A maximum property for the characteristic initial value problem of the wave equation
In: Proceedings of the American Mathematical Society, Jg. 46 (1974), S. 77-78
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Zugriff:
A maximum property is obtained for the solutions of the odd-dimensional wave equation in the interior of the characteristic cone. In this note we will establish a maximum principle for the odd-dimensional wave equation O u =: E 1 u _ 2u= ol n odd > 39 in the interior of the characteristic cone t = |Xj, X = (xl, x21 . x ), under hypotheses on the behavior of u and its derivatives on the cone. A maximum principle for the one-dimensional wave equation uxx utt 0 O in the cone is given in [21. The ellipsoidal means EM of a continuous function /(X) is defined as follows [1], [3]: EM[/; X, t] = (-l)kt(c__ 1r') 1(t2 r2)kf dpf pn-3(1 p2)/i(py) dc, where X = ra, r= |X, t > r, c = (t + r)/2, b = (t r)/2, k = (n3)V2, p = t/r+ (r2 _ t2)/2pr, y pa+ (1 _ p2)/28, 8 a unit vector perpendicular to a, o1 =2Z77(nY2 /F((n 1)/2,. It is easily seen that (1) (1 _ p2) = (r2 _ t2)(4p2r2) 1(2p + r t) (2p r t). Let ?>(X) be a smooth function in Rn, and let TOW(x) = (1)k(k!)ik X, r] is the solution of the problem Eu = 0 in t > |XI with u(X, |X|) = ?(. Using (1), it can be shown t-hat Received by the editors October 31, 1973 and, in revised form, December 5, 1973. AMS (MOS) subject classifications (1970). Primary 35L05.
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A maximum property for the characteristic initial value problem of the wave equation
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Autor/in / Beteiligte Person: | Rhee, H. |
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Zeitschrift: | Proceedings of the American Mathematical Society, Jg. 46 (1974), S. 77-78 |
Veröffentlichung: | American Mathematical Society (AMS), 1974 |
Medientyp: | unknown |
ISSN: | 1088-6826 (print) ; 0002-9939 (print) |
DOI: | 10.1090/s0002-9939-1974-0377289-5 |
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