3.3 Well-posed problems
Hadamard (1902) called a problem well posed if a solution exists, it is unique, and it depends continuously on the data. The third condition matters because data are measured: a problem in which an error of \(10^{-10}\) in the data can change the solution by \(10^{10}\) is useless for prediction, whatever its existence theorems. The type of the equation decides which data are appropriate:
- hyperbolic equations take Cauchy data, \(u\) and \(u_t\) at \(t = 0\), and boundary values on the sides of a region (the next two sections);
- parabolic equations take \(u\) at \(t = 0\) only, with boundary values, and are solved forwards in time;
- elliptic equations take boundary values, \(u\) or its normal derivative, on the whole boundary of a region, and no Cauchy data at all.
Example 3.3.1 (Hadamard’s example). For Laplace’s equation, try the Cauchy problem with \(u(x, 0) = 0\), \(u_y(x, 0) = \sin (nx)/n^2\). The solution is \[ u_n(x, y) = \frac {\sin (nx)\sinh (ny)}{n^3} , \] which is unique by the Cauchy–Kovalevskaya and Holmgren theorems below. As \(n \to \infty \) the data tend uniformly to \(0\), whose solution is \(u = 0\), but for every \(y > 0\), \(\sup _x|u_n(x, y)| = \sinh (ny)/n^3 \to \infty \). There is no continuous dependence; and replacing \(1/n^2\) by \(e^{-\sqrt n}\) makes the data tend to \(0\) with all their derivatives, while the solutions still blow up.
The same instability appears in the backward heat equation. On \(0 < x < \pi \) with \(u = 0\) at the ends, the solution of \(u_t = u_{xx}\) with \(u(x, 0) = \sum b_k\sin kx\) is \(\sum b_ke^{-k^2t}\sin kx\), and running time backwards multiplies the \(k\)th mode by \(e^{k^2t}\). A final state known to any finite accuracy does not determine the initial state: the heat equation forgets its past, and the backward problem is ill posed.
For equations with analytic coefficients there is a general local existence theorem for the Cauchy problem, which does not distinguish types.
Theorem 3.3.2 (Cauchy–Kovalevskaya). If the coefficients of a linear equation of order \(m\) are analytic near a point of an analytic hypersurface \(S\) that is non-characteristic there, and the Cauchy data (the values of \(u\) and its first \(m - 1\) normal derivatives on \(S\)) are analytic, then there is a unique analytic solution in a neighbourhood of the point.
The proof expands \(u\) in a power series, computes the coefficients from the equation and the data, and shows convergence by comparing with a majorant problem whose solution is known, the method used for series solutions of ordinary differential equations. Holmgren (1901) showed that the solution is unique even among non-analytic solutions. But the theorem says nothing about stability, and Hadamard’s example shows that for elliptic equations there is none: analytic Cauchy data give a solution, but it cannot be trusted. Analyticity is also essential: Lewy (1957) found a linear equation with smooth (non-analytic) right-hand side that has no solution at all near any point.
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