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Trace identities for solutions of the wave equation with initial data supported in a ball
Author(s) -
Finch David
Publication year - 2005
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.647
Subject(s) - mathematics , ball (mathematics) , norm (philosophy) , trace (psycholinguistics) , multiplier (economics) , boundary value problem , mathematical analysis , pure mathematics , law , linguistics , philosophy , political science , economics , macroeconomics
Abstract Suppose u is the solution of the initial value problem$$u_{tt} - \Delta_{x} u = 0, \quad (x,t) \in R^{n} {\times}[0, \infty)$$$$u(x, t = 0)=f(x), \quad u_{t}(x, t = 0) = g(x), \quad x \in R^{n}$$ Suppose n ≥ 1 is odd, f and g are supported in a ball B with boundary S , and one of f or g is zero. We derive identities relating the norm of f or g to the norm of the trace of u on S × [0,∞) . These identities are derived using integral geometric and multiplier methods. Copyright © 2005 John Wiley & Sons, Ltd.