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

DOI:
https://doi.org/10.1090/S0002-9947-98-01961-8

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 , with the integral taken over the domain D. We prove that the solution is a biharmonic function in except on the interior boundaries , and satisfies some matching conditions on . 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 a positive integer, then the solution is a polyharmonic function of order for , satisfying matching conditions on , and is called a polyspline of order . Uniqueness and existence for polysplines of order , provided that the interior boundaries are sufficiently smooth surfaces and , is proved. Three examples of data sets possessing symmetry are considered, in which the computation of polysplines is reduced to computation of one-dimensional 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