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Additional info for V -cycle multigrid method for a viscoelastic fluid flow satisfying an Oldroyd-B-type constitutive equation
Thus we have proved the following proposition. 3 Proposition. If f E A(11) and f' 0, then f is a conformal mapping. Conversely, if f is a conformal C1 mapping in c, then f is analytic and f'#0. Two domains 0 and 0' in C are conformally equivalent if there is a ¢ E A(c) that maps Sl bijectively onto 1z'. We then find that 0-1 E A(c') and 2. Properties of Analytic Mappings 30 both 0 and 45-1 are conformal mappings. If 11 is conformally equivalent to U, then S2 is homeomorphic to U and hence simply connected.
In Cl (the limit function is not necessarily in '1). Sometimes it is convenient to make the definition less 32 2. c. For our purposes one can adopt either definition. 7 Proposition. If 4) C A(Q) and 4) is uniformly bounded on compacts, then it is a normal family. Proof. 8 of Ch. 1 that the gradients of the functions in 4) are uniformly bounded on compacts and hence 4) is equicontinuous on compacts. The conclusion then follows from the Arzela-Ascoli theorem. 8 Lemma. , all f" are injective, and f is not constant, then f is also injective.
Now take 0 E CO '(P) such that 0 = I in a neighborhood of H. ) 27ri f (z)dz = 7r f aT'f (T) Indr(()dA(() = 0 since Indr is supported on H and 80/8c = 0 there. 4 Remark. The preceding proof can be interpreted in the sense of distributions in the following way. Let the distribution (measure) u be defined by u(q) _ (i/2) fl, q5dS' for 0 E CO. Then u has compact support in Sl and (8/8() Indr = u, cf. 7 (c). The assumption on r implies as before that Indr is compactly supported in Q. 5 Remark. , d(fdz) = 0 in Q.