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By Giovanni P. Galdi (auth.)

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R• :~(e,x2,x3)£1e ~ hl 3 :u(Yli'7•X3)dq '7 Integrating over they-variables and raising to the qth power, we deduce ju(x1,x2,x3) -ulq ~ ICI-q [a foa IVu(e,x2,x3)lde 3 +a2 foa loa jVu(Yli'7•X3)Idyld'7 +a Ia IVuldC] q. 4. v 53 which completes the proof. 3. 2). 1. Suppose 0 is a cube of side a and subdivide it into N equal cubes Ci, each having sides of length a/N 1fn. -q)/q Xi(x), with Xi characteristic function of the cube Ci, from the previous inequality one has the following result due to Friedrichs {1933).

Basic Function Spaces and Related Inequalities and we shall write Um -+ u strongly. On the other hand, we say that {Um} converges weakly to u if lim i(um- u) = 0 m-+oo for any bounded linear functional i in Lq (0) and we shall write u weakly. VII); this amounts to saying that Um converges weakly to u E Lq if and only if for all e > 0 we can choose m and m' sufficiently large so that li(um - Um 1 )I < E, for any bounded linear functional i in Lq. We wish now to give some classical results concerning weak convergence.

E. in 0 with a. (uniquely determined) function of C(ft). 14) llullc $ C3llullt,q q > n with Ci = C;(n,q, r). 13) to functions from wm,q(O), to obtain the following embedding theorem whose proof is left to the reader as an exercise. 2. : 0. q J if mq < n, and for all r E (q, oo) if mq = n. e. in 0 to a unique function in Ck(fi), for all k E [0, m- (nfq)) and the followin g inequal ity holds llullr $ c2llullm,q for all r E [q,oo), ifmq with C3 = c3(m, q, r, n). 2 to the spaces wm,q(O), 0 =/: R n.

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