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Transactions of the American Mathematical Society

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


Author: Ognyan Iv. Kounchev
Journal: Trans. Amer. Math. Soc. 350 (1998), 2105-2128
MSC (1991): Primary 35J40; Secondary 41A15, 65D07
MathSciNet review: 1422610
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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.


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Additional Information

Ognyan Iv. Kounchev
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. 8, 1113 Sofia, Bulgaria; Department of Mathematics, University of Duisburg, Lotharstr. 65, 4100 Duisburg, Germany
Email: kounchev@math.uni-duisburg.de

DOI: https://doi.org/10.1090/S0002-9947-98-01961-8
Keywords: Polyharmonic functions, a priori estimates, multivariate splines, variational problem
Received by editor(s): January 19, 1993
Received by editor(s) in revised form: September 17, 1996
Additional Notes: Partially sponsored by the Alexander von Humboldt Foundation and by the NFSR of the Bulgarian Ministery of Education and Science under grant number MM21/91
Article copyright: © Copyright 1998 American Mathematical Society