Existence of set-interpolating and energy-minimizing curves
We consider existence of curves c : [0, 1] --> R-n which minimize an energy of the form integral \\c((k))\\(p) (k = 1, 2,..., 1 < p < infinity) under side-conditions of the form G(j) (c(t(1,j)),..., c((k-1))(t(k,j))) epsilon M-j, where G(j) is a continuous function, t(i,j) epsilon [0, 1], M-j is some closed set, and the indices j range in some index set J. This includes the problem of finding energy minimizing interpolants restricted to surfaces, and also variational near-interpolating problems. The norm used for vectors does not have to be Euclidean. It is shown that such an energy minimizer exists if there exists a curve satisfying the side conditions at all, and if among the interpolation conditions there are at least k points to be interpolated. In the case k = 1, some relations to arc length are shown. (C) 2004 Elsevier B.V. All rights reserved.
variational spline interpolation existence of energy-minimizing curves