Minimizing the Laplacian of a function squared with prescribed values on interior boundaries- Theory of polysplines

Abstract:

In this paper we consider the minimization of the integral of the Laplacian of a real-valued function squared (and more general functionals) with prescribed values on some interior boundaries $\Gamma$, with the integral taken over the domain D. We prove that the solution is a biharmonic function in $D$ except on the interior boundaries $\Gamma$, and satisfies some matching conditions on $\Gamma$. There is a close analogy with the one-dimensional cubic splines, which is the reason for calling the solution a polyspline of order 2, or biharmonic polyspline. Similarly, when the quadratic functional is the integral of $(\Delta ^{q}f)^{2}, q$ a positive integer, then the solution is a polyharmonic function of order $2q, \Delta ^{2q}f(x) = 0,$ for $x \in D\setminus \Gamma$, satisfying matching conditions on $\Gamma$, and is called a polyspline of order $2q$. Uniqueness and existence for polysplines of order $2q$, provided that the interior boundaries $\Gamma$ are sufficiently smooth surfaces and $\partial D \subseteq \Gamma$, is proved. Three examples of data sets $\Gamma$ possessing symmetry are considered, in which the computation of polysplines is reduced to computation of one-dimensional $L-$splines.