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On the basis property of root vectors related to a non-self-adjoint analytic operator and applications

Ellouz Hanen, Feki Ines, Jeribi Aref

Journal of Mathematics and Applications

2022

Vol. 45

59--86

In the present paper, based on a separation condition on the spectrum of a self-adjoint operator T0 on a separable Hilbert space H, we prove that the system of root vectors of the perturbed operator T (ε) given by T (ε) := T0 + εT1 + ε2T2 + . . . + εkTk + . . . is complete and forms a basis with parentheses in H, for small enough |ε|. Here ε ∈ C and T1, T2, . . . are linear operators on H having the same domain D ⊃ D(T0) and satisfying a specific growing inequality. The obtained results are of importance for applications to a non-self-adjoint Gribov operator in Bargmann space and to a non-self-adjoint problem deduced from a perturbation method for sound radiation.

Effective energy integral functionals for thin films in the Orlicz-Sobolev space setting

Laskowski W., Nguyen H. T.

Demonstratio Mathematica

2013

Vol. 46, nr 3

585--604

We consider an elastic thin film as a bounded open subset ω of R2. First, the effective energy functional for the thin film ω is obtained, by Γ-convergence and 3D-2D dimension reduction techniques applied to the sequence of re-scaled total energy integral functionals of the elastic cylinders (…) as the thickness ε goes to 0. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function for the elastic cylinders has the growth prescribed by an Orlicz convex function M. Here M is assumed to be non-power-growth-type and to satisfy the conditions (…) and (…) (that is equivalent to the reflexivity of Orlicz and Orlicz–Sobolev spaces generated by M). These results extend results of H. Le Dret and A. Raoult for the case M(t) = (…) for some (…).