Effective energy integral functionals for thin films with bending moment in the Orlicz–Sobolev space setting

Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

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REITERATED HOMOGENIZATION OF INTEGRAL FUNCTIONALS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202500000057 ◽

2000 ◽

Vol 10 (01) ◽

pp. 47-71 ◽

Cited By ~ 19

Author(s):

ANDREA BRAIDES ◽

DAG LUKKASSEN

Keyword(s):

Energy Density ◽

Quadratic Form ◽

Phase Structure ◽

Effective Properties ◽

Effective Property ◽

Integral Functionals ◽

Effective Energy ◽

Two Phase ◽

Wide Range ◽

Reiterated Homogenization

We consider the homogenization of sequences of integral functionals defined on media with several length-scales. Our general results connected to the corresponding homogenized functional are used to analyze new types of structures and to illustrate the wide range of effective properties achievable through reiteration. In particular, we consider a two-phase structure which is optimal when the integrand is a quadratic form and point out examples where the macroscopic behavior of this structure underlines an effective energy density which is lower than that of the best possible multirank laminate. We also present some results connected to a reiterated structure where the effective property is extremely sensitive of the growth of the integrand.

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Composition and Phase Control for Molybdenum Nitride thin Films

MRS Proceedings ◽

10.1557/proc-354-69 ◽

1994 ◽

Vol 354 ◽

Cited By ~ 3

Author(s):

Mandar S. Mudholkar ◽

Levi T. Thompson

Keyword(s):

Thin Films ◽

Energy Density ◽

Rutherford Backscattering Spectrometry ◽

Ion Beam ◽

Arrival Rate ◽

Molybdenum Nitride ◽

Effective Energy ◽

Film Composition ◽

Ion Beam Assisted Deposition ◽

Molybdenum Nitrides

AbstractMolybdenum nitrides are active and selective hydrodenitrogenation (HDN) catalysts. The catalytic properties of molybdenum nitrides were found to be dependent on the structural properties. The purpose of research described in this paper was to synthesize molybdenum nitride thin films with well defined structures and stoichiometries using ion beam assisted deposition. The films were deposited by evaporating Mo metal, and simultaneously bombarding the growing film with low energy nitrogen ions. The phase constituents of the films were determined using x-ray diffraction and the film composition was obtained by Rutherford backscattering spectrometry.The film composition and phase constituents were strong functions of the ion-to-atom arrival rate ratio, ion energy and ion angle of incidence. Differences in the film composition for different arrival rate ratios and ion angles of incidence were interpreted based on reflection and sputtering effects. Our results suggest that phase formation was governed by the effective energy density per deposited atom. Evaluation of the effective energy density per deposited atom and its physical significance in ion beam assisted deposition is discussed.

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