By Scot. 1970 Symposium on the Theory of Numerical Analysis Dundee, Dold Heidelberg, B. Eckmann, John Ll Morris, John Ll Morris
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0 For by Lemma 2 we have for ~ t ~ T, which is clearly bounded. In particular, the heat equation clearly falls into this category. Solutions of parabolic systems are smooth for t ~ Theorem 2 0; we have Assume that (1) is parabolic in Petrovsk~'s sense. <. -i" Proof Via the Fourier transform and Parseval's relation this reduces to But this follows at once by (6). Ex. Consider the Schrodinger equation (N=I) 4 The initial-value problem for this equation is also correctly posed in L 2. Ex, 4 For The Cauchy-Riemann equations can be written (d=l) For these we shall prove the negative result, that the corresponding initial-value problem is not correctly posed in L z.
Let v ~ a o~-~ ~ T ~ IL ~ and consider (~) By differentiation under the integral sign we find that u(x,t) satisfies (i), and so is a solution te (i), (2). Since for t >I O, u(x,t) ~ in the sense of Lecture land is also unique. ~ ) it is a genuine solution Thus Eo(t)v = u(x,t) with D = O Fourier's inversion formula smd Parseval's theorem 2 . By 54 Since S is dense in L ~ it follows that the initial-value problem is correctly posed° Jk We now want to prove the necessity of (3) for correctness. define u(x,t) by (4).
Such a problem might arise in the following manner. Suppose we wish to approximate given aata by the polynomial y(t) = ~t ~ + @t 2 such that y(t) is convex. y(')(t) + yt + This implies = 6at + 2~ ) 0 . Thus, we require 6 a t i + 2~ ) 0 where t. are the data points, (This aces not necessarily guarantee that the polyl hernial will be convex throughout the interval. ) that Gx - w = h where w ~ _O . Introduce slack variables w such 36 Introducing Lagrange multipliers as before, we may write the system as: h i O 0 G -I 0 I A 0 r b GT AT 0 0 x 0 w At the solution, we must have _• ~ o , w~o, T _z_w=0.