One technique which is useful in the calculus of variations is that of “blowing up”. This technique can contribute to the understanding of the boundary behavior of solutions of boundary value problems, especially when they involve mean curvature and a contact angle boundary condition. Our goal in this note is to investigate the structure of “blown up” sets of the form P ×R and N ×R when P, N ⊂ R2 and P (or N) minimizes an appropriate functional; sets like P × R can be the limits of the blow ups of subgraphs of solutions of mean curvature problems, for example. In Part One, we investigate “blown up” sets when the domain has a convex corner. As an application, we illustrate the second author’s proof of the Concus-Finn Conjecture by providing a simplified proof when the mean curvature is zero.