Chapter 13.2, Problem 16P

Show that there is only one function $u$ which tsakes given values on the (closed) boundary of a region and satisfies Laplace’s equation ${\nabla }^{2}u=0$ in the interior of the region. Hints: Suppose $u_1$ and $u_2$ are both solutions with the same boundaryconditions so that $U=u_1-u_2=0$ on the boundary. In Green’s first identity (Chapter 6, Problem 10.16), let $\varphi =\Psi =U$ to show that $\nabla U\equiv 0.$ Thus show $U\equiv 0$ everywhere inside the region.