On Some Elliptic Problems with Nonlocal Boundary Coefficient-operator Conditions in the Framework of Hölderian Spaces

In this paper we give some new results on second order differentialoperator equations of elliptic type with nonregular boundary conditions with coefficient-operator. The study is developped in H¨ spaces and uses the reduction method of S. G. Krein. Necessary and sufficient conditions of compatibility are established to obtain different types of solutions. Maximal regularity properties are also studied.


Introduction and hypotheses
Many authors have studied nonlocal boundary value problems: we can first refer to the pioneering works by T. Carleman [4] and J. D. Tamarkin [19], see also A. V. Bitsadze and A. A Samarskii [3] who introduce some nonlocal boundary conditions, to study elliptic problems coming from plasma theory.The case of a non linear elliptic equation with a nonlocal boundary condition has been treated by Y. Wang [22].More bibliographic details on nonlocal elliptic problems can be found in the monograph of A. L. Skubachevskii [18].Such nonlocal problems have been also considered in the framework of elliptic differential-operator equations, studying coerciveness and Fredholmness, see S. Yakubov [20] and also more recently A. Favini and Y. Yakubov [10], [11], B. A. Aliev and S. Yakubov [1].
In this work we consider the following second order differential-operator problem: where X is a complex Banach space, f ∈ C θ ([0, 1] ; X) with 0 < θ < 1, u 0 , u 1,0 are given elements of X, A is a closed linear operator with domain D (A) not necessarily dense in X and H is a closed linear operator with domain D (H).Recall that, for any interval J C θ (J; X) = h : J −→ X, sup x,y∈J,x =y h(x) − h(y) |x − y| θ < +∞ .
2. Due to (2), there exists ε A > 0, β A ∈ 0, π 2 such that ρ (A) contains a sectorial domain Moreover ρ (−A) ⊃ {z ∈ C\ {0} : |arg (z)| > π − β A }, thus, we obtain ρ (−A) and setting We adapt to our situation the definitions of strict and classical solutions given by E. Sinestrari in [17], Section 2, p. 34: We first notice that here D(A) is endowed with the graph norm that is and then for an interval J, we define C (J; D(A)) in the following manner (see Proposition 2 below).
• a strict solution u of problem ( 1) is a function u such that and which satisfies (1).This strict solution satisfies the maximal regularity property if When H = 0, which means that we consider Dirichlet boundary conditions, it is known that, under assumption (2), problem (1) has a strict solution u if and only if u 0 , u 1,0 ∈ D(A) and Moreover u has the maximal regularity property if and only if u 0 , u 1,0 ∈ D(A) and f (0) − Au 0 , f (1) − Au 1,0 ∈ D A (θ/2, +∞), see R. Labbas [13].
When H = 0, the nonregular boundary condition u (1) + Hu ′ (0) = u 1,0 , involves in general, a loss of regularity for the solution u at point 1, but we must also take into account the fact that this nonregular boundary condition make sense if u is continuous at 1, with u ′ continuous at 0. This leads us to introduce new types of solutions of problem (1): and which satisfies (1); moreover we say that this semiclassical solution satisfies the maximal regularity property if EJQTDE, 2013 No. 36, p. 4

It is well known that any
) .We will say this semistrict solution satisfies the maximal regularity property if it satisfies (7) together with Note that a particular case of Problem (1), that is H = αI, has been studied by Labbas-Maingot (see [14]).These authors used a direct method based on the techniques of Dunford integrals to build a representation formula of the solution.
In this work, a representation formula of problem ( 1) is found by using analytic semigroups and fractional operators theory.
This work is organized as follows: Section 2 is devoted to Problem (1) and contains our main result (Theorem 13): we first recall classical results on generalized analytic semigroup, then, under assumptions (2)∼(5), we build a representation formula for the solution of (1) and study the regularity of this representation.Finally we consider some particular cases in which our invertibility assumption (5) is satisfied.
In Section 3 we introduce a spectral parameter ω ≥ 0 which allows us to apply the results of section 2.
In section 4, a concrete problem is considered to illustrate our results.
Let L be a linear operator in X such that EJQTDE, 2013 No. 36, p. 5 for some given µ ∈ R and δ ∈ 0, π 2 .This says exactly that L is the infinitesimal generator of a generalized analytic semigroup e xL x≥0 , "generalized" in the sense that L is not supposed to be densely defined.
Proposition 2 Let L is the infinitesimal generator of a generalized analytic semigroup e xL x≥0 .
Then the two following assertions are equivalent Let us recall that for an operator P in X satisfying ρ(P ) ⊃ ]0, +∞[ and we define the interpolation space D P (θ, +∞) by  Then the two following assertions are equivalent For these two propositions see, for instance, E. Sinestrari [17].
Lemma 6 One has where and Proof.As in [5] (see also S. Yakubov and Y Yakubov [21]), we immediately deduce that u has the representation where ξ 0 , ξ 1 ∈ X and I x , J x satisfy (10).
To obtain the final representation of u, it is enough to find ξ 0 and ξ 1 by taking into account the data u 0 , u 1,0 , f and A. A formal computation gives and from which we deduce (9) by using e Q Λ −1 = Λ −1 e Q (which is a consequence of ( 4)).We need to justify the terms HQu 0 , HQJ 0 in (9) : EJQTDE, 2013 No. 36, p. 8 so Qu 0 ∈ D (Q) ⊂ D (H), moreover, using Proposition 3, assertion 3, we can write and thus In order to simplify representation (9) we first show the following Lemma.
Lemma 7 Proof.For statement 1 we write EJQTDE, 2013 No. 36, p. 9 We conclude by noting that For statement 2, it is enough to remark that for any g ∈ C θ ([0, 1] ; X) . Now, using ( 9) and Lemma 7, we can write we can rearrange the terms of u to obtain the decomposition EJQTDE, 2013 No. 36, p. 10 with the regular part u R in [0, 1] given by the terms which gives the behavior near 0 and the one concerning the nonlocal behavior in 0 and 1 (note that since u 0 ∈ D(A) then HQu 0 = −HQ −1 Au 0 is well defined).

Regularity results
To study the regularity of the solution we need some technical lemmas.First EJQTDE, 2013 No. 36, p. 11 and (see Propositions 3 and 5). Proof.
and when HQ −1 Au 0 ∈ D(Q) we have and when HQ −1 Au 0 ∈ D(Q 2 ) we have
1.Here f (0) ∈ D(Q) and then, from Lemma 9, statement 3, we get and we conclude noting that 2. Here f (0) ∈ D(Q 2 ) and then, from Lemma 9, statement 5, we get Now, when HQ −1 Au 0 ∈ D(Q 2 ), Lemma 9, statement 6, furnish and we conclude noting that By similar arguments, we can also prove the following Lemma.

Main results
Theorem 13 Assume (2)∼( 5), suppose that u 0 , u 1,0 ∈ D (A) and 1. there exists a semiclassical solution u of problem (1) if and only if 2. there exists a semiclassical solution u of problem (1) having the maximal regularity property (7) if and only if

there exists a semistrict solution u of problem (1) if and only if
4. there exists a semistrict solution u of problem (1) having the maximal regularity property ( 7)-( 8) if and only if

there exists a strict solution u of problem (1) if and only if
6. there exists a strict solution u of problem (1) having the maximal regularity property (6) if and only if EJQTDE, 2013 No. 36, p. 18 Moreover, in the 6 cases u is unique and given by u = u R + v + w where u R , v, w are defined in ( 13),( 14) and (15).
Proof.For statements 1 and 2, we first remark that, from subsection 2.2, if there is a semiclassical solution u of problem (1) then u is uniquely determined by u = u R + v + w.We conclude by applying Lemmas 8 and 9 and noting that, since u ′′ + Au = f , then Statements 3∼6 are similarly proved.
We now study some situations where more regularity is given on H or f which allow us to drop the conditions on I f .
1. Suppose that H ∈ L(X) then: there exists a semistrict solution u of problem (1) if and only if 2. Suppose that H ∈ L(X) with H(X) ⊂ D(Q) then: there exists a strict solution u of problem (1) if and only if 3. Suppose that f ∈ C θ ([0, 1] , D(Q)) then: there exists a semistrict solution u of problem (1) if and only if ) then: there exists a strict solution u of problem (1) if and only if EJQTDE, 2013 No. 36, p. 19 5. Suppose that H ∈ L(X) and f ∈ C θ ([0, 1] , D(Q)) then: there exists a unique strict solution u of problem (1) if and only if Proof.For statement 1 and 2, we apply Lemmas 8 and 10, noting that For statement 3, we use Lemmas 8, 11 and also the fact that which gives Statement 4 and 5 are similarly treated.
In the previous corollary, we will obtain, in each case, maximal regularity for the solution u if we replace D (A) by D A (θ/2, +∞).

Particular case for Problem (1)
We first study the particular case The main difficulty is assumption (5) and we need some results of functional calculus.
Lemma 15 Setting S = S π/4 , we get: 1. F, G are holomorphic on a neighborhood of S.

lim
Re z→+∞, z∈S 2αze −z + e −2z = 0 and then (a) there exists x 0 > 0 such that z ∈ S and Re z x 0 imply .
1.By the same techniques we can consider H = −αQ under hypothesis (22), study functions F , G defined by and thus prove that Λ = I + 2αQ 2 e Q − e 2Q is boundedly invertible with since second boundary condition can be written 3 Problem with a spectral parameter In order to provide results for general H satisfying ( 5), we will consider some large positive number ω and the problem

Study of Problem (24)
We consider some fixed ω 0 ≥ 0 and we set, for ω ≥ ω 0 then Problem (24) is Problem (1) with A replaced by A ω .
Our main assumptions on the operators are this assumption implies that , is the infnitesimal generator of a generalized analytic semigroup on X.

∀ζ ∈ D(H)
, is the infnitesimal generator of a generalized analytic semigroup on X.Note that and then c 0 does not depend of ω.Proof.We can write Λ ω = I − T ω with T ω = 2HQ ω e Qω + e 2Qω .Thus, to show that the operator Λ ω has a bounded inverse, it is enough to have T ω L(X) < 1.
By using Lemma p. 103 in G. Dore and S. Yakubov [8], we have and, since HQ −2 ω 0 is bounded, then Moreover, in the 3 cases u is unique and given by u ω = u ω,R + v ω + w ω where u ω,R , v ω , w ω are defined as in ( 13),( 14) and ( 15
∼(5) will be satisfied and we can apply Theorem 13.