Show that there is only one function u which takes given values on the (closed) boundary of a region and satisfies Laplace's equation ∇2u=0 in the interior of the region. Hints: Suppose u1 and u2 are both solutions with the same boundary conditions so that U=u1−u2=0 on the boundary. In Green's first identity (Chapter 6, Problem 10.16), let ϕ=ψ=U to show that ∇U≡0. Thus show U≡0 everywhere inside the region.