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L. 13). We denote by P' the class of all such operators L . The main result on G-convergence of parabolic operators is the following.

Let J; E v· and Uik = Bkj;, i = 1, 2. Then Hence, weakly in V* . )/P X (Akulk- Aku2k, u1k- U2k)o:fp = ]o:/p 1 x [ (h- h v1k- v·2k)- 2lulk(T)- v·2k(T)I 2 Passing to the limit and using weak lower semicontinuity of the norm of V* we get llh- h- [(Bh)'- (Bh)'JIIv·:::; :S: cJ(m + (h- (B h)', Bh) X + (h- (Bf2)', B h) )(p- 1-o:)jp x ] o:fp 1 [ (h- h,Bh- Bh)- 21(Bh)(T)- (Bh(T)I 2 . 26) 35 G-CONVERGENCE OF ABSTRACT OPERATORS This implies that h = h provided B h = B h Next we show that the image of B is dense in V.

The general case may be covered by trivial passage to the limit. Set Wo = {u E W : u(O) = 0}. The subspace W 0 is dense in V. ) 2: 0, u. E Wo. Now let us introduce the main class of abstract parabolic operators we shall consider. We fix nonnegative functions m, m1, m 2 E £ 1 (0 , T), constants c 1 > 0, cz > 0, c3 > 0, c4 > 0 and constants a and fJ such that 0

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