= Mixed boundary data distinguish heat, Laplace and wave uniqueness
{c}
{title2=$u(x,0)=u(0,y)=u_x(1,y)=0$}
On the unit square, these data determine only the zero solution of the <heat equation> $u_y=u_{xx}$. The squared $L^2$ <norm> decreases because its derivative is $-2\int u_x^2\,dx$, and its initial value is zero. For <Laplace's equation>, $\sin(\pi x/2)\sinh(\pi y/2)$ is a nonzero solution with the same three data; adding zero <Dirichlet data> on the top edge restores <uniqueness> by <Green's first identity>. For the <wave equation> $u_{yy}=u_{xx}$, $\sin(\pi x/2)\sin(\pi y/2)$ is a counterexample; the missing datum is <initial velocity>. Adding $u_y(x,0)=0$ restores <uniqueness> by conservation of $\frac12\int(u_x^2+u_y^2)\,dx$. Thus the type of equation determines what information is needed on an unprescribed edge.
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