On monotone extensions of boundary data |
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Authors: | Wolfgang Dahmen Ronald A DeVore Charles A Micchelli |
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Institution: | (1) Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2-6, W-1000 Berlin 33, Federal Republic of Germany;(2) Department of Mathematics and Statistics, University of South Carolina, 29208 Columbia, SC, USA;(3) IBM T.J. Watson Research Center, P.O. Box 218, 10598 Yorktown Heights, NY, USA |
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Abstract: | Summary A functionf C (),
is called monotone on if for anyx, y the relation x – y
+
s
impliesf(x)f(y). Given a domain
with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square 0, 1]2 in 2, strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323 |
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Keywords: | 65D15 |
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