On the existence of solutions to non-parametric fairing problems

Let s be the graph of a function s: Omega --> R over the bounded set Omega subset of R-2. The fairness of s can be measured by means of functionals de pending on the principal curvatures kappa(1), kappa(2) and the area a of s. Here we consider functionals of type F-psi,F- p(s) := integral(Omega)(kapp a(1)(2) + kappa(2)(2))(p) n d mu + psi(a), where psi is a non-negative lowe r semi-continuous function, n := (1 + parallel to del s parallel to(2))(1/2 ), and 1 < p < infinity. Given s(0) is an element of H-1,H- infinity(Omega) and some constant M > 0, we show that there exists a function s* is an ele ment of H-2.2p(Omega) which minimizes F-psi,F- p on the set {s is an elemen t of H-2,H- 2p(Omega) : parallel to s - s(0)parallel to(1, infinity) less t han or equal to M}. The set of competing functions can also be restricted b y additional constraints involving in particular point and gradient data. ( C) 1999 Academic Press.