On the Existence of Solutions to Non-parametric Fairing Problems*

Let s be the graph of a function s: VªR over the bounded set V;R2. The fairness of s can be measured by means of functionals depending on the principal curvatures k1, k2 and the area a of s. Here we consider functionals of type F s.[H k2qk2. pn dm qc a., where c is a non-negative lower semi- c , p V 1 2 continuous function, n[ 1q5=s52 .1r2, and 1-p-`. Given s0gH1, ` V.and some constant M)0, we show that there exists a function s*gH2, 2 p V. which minimizes Fc , p on the set sgH2, 2 p V.:5sys051, ` FM4. The set of competing functions can also be restricted by additional constraints involving in particular point and gradient data.