β(1

ξ

)

∂ξ v + α (α + 1) θ−2/βη

(1

ξ

)2

∂2ηv

)

It follows that Problem (12) becomes{

θ−2ξ−2β∂2υv + Δ(ξ,η)v − λv + 1ξ[Pv] = f (ξ, η, υ) ∈ Q

v (ξ, η, υ) = 0 (ξ, η, υ) ∈ ∂ Q,(14)

with

f (ξ, η, υ) = θ−2ξ−2βg (ξ, η, υ) , (ξ, η, υ) ∈ Π∞,

here,

β = α/ (α − 1) , θ = (α − 1)β and Δ(ξ,η) = ∂2ξ + ∂2η .

We have also

f (ξ0, η, υ) = 0 on ∂ D, (15)

here P is the second-order differential operator with C∞ -bounded coefficients on Π∞ given by

[P (ξ, η, υ) v] (ξ, η, υ)

= α2θ−2/βη2(1

ξ

)

∂2ηv (ξ, η, υ) + 2αθ−1/βη∂2ξηv (ξ, η, υ)

+αθ−1/β∂ξ v (ξ, η, υ) + α (α + 1) θ−2/βη

(1

ξ

)

∂ηv (ξ, η, υ) .

In the sequel, we will focus ourselves on the study of the concrete problem{

ξ−2β∂2υv + Δ(ξ,η)v − λv = f (ξ, η, υ) ∈ Π∞v (ξ, η, υ) = 0 (ξ, η, υ) ∈ ∂Π∞,

(16)

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338 Arab J Math (2014) 3:325–340

Remark 3.2 Observe that the right-hand side f , is necessarily bounded on Π∞ , since

| f (ξ, η, υ)| =∣∣∣∣θ

−2ξ−2βh

(

exp

(ln [(α − 1)ξ ]

1 − α

)

, η exp

(α ln [(α − 1)ξ ]

1 − α

)

, υ

)∣∣∣∣

≤ θ−2 (ξ0)−2β

∣∣∣∣h

(

exp

(ln [(α − 1)ξ ]

1 − α

)

, η exp

(α ln [(α − 1)ξ ]

1 − α

)

, υ

)∣∣∣∣

≤ C max(x1,x2,x3)∈Π

|h (x1, x2, x3)| .

Remark also that our change of variables leads to the following property on the behavior on f at +∞. Forall (η, υ) ∈ D

limξ→+∞ f (ξ, η, υ)

= θ−2 limξ→+∞ ξ−2βh

(

exp

(ln [(α − 1)ξ ]

1 − α

)

, η exp

(α ln [(α − 1)ξ ]

1 − α

)

, υ

)

= θ−2 limξ→+∞ ξ−2βh (0, 0, x3)

= 0.h (0, 0, x3)

= 0.

Using the same arguments as in [5, Proposition 3.1], we get

Lemma 3.3 Let 0 < 2σ < 1. Then

1. h ∈ C2σ (Π) ⇒ g ∈ BUC2σ (Π∞).2. g ∈ BUC2σ (Π∞) ⇒ (x1)4σα h ∈ C2σ (Π) .

3.3 The abstract formulation of Problem (16)

Set E = BUC(D

). Define the vector-valued following functions:

v : [ξ0, +∞[→ E; ξ −→ v(ξ); v(ξ)(η, υ) = v(ξ, η, υ),

f : [ξ0, +∞[→ E; ξ −→ f (ξ); f (ξ)(η, υ) = f (ξ, η, υ).

Consider the family of closed linear operators (A (ξ))ξ≥0 defined by

{D (A (ξ)) = {

ϕ ∈ BUC0(D

) ∩ W 2,q (D) , q > 2, A (ξ) ϕ ∈ BUC(D

)},

A (ξ) ϕ (η, υ) =(∂2η + ξ−2β∂2υ

)ϕ, ξ ≥ ξ0.

where

BUC0(D

) = {φ ∈ BUC

(D

)/ φ = 0 on ∂ D

}.

Then, the concrete problem (16) is written in the following operational form{

v′′ (ξ) + A (ξ) v (ξ) − λv (ξ) = f (ξ) , ξ ≥ ξ0,v (ξ0) = 0, v (+∞) = 0, (17)

where

f ∈ L∞(]ξ0, +∞[ ; BUC2σ (D

)) ∩ BUC2σ ([ξ0,+∞[ ; BUC(D)).

It will be more convenient to work on [0,+∞[ instead of [ξ0, +∞[. So, we consider the natural change offunction: for ξ ∈ [0, +∞[, set

V (ξ) = v(ξ + ξ0), F(ξ) = f (ξ + ξ0).

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Arab J Math (2014) 3:325–340 339

Therefore, it is clear that

F ∈ L∞(]0,+∞[ ; BUC2σ (D

)) ∩ BUC2σ ([0,+∞[ ; BUC(D)).

Now the complete analysis of (17) on [ξ0, +∞[ is equivalent to the one done for the following problem{

V ′′ (ξ) + A0 (ξ) V (ξ) − λV (ξ) = F (ξ) ξ ≥ 0,V (0) = 0, V (+∞) = 0. (18)

where{

D (A0 (ξ)) = {ϕ ∈ BUC0

(D

) ∩ W 2,q (D) , q > 2, A0 (ξ) ϕ ∈ BUC(D

)},

A0 (ξ) ϕ (η, υ) =(∂2η + (ξ + ξ0)

−2β ∂2υ

)ϕ, ξ ≥ 0.

(19)

For simplicity, in the sequel , we write (18) as{

V ′′ (ξ) + Q (ξ) V (ξ) = F (ξ) ξ ≥ 0,V (0) = 0, V (+∞) = 0.

where

Q (ξ) := A0 (ξ) − λ, λ > 0.

3.4 Optimal results for Problem (12)

At this level, it is important to recall that the spectral properties of the family (19) in its most general formwere deeply discussed in [1], [12] and [14]. From which, we can deduce that

Lemma 3.4 (A (ξ))ξ≥0 is a family of closed linear operators with non-dense domains in E verifying theassumptions (3)–(6).

Remark 3.5 Observe that in our situation thanks to (13), it is well known that

DA(.) (σ, +∞) = {φ ∈ BUC2σ (D) : φ = 0 on ∂ D

}. (20)

For more details, see [13].

Then, our abstract results (2.5) and (2.6 ) applied for Problem (16) give

Proposition 3.6 Let f ∈ BUC2σ (Π∞), 0 < 2σ < 1. Then, there exists λ∗ > 0 such that for all λ ≥ λ∗ , theproblem

{ξ−2β∂2υv + Δ(ξ,η)v − λv = f, (ξ, η, υ) ∈ Π∞,v (ξ, η, υ) = 0, (ξ, η, υ) ∈ ∂Π∞.

admits a unique strict solution w satisfying

∂2ξ v,(∂2η + ξ−2β∂2υ

)v ∈ BUC2σ (Π∞).

Now, adapting the same argument of perturbation used in [2] and [9], we are able to say that

Proposition 3.7 Let k ∈ BUC2σ (Π∞), 0 < 2σ < 1. Then, there exist λ∗ > 0 and ξ∗ > ξ0 such that for allλ ≥ λ∗ and ξ ≥ ξ∗, the problem

{ξ−2β∂2υv + Δ(ξ,η)v − λv + 1

ξPv = f, (ξ, η, υ) ∈ Π∞,

v (ξ, η, υ) = 0, (ξ, η, υ) ∈ ∂Π∞.

admits a unique strict solution v satisfying

∂2ξ v,(∂2η + ξ−2β∂2υ

)v ∈ BUC2σ (Π∞ ξ∗).

where

Π∞ ξ∗ = [ξ∗, +∞[ × D

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340 Arab J Math (2014) 3:325–340

Let x∗1 := T −1 (ξ∗) where T −1 is the inverse change of variables given by

T −1 : Π∞ → Π

(ξ, η, υ) → (x1, x2, x3)(21)

with

(x1, x2, x3) :=(

exp

(ln [(α − 1)ξ ]

1 − α

)

, η exp

(α ln [(α − 1)ξ ]

1 − α

)

, υ

)

First, it is easy to see that

x∗1 < x1.

Using (21), we are then in position to state our main result describing the regularity of the unique solution uof Problem (12) near the cuspidal edge

Theorem 3.8 Let h ∈ C2σ (Π), 0 < 2σ < 1, satisfying ( 13). Then, there exist x∗1 > 0 and λ∗ > 0 such that,

for all λ > λ∗ Problem (12) admits a unique strict solution u satisfying

(x1)4σα ∂2x1u and (x1)

4σα(Δ(x2,x3) − λ

)u ∈ C2σ (Πx∗

1),

where

Πx∗1

={(x1, x2, x3) ∈ R3 : (x1, x2) ∈ Ωx∗

1, 0 < x3 < b

},

with

Ωx∗1

= {(x1, x2) ∈ R2 : 0 < x1 < x∗

1 , − (x1)α < x2 < (x1)

α}.

Acknowledgments The authors would like to thank the referees for their interesting comments and suggestions.

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use,distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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