By D. Hodges, et al.,
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An example of the formulae for matrices Aij IS gIven below . £111 = C2, £112 = C2 (A4A31 \ ' £1 21 = £1 22 = (A3 1)1 (A3 1A4)1 C2 + (A3 1A4)1 C2 ( A4A3 1)1 Here matrices AI, A 2, A 3 , A4 and C2 are defined by the formulae A2 = (A12A221)2 ' A4 = (A221 A21 A 1)2' J 1 (fh = jdYk· o It has been proved that the following estimates are valid IIAij - Aij II :s; (3(w), (3(w) Ilu - vll£2(G o ) :s; Q V3 (VEl + (3(w)) , -t 0, at w -t 00 Here Aij are the effective moduli obtained by the exact solution of the cell-problems, u is a solution of the original problem, v is the solution of the problem with the coefficients Aij , Q is a constant independent of E1, w, and V3 = Ilv11H3' 5.
The matrix material moduli are A}j. The moduli of the inclusions are supposed to have an order 'T/ in comparison to the matrix moduli. The problem B deals with the same geometrical structure but the inclusions are replaced by empty pores. Let u~ be a solution of the boundary value problem A in G a ( aXi aue) = Aij ax; (u~ f, - g)18G = 0, Aij = ATj' x E G e , Aij = Atj = 'T/A~j' x E G/G e , and U o be a solution of the boundary value problem B in G e ~ aXi auo) = f, (At. (':) J E aXj (uo - g)lr.
1. Applications to waveguide systems. Phys. Rev. B 55(1997), 9842-9851. : Bound states in waveguides and bent quantum wires. II. Electrons in quantum wires. Phys. Rev. B 55(1997), 9852-9859. : Hill's equation for a homogeneous tree. Electronic J. Diff. Equations (1997), no. : Adjoint and self-adjoint operators on graphs. Electronic J. Diff. Equations (1998), no. : Inverse eigenvalue problems on directed graphs. Trans. Amer. Math. Soc. 351 (1999), no. 10, 4069-4088. : Nonclassical Sturm-Liouville problems and Schriidinger Operators on Radial Trees.
Analysis and Design of Digital IC's in Deep Submicron Tech. by D. Hodges, et al.,