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

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


Introduction
In this work, we deal with non-autonomous problems of the form subject to the following boundary conditions where: where E is a complex Banach space. (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 strict solution, that means a function u such that ⎧ ⎨  (1) and (2). Just, we recall here that for k ∈ N , BU C k ([0, +∞[; E) is the space of vectorvalued 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 concrete boundary value problem set on a singular domain.
It should be noted that the solvability of boundary value problems for differential operator equations on bounded domains 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 For an overview on these kind of problems and some historical references, see [11,15,16].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 these methods, 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 methods applied in [1] and [12]. Note that besides being complementary to [5], the present paper contains important new results. In fact, we present some new Hölder continuous regularity results for an elliptic equation set on nonsmooth cylindrical domains. These results can be hardly obtained using the classical standard techniques such 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. We prove also some results, which allow us to justify the optimal smoothness of the previous representation. In Sect. 3, we give a concrete example to which our abstract results can be apply.

Assumptions and representation of the solution
For simplicity of notation, we set Throughout this work, we assume that the the family of linear closed operators (Q(t), D(Q(t))) t≥0 enjoys the following properties: here, ρ(Q(t)) is the resolvent set of Q(t).
3. For all z ∈ δ 0 , the mapping t → (Q(t) − z I ) −1 defined on R + , is of class C 2 . Furthermore, we suppose also that: There exist C 2 > 0, σ ∈]0, 1/2[ such that for all z ∈ δ 0 and all t, τ ≥ 0, and 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. (3) and (4) express the ellipticity of (1). Moreover, all the previous assumptions remain true if we replace z by z + √ λ.

Hypotheses
Throughout the rest of this paper, C stands for a generic constant and σ ∈]0, 1/2[, We know that in the case when is a constant operator satisfying the hypothesis (3) and (4), the representation of the solution u of (1) and (2) is given by where Here, the curve γ is the boundary of the sector δ 0 oriented from ∞e +iδ 0 to ∞e −iδ 0 and √ −z is the analytic 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 We are then concerned with the determination of the unknown function f * in order that (9) is a strict solution of Problems (1) 2. For all t ≥ 0, the function f * (introduced in 9) satisfies the following equation where Proof Statement 1 follows from Proposition 3.1 in [3].
Concerning Statement 2, one has To calculate u (t), we follow the same reasoning as in [12]. We set On the other hand, one has where and and by the dominated convergence theorem, we get It remains to treat the quantity I (ε) ( f * )(t), using the identity we get where Now for the quantity I (ε) ( f * ) (t), one has Concerning the term I (ε) ( f * ) (t), we write On the other hand, one has Summing up, we deduce that, for all t ≥ 0 : Now, we need the following important result concerning f * .
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 * ∈ BU C 2σ ([0, +∞[ ; E). Then, the vector-valued function t → Op ( f * ) (t) belongs to the space BU C 2σ ([0, +∞[ ; E).
Proof Let t > τ ≥ 0, thus where (I) and (II) can be treated similarly. So, we restrict ourselves to treat the first quantity. One has where and We can write I 1 as where thanks to the differentiability properties of the resolvent, that is, (5 )and (6), we conclude that The same arguments applied for I 2 give Summing up, we are in position to give our main maximal regularity results concerning Problems (1) and (2). ([0, +∞[ ; E). Then, there exists λ * > 0, such that , for all λ ≥ λ * , Problems (1) and (2) have a unique strict solution

Proposition 2.5 Let f ∈ BU C 2σ
Moreover, one has 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 in for more details about these spaces, see [7] and [13].
Proof We are interesting with the convergence of the integral 1 2iπ Therefore, we can deduce the following result.

Position of the problem
Now, we will apply the abstract regularity results obtained in the previous section for the study of a concrete elliptic problem. We consider the following problem: Let Π be an open set of R 3 defined by where b is a finite positive number and Ω is the planar cusp domain defined by a is a finite positive number small enough.
In Π , we consider the boundary value problem The right-hand side h belongs to the Hölder space C 2σ (Π ) and satisfies the following condition where ∂Γ (a) denotes the the boundary of the lateral surface It should be noted that Problem (12) is a particular case of some elliptic equations frequently encountered in engineering application. In fact, applications of such equation are abundant in fluid dynamics and the modelization of weather prediction. It is well known that the solvability of elliptic problems posed in singular domains was intensively investigated by numerous authors via several techniques, see, for example [8,10]. Most of these studies have focused on 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 Dirichlet problem for Laplace equation posed in planar cusp domain. These authors have used the abstract differential equation theory which seems more adapted for this kind of problems. In our situation, the study of the concrete problem (12) will be reduced to the study of an abstract differential equation of elliptic type with variable operator coefficients., that is, Problems (1) and (2).

Change of variables
As in [5], we use the following change of variables which means that the cuspidal edge (0, 0, Note that for any Now, define the following change of functions From the above changes of variables, one has It follows that Problem (12) becomes with f (ξ, η, υ) = θ −2 ξ −2β g (ξ, η, υ) , (ξ, η, υ) ∈ Π ∞ , here, We have also here P is the second-order differential operator with C ∞ -bounded coefficients on Π ∞ given by In the sequel, we will focus ourselves on the study of the concrete problem Therefore, it is clear that Now the complete analysis of (17) on [ξ 0 , +∞[ is equivalent to the one done for the following problem where For simplicity, in the sequel , we write (18) as (12) At this level, it is important to recall that the spectral properties of the family (19) in its most general form were deeply discussed in [1], [12] and [14]. From which, we can deduce that