Computational Methods for Optimizing Distributed Systems by K.L. Teo and Z.S. Wu (Eds.)

By K.L. Teo and Z.S. Wu (Eds.)

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Iii) 1 . 4 E LP(W,where 1 5 P < 03, then 4&E Lp(Q), I14&llp,n 5 l1411p,a and lim,,o 114, - 411p,n = 0. (iv) I f 4 E C(Q) and G is a subdomain of R such that G is compact and contained in Q, then lim,lo, $,(x) = 4 ( x ) uniformly on G. (v) I f 4 E C(@, then limELo 4,(x) = 4 ( x ) uniformly on R. Note that Co(Q) is dense in L,(Q) for p E [1, co). 1, C$(Q) is also dense in L,(R) for p E [l, co). 2. 1, pp. 1 and suppose thut #, DB#cL1(Q),where DO4 denotes the 8th generalized partial derivative of 4.

1. When n 2 pm, more care is necessary in interpreting the trace 4 Ian of the corresponding function Cp E WF(Q)on dQ. To explain this concept we note that C"'(a) is dense in W;(Q). Thus there exists a sequence {&} c Cm@) such that \lCpk - CpllLmh -+ 0 as k -+ 00. Clearly, the function 4k have traces CpklaR on 8R. 4a) signifies that these traces converge in L,(dR) to a function such that 6 l1611q,aR 5 K , Il4llbmh? 6 is called the trace of the function 4 E W;(R) on 8 0 and denoted by 4 Ian. Note that the preceding inequality with I1611q,aR( k .

A Priori Estimates 49 in ti)) n Lnl,Nl(Q(O,ti)) and LA3,,,3(r(O> ti)), respectively, as m,k 00, independent of the order with respect to m and k. Then the functions 4p,p > 0, as defined by ( 4 4 , are also in V;>O(Q). 1 that J':l0(Q(O, -+ (iii) (i (iv) (Fi)pE L2(Q), i= 1 (v) f p (4 E L ~ ~ , ~ ~ ( Q ) ; i = 1, . . ) y p E E L*3,p3(r); Lgd, r 403. 2. For any q, r E [l, co), let g be an element in Lq,r(Q). 40). (Q(O,T - d)), as p + 0, where 6, 0 < 6 < T, is arbitrary. 51) 4 9, 0. 8, respectively.

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