On the characterization of non-negative volume-matching surface splines (Q1103134)

The author considers the surface spline defined as the solution of the variational problem \(_{w\in H^ 1(\Omega)}\int_{\Omega}(u^ 2_ x+u^ 2_ y)dxdy\) subject to a finite number of volume-matching constrains: \(\int_{\Omega}uf_ i=\alpha_ i,\quad i=1,...,s\) and \(u\geq 0\) a.e. in \(\Omega\), where \(\Omega\) is a smooth bounded region in \(R^ 2\), \(f_ i\in L^ 2(\Omega)\), \(i=1,...,s\). Existence and uniqueness of this surface spline are proved. The author gives a characterization of the surface splines in terms of a variational inequality and studies their local properties. Surface splines of higher- order are also considered.