Minimal Graphs

n 1 (0) ! R and ⌦ open ⇢ B, define A(u; ⌦) = Area (graph u| ⌦) = Z ⌦ p 1 + |Du| 2 , where, by 'area', we mean n-dimensional Hausdor↵ measure. The notation A(u) simply means Area (graph u). It can be established (firstly for C 1 functions using integration by parts and duality and then by approximating a Lipschitz function in the L 1 norm by a sequence of C 1 functions) that for any Lipschitz u : B ! R, we have (1) A(u) = sup g2C 1 c (B,R From here one can easily deduce that A is strictly convex on L and lower semicon-tinuous with respect to weak L 1 convergence. Given f 2 C 2 (@B), if u 2 C 2 (B) is such that u| @B = f and A(u)  A(˜ u) for any˜u 2 C 2 (B) with˜u| @B = f , then u solves the Dirichlet problem for the Minimal Surface Equation, i.e. Mu := div Du p 1 + |Du| 2 ! = 0 in B u = f on @B Write L L for the set of all Lipschitz functions u : B ! R with u| @B = f and Lip u  L and write a L := inf˜u2L L A(˜ u). The symbols L and a are defined similarly, but without the condition that Lip u  L. 1 j=1 2 L L with A(u j) ! a L as j ! 1. The fact that Lip u j  L for every j means that {u j } 1 j=1 is equicontinuous and for suciently large j, this sequence must also be uniformly bounded (why?). Therefore the Arzela – Ascoli theorem implies that there exists v 2 L L and a subsequence {j 0 } of {j} such that u j 0 ! v uniformly as j ! 1. Setting u L := v, the first claim follows from the lower semicontinuity of A with respect to weak L 1 convergence. Now giveñ u 2 L observe that (tu L + (1 t)˜ u)| @B = f for all t 2 (0, 1) and that Lip(tu L + (1 t)˜ u)  k for suciently small t > 0. Thus for suciently small t > 0 we have (tu L + (1 t)˜ u) 2 L L whence A(u L)  …