By Louis Boutet de Monvel (auth.), Prof. Jacques Chazarain (eds.)
Read or Download Fourier Integral Operators and Partial Differential Equations: Colloque International, Université de Nice, 1974 PDF
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Extra resources for Fourier Integral Operators and Partial Differential Equations: Colloque International, Université de Nice, 1974
Now we have + #j(P) < j >= o, with equality for just one j, so gp(O~) = D~J + /~j(P) in a neighborhood of 2~ . 2) is valid with g p ( ~ ) = g(P) and d(P) = dp. The second part follows in the same way. Replacing t by an integral power of t we can make ~ an integer, which completes the proof. Proof of Theorem ~,I~ To construct u we shall piece together functions u~ which are appropriately chosen in the slabs Here we take b~ = I/~2 , for example, and 12 will be large. ~) we set u (x) = exp i(
3 is already applicable). 5O The only examples of non-uniqueness for C oo perturbations of elliptic operators have been given by Pli~ ~8]. 1. 3) rather arbitrarily, just making sure that Im ~ and ~ were of the right order of magnitude and exercising some care at the points b' switch was made from u~ where a to u~+ I. In the case of complex chazacteristics, on the other hand, the analogue of the function ~ has to be chosen so that it fits an initial analytic perturbation of the operator. We shall do so by means of the asymptotic expansions of geometrical optics which seem to have greater generality than the arguments used by Pli~ .
59 Here C(S, 8) -K(6~)(I-A(sF))/A(8~)--~-~JoS , 6--~0. It is clear that C(S, 6) is analytic for small 8. 22)" become smooth after multiplication by some power of 6. 23) r(z) - ~ jO S q(z) when 6--YO. 17)' if ~ > 1, as we assume from now on. 12 (and the remark following its proof). 18)" is fulfilled for some z O. 23) has a unique C OO zero z(S) with z(O) ~ z0 defined for S in a neighborhood of O. 18)" that Im dz/dS = f JO Im q(Zo)/r'(Zo) > 0 when S = O. 25) Im dz/dS > O, S~-I. 22)" with = z(S). 25) it follows that for small 8 ~2 Im ~ ( S , 8)/~S 2 ~ c > O, S e I .
Fourier Integral Operators and Partial Differential Equations: Colloque International, Université de Nice, 1974 by Louis Boutet de Monvel (auth.), Prof. Jacques Chazarain (eds.)