By C. Harper
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Hint: If u(x,O) = 0 and ut(x,O) = t > 0 for x t (-cm,cm), then u grows arbitrarily large with time. Figure out conditions on the initial data that assure that u stays bounded. 27. Suppose f and g are identically zero outside the interval [-I, 11. In what region in ( - c m , ~ )x [O,cm) can you ensure that the solution u of the Cauchy problem is identically zero. 28. Is there a similar result to the previous problem for the heat equation? Hint: Use as initial datum. 2. 29. 142) for every 4 t Ci(R2).
By repeated application of Cauchy's formula for functions of a single complex variable, we find that < where ri is the positively oriented circle zi x i = T. We now write and expand in a geometric series with respect to powers of (yi-xi)/(zi-xi). 61) where The characterization of real analytic functions as restrictions of complex differentiable functions allows a simple proof of the implicit function theorem for real analytic functions. 18. Let the functions F,(xl,. . , x n , y l , . . ,ym), i be real analytic at the point ( x Oy, o ) t Rn x Rm.
Then we say that f is majorized by F , f 5 F if IDa f ( 0 ) D a F ( 0 ) for every or. 20. I f f t CM,,(O),then f is majorized by the function This is clear, since D a d ( 0 ) = M o r ! r a l We need the following result concerning composite functions. 21. Let f , F be vector-valued real analytic functions from a neighborhood of the origin i n Rn into Rm such that f ( 0 ) = F ( 0 ) = 0 . Let g , G be real analytic functions from a neighborhood of the origin i n Rm into R . Assume that f , 5 F, for i = 1,.