Existence of the Solution of Classical Smoothing Problems with Obstacles and Weights

Abstract

We show that classical smoothing problems with obstacles and weights have always the solution. These problems are considered in quite general case, namely, we allow arbitrary dimension of variable and arbitrary degree in derivative part of functional to minimize. While the existence is proved without any assumption about the uniqueness of solution, some conditions assuring the uniqueness are also analyzed.

Keywords:

• smoothing problems
• natural spline
• existence of solution

AMS Subject Classification::

• 49J30
• 65K10
• 90C20