$ G$-Convergence and Homogenization of Nonlinear Elliptic Operators in Divergence Form With Variable Domain

Abstract

Particle losses much larger than given by the equilibrium theory have always been observed in the low-density range of thermally ionized magnetoplasmas in a Q-device. This could be attributed either to anomalous radial losses or to enhanced end-plate recombination. Our previous experiments with Ba in "narrow-beam illumination" and quiescent state have indicated an enhanced surface recombination rather than anomalous diffusion to be responsible for the additional losses.In this paper, a series of results with Cs, Ba, Sr, and Li plasmas is presented and compared with the theory of Part I, which deals with enhanced surface recombination. It is found that the intrinsic loss processes are surface recombination -as described in Part I -and classical diffusion. Additionally, enhanced losses across the magnetic field may occur, e.g. in a non-quiescent state, or in the presence of azimuthal temperature gradients on the end plates in broad-beam illumination.

G*-Convergence of Nonlinear Operators in the Theory of Homogenization of Control Objects

“…Then the sought-for relationship (13) is a direct consequence of the above-mentioned inequality, estimate (18), and a similar estimate for g. In this case, as follows from [18], the truth of inequalities (15) and (16) and satisfies relationships (19) and (20) for all u Y , u Î . Proof.…”

“…We introduce the following notations: let w Y be the weak topology of Y , let w Y * * be the *-weak topology of Y * , and It is well known (see [5,15] …”

“…As is well known, the practical significance of the solution of the problem of homogenization of such objects follows from the needs of mechanics of composite materials and the theory of filtration in porous media. In particular, as is shown in works on problems of limit analysis of boundary-value problems in perforated domains (see, for example, [4,[11][12][13][14][15][16][17]), their solutions converge to the solution of a boundary-value problem whose associated operator is the sum of a G-limit operator and some additional lowest term that characterizes the perforation structure of the initial domain.…”

“…We additionally is consistent with the scale of Banach spaces { } Y E e eÎ in the sense of Definition 11. The solution of the problem formulated is preceded by some concepts and results from[15]. always implies its weak Y e -convergence to the same limit.…”

Convergence of Solutions of Bilateral Problems in Variable Domains and Related Questions

“…Concerning the proof of this result, see, for instance, [10]. We also remark that the condition of Proposition 11 is satisfied in the case where the domains Ω s have a perforated structure of the same kind as the structure of the domains considered in [16,Section 2].…”