Hadamard's example of ill-posedness

Counterexample

Consider the Cauchy problem for the Laplace equation in the half-plane $y > 0$: $$\partial_x^2 u + \partial_y^2 u = 0 \ \text{ for } y > 0, \qquad u(x, 0) = 0, \qquad \partial_y u(x, 0) = g(x).$$ For an integer $n \ge 1$, take the datum $g_n(x) = \frac{1}{n} \sin(n x)$. The function $$u_n(x, y) = \frac{1}{n^2} \sinh(n y) \sin(n x)$$ solves the problem. Indeed $\partial_x^2 u_n = -n^2 u_n$ and $\partial_y^2 u_n = n^2 u_n$, so the Laplace equation holds. Also $u_n(x, 0) = 0$, since $\sinh 0 = 0$, and $\partial_y u_n(x, 0) = \frac{1}{n} \cosh(0) \sin(n x) = g_n(x)$. The data tend to $0$ uniformly: $\sup_x \abs{g_n(x)} = 1/n \to 0$. For the zero datum $g = 0$ the solution is $u = 0$. But for every fixed $y > 0$, $\sup_x \abs{u_n(x, y)} = \sinh(n y) / n^2 \to \infty$. So the solution does not depend continuously on the data, and the problem is not well posed.