78 2. FOUR IMPORTANT LINEAR PDE Remarks. (i) Notice that to compute u(x, t) we need only have information on g, h and their derivatives on the sphere ∂B(x, t), and not on the entire ball B(x, t). (ii) Comparing formula (31) with d’Alembert’s formula (8) (n = 1), we observe that the latter does not involve the derivatives of g. This suggests that for n 1, a solution of the wave equation (11) need not for times t 0 be as smooth as its initial value g: irregularities in g may focus at times t 0, thereby causing u to be less regular. (We will see later in §2.4.3 that the “energy norm” of u does not deteriorate for t 0.) (iii) Once again (as in the case n = 1) we see the phenomenon of finite propagation speed of the initial disturbance. (iv) A completely different derivation of formula (31) (using the heat equation!) is in §4.3.3. e. Solution for even n. Assume now n is an even integer. Suppose u is a Cm solution of (11), m = n+2 2 . We want to fashion a repre- sentation formula like (31) for u. The trick, as above for n = 2, is to note that (33) ¯(x1,...,xn+1,t) u := u(x1,...,xn,t) solves the wave equation in Rn+1 × (0, ∞), with the initial conditions ¯ u = ¯ g, ¯t u = ¯ h on Rn+1 × {t = 0}, where (34) ¯(x1,...,xn+1) g := g(x1,...,xn) ¯(x1,...,xn+1) h := h(x1,...,xn). As n + 1 is odd, we may employ (31) (with n + 1 replacing n) to secure a representation formula for ¯ u in terms of ¯ g, ¯. h But then (33) and (34) yield at once a formula for u in terms of g, h. This is again the method of descent. To carry out the details, let us fix x Rn, t 0, and write ¯ x = (x1,... , xn, 0) Rn+1. Then (31), with n + 1 replacing n, gives (35) u(x, t) = 1 γn+1 ∂t 1 t ∂t n−2 2 tn−1 ¯(¯ B x,t ) ¯ g S + 1 t ∂t n−2 2 tn−1 ¯(¯ B x,t ) ¯ h S ,
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