New Results on Complete Elliptic Equations with Robin Boundary Coefficient-Operator Conditions in Non Commutative Case

Abstract

In this paper, we prove some new results on operational second order dierential equations of elliptic type with general Robin boundary conditions in a non-commutative framework. The study is performed when the second member belongs to a Sobolev space. Existence, uniqueness and optimal regularity of the classical solution are proved using interpolation theory and results on the class of operators with bounded imaginary powers. We also give an example to which our theory applies. This paper improves naturally the ones studied in the commutative case by M. Cheggag, A. Favini, R. Labbas, S. Maingot and A. Medeghri: in fact, introducing some operational commutator, we generalize the representation formula of the solution given in the commutative case and prove that this representation has the desired regularity. Keywords: second-order elliptic dierential equations; Robin boundary conditions in non commutative cases; analytic semigroup; maximal regularity. Dedicated to Professor Angelo Favini on his 70th birthday.

Elliptic differential-operator with an abstract Robin boundary condition containing two spectral parameters, study in a non commutative framework

“…When ( 19) is satisfied, we can check that this representation formula coincides with the one given in [8] p. 528. We can also, after computations, verify that ( 24) is the same formula as the one in [9], p. 92 (with L = M = Q) and also compare it with (34)∼(38) pp. 54-55, in [19].…”

“…Assume (H 2 ) and (H 4 ). If we suppose that QΛ −1 ξ ∈ D(Q), then we have (H 5 ), see Lemma 5 p. 76 in [9]. Similarly, when Q − H is boundedly invertible, using (20) we have that…”

“…For ω large enough, under some geometrical assumptions on the space X and hypotheses on operators A−ωI and H, including the fact that they commute in the resolvent sense, the authors furnish necessary and sufficient conditions on the data d 0 , u 1 to obtain the existence and uniqueness of a solution u of ( 1)-( 2) with maximal regularity. Recently, in [9], the authors develop an interesting new approach in a non commutative framework, concerning some general Sturm-Liouville problems with the same Robin boundary condition in 0.…”

New Results on the Solvability of Abstract Sequential Caputo Fractional Differential Equations with a Resolvent-Operator Approach and Applications

“…Over the past two decades, several researchers have developed the theory of abstract impulsive and abstract fractional differential equations with nonlocal conditions; see, for instance, references [10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the studies cited therein.…”

Sturm-Liouville Abstract Problems for the Second Order Differential Equations in a Non Commutative Case

“…Recently, in [3], the authors have developed an interesting new approach concerning some general SturmLiouville problems with the same Robin boundary condition in 0. They have assumed that:…”