On the characterization of non-negative volume-matching surface splines

Abstract In this paper we study the surface spline which minimizes the Dirichlet Integral over a two-dimensional bounded domain, among all non-negative functions satisfying a finite number of volume-matching constraints. Existence and uniqueness of this surface spline are proved. A characterization by a variational inequality is given, revealing local and boundary behaviour of the surface spline. This characterization is of importance in the construction of numerical algorithms for the production of non-negative smooth surfaces from aggregated data.